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Interactive Notebook Foldables, Organizers and Templates

Interactive Notebook Foldables, Organizers and Templates

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    Algebra 2 notebook foldables ebook Algebra 2 notebook foldables ebook Document Transcript

    • Contributing AuthorDinah ZikeConsultantDouglas Fisher, Ph.D.Director of Professional DevelopmentSan Diego, CA
    • Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. Printed in theUnited States of America. Except as permitted under the United States CopyrightAct, no part of this book may be reproduced in any form, electronic or mechanical,including photocopy, recording, or any information storage or retrieval system,without prior written permission of the publisher.Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027ISBN-13: 978-0-07-875695-5 Algebra 2 (Texas Student Edition)ISBN-10: 0-07-875695-2 Noteables™: Interactive Study Notebook with Foldables™1 2 3 4 5 6 7 8 9 10 047 11 10 09 08 07 06
    • Glencoe Algebra 2 iiiContentsFoldables. . . . . . . . . . . . . . . . . . . . . . . . . . . 1Vocabulary Builder . . . . . . . . . . . . . . . . . . 21-1 Expressions and Formulas . . . . . . . . 41-2 Properties of Real Numbers. . . . . . . 61-3 Solving Equations. . . . . . . . . . . . . . . 91-4 Solving Absolute Value Equations . 131-5 Solving Inequalities . . . . . . . . . . . . 151-6 Solving Compound and AbsoluteValue Inequalities. . . . . . . . . . . . . . 18Study Guide . . . . . . . . . . . . . . . . . . . . . . . 22Foldables. . . . . . . . . . . . . . . . . . . . . . . . . . 26Vocabulary Builder . . . . . . . . . . . . . . . . . 272-1 Relations and Functions. . . . . . . . . 292-2 Linear Equations. . . . . . . . . . . . . . . 332-3 Slope . . . . . . . . . . . . . . . . . . . . . . . . 372-4 Writing Linear Equations. . . . . . . . 402-5 Statistics: Using Scatter Plots. . . . . 432-6 Special Functions . . . . . . . . . . . . . . 462-7 Graphing Inequalities . . . . . . . . . . 50Study Guide . . . . . . . . . . . . . . . . . . . . . . . 52Foldables. . . . . . . . . . . . . . . . . . . . . . . . . . 56Vocabulary Builder . . . . . . . . . . . . . . . . . 573-1 Graph Systems of Equations byGraphing. . . . . . . . . . . . . . . . . . . . . 593-2 Solving Systems of EquationsAlgebraically. . . . . . . . . . . . . . . . . . 633-3 Graph Systems of Inequalities byGraphing. . . . . . . . . . . . . . . . . . . . . 673-4 Linear Programming . . . . . . . . . . . 713-5 Solving Systems of Equations inThree Varaiables . . . . . . . . . . . . . . . 76Study Guide . . . . . . . . . . . . . . . . . . . . . . . 81Foldables. . . . . . . . . . . . . . . . . . . . . . . . . . 84Vocabulary Builder . . . . . . . . . . . . . . . . . 854-1 Introduction to Matrices . . . . . . . . . 874-2 Operations with Matrices . . . . . . . . 904-3 Multiplying Matrices. . . . . . . . . . . . . 944-4 Transformations with Matrices . . . . 984-5 Determinants. . . . . . . . . . . . . . . . . . 1034-6 Cramer’s Rule . . . . . . . . . . . . . . . . 1064-7 Identity and Inverse Matrices . . . . 1084-8 Using Matrices to Solve SystemsOf Equations . . . . . . . . . . . . . . . . . 113Study Guide . . . . . . . . . . . . . . . . . . . . . . 116Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 120Vocabulary Builder . . . . . . . . . . . . . . . . 1215-1 Graphing Quadratic Functions . . 1235-2 Solving Quadratic Equations byGraphing. . . . . . . . . . . . . . . . . . . . 1285-3 Solving Quadratic Equations byFactoring. . . . . . . . . . . . . . . . . . . . 1325-4 Complex Numbers . . . . . . . . . . . . 1355-5 Completing the Square . . . . . . . . 1405-6 The Quadratic Formula and theDiscriminant . . . . . . . . . . . . . . . . . 1445-7 Analyzing Graphs of QuadraticFunctions. . . . . . . . . . . . . . . . . . . . 1485-8 Graphing and Solving QuadraticInequalities . . . . . . . . . . . . . . . . . . 152Study Guide . . . . . . . . . . . . . . . . . . . . . . 155Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 160Vocabulary Builder . . . . . . . . . . . . . . . . 1616-1 Properties of Exponents. . . . . . . . 1636-2 Operations with Polynomials . . . 1666-3 Dividing Polynomials . . . . . . . . . . 1686-4 Polynomial Functions . . . . . . . . . . 1716-5 Analyzing Graphs of PolynomalsFunctions. . . . . . . . . . . . . . . . . . . . 1756-6 Solving Polynomial Functions . . . 1796-7 The Remainder and FactorTheorems . . . . . . . . . . . . . . . . . . . 1836-8 Roots and Zeros . . . . . . . . . . . . . . 1866-9 Rational Zero Theorem . . . . . . . . 190Study Guide . . . . . . . . . . . . . . . . . . . . . . 193
    • iv Glencoe Algebra 2Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 197Vocabulary Builder . . . . . . . . . . . . . . . . 1987-1 Operations on Functions . . . . . . . 2007-2 Inverse Functions and Relations . . 2047-3 Square Root Functions andInequalities . . . . . . . . . . . . . . . . . . 2067-4 nth Root . . . . . . . . . . . . . . . . . . . . 2097-5 Operations with RadicalExpressions . . . . . . . . . . . . . . . . . . 2117-6 Rational Exponents . . . . . . . . . . . 2157-7 Solving Radical Equations andInequalities . . . . . . . . . . . . . . . . . . 219Study Guide . . . . . . . . . . . . . . . . . . . . . . 221Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 225Vocabulary Builder . . . . . . . . . . . . . . . . 2268-1 Multiplying and Dividing RationalExpressions . . . . . . . . . . . . . . . . . . 2288-2 Adding and Subtracting RationalExpressions . . . . . . . . . . . . . . . . . . 2328-3 Graphing Rational Functions. . . . 2368-4 Direct, Joint, Inverse Variation . . 2398-5 Classes of Functions . . . . . . . . . . . 2418-6 Solving Rational Equationsand Inequalities . . . . . . . . . . . . . . 243Study Guide . . . . . . . . . . . . . . . . . . . . . . 249Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 253Vocabulary Builder . . . . . . . . . . . . . . . . 2549-1 Exponential Functions . . . . . . . . . 2569-2 Logarithms and LogarithmicFunctions. . . . . . . . . . . . . . . . . . . . 2609-3 Properties of Logarithms . . . . . . . 2639-4 Common Logarithms. . . . . . . . . . . 2669-5 Base e and Natural Logarithms. . 2699-6 Exponential Growth and Decay . 273Study Guide . . . . . . . . . . . . . . . . . . . . . . 278Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 282Vocabulary Builder . . . . . . . . . . . . . . . . 28310-1 Midpoint and Distance Formulas . 28510-2 Parabolas . . . . . . . . . . . . . . . . . . . 28710-3 Circles . . . . . . . . . . . . . . . . . . . . . . 29110-4 Ellipses. . . . . . . . . . . . . . . . . . . . . . 29410-5 Hyperbolas . . . . . . . . . . . . . . . . . . 29910-6 Conic Sections. . . . . . . . . . . . . . . . 30210-7 Solving Quadratic Systems. . . . . . 304Study Guide . . . . . . . . . . . . . . . . . . . . . . 307Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 311Vocabulary Builder . . . . . . . . . . . . . . . . 31211-1 Arithmetic Sequences . . . . . . . . . 31411-2 Arithmetic Series . . . . . . . . . . . . . 31711-3 Geometric Sequences. . . . . . . . . . 32111-4 Geometric Series. . . . . . . . . . . . . . 32411-5 Infinite Geometric Series . . . . . . . 32611-6 Recursion and Special Sequences . 32811-7 The Binomial Theorem. . . . . . . . . 33011-8 Proof and MathematicalInduction. . . . . . . . . . . . . . . . . . . . 333Study Guide . . . . . . . . . . . . . . . . . . . . . . 336Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 340Vocabulary Builder . . . . . . . . . . . . . . . . 34112-1 The Counting Principle . . . . . . . . 34312-2 Permutations and Combinations . 34612-3 Probability. . . . . . . . . . . . . . . . . . . 34912-4 Multiplying Probabilities . . . . . . . 35212-5 Adding Probabilities . . . . . . . . . . 35412-6 Statistical Measures . . . . . . . . . . . 35712-7 The Normal Distribution . . . . . . . 35912-8 Binomial Experiments . . . . . . . . . 36112-9 Sampling and Error . . . . . . . . . . . 364Study Guide . . . . . . . . . . . . . . . . . . . . . . 366
    • Make this Foldable to help you organize andstore your chapter Foldables. Begin with onesheet of 11” ϫ 17” paper.FoldFold the paper in half lengthwise. Then unfold.Fold and GlueFold the paper in half widthwise and glue all of the edges.Glue and LabelGlue the left, right, and bottom edges of the Foldableto the inside back cover of your Noteables notebook.Foldables OrganizerReading and Taking Notes As you read and study each chapter, recordnotes in your chapter Foldable. Then store your chapter Foldables insidethis Foldable organizer.Glencoe Algebra 2 vOrganizing Your FoldablesCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • This note-taking guide is designed to help you succeed in Algebra 2.Each chapter includes:Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.120 Glencoe Algebra 2C H A P T E R5 Quadratic Functions and InequalitiesUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.NOTE-TAKING TIP: When you take notes, you maywish to use a highlighting marker to emphasizeimportant concepts.Fold in half lengthwise. Then fold infourths crosswise. Cut along the middlefold from the edge to the last crease asshown.Staple along the lengthwise fold andstaple the uncut section at the top. Labeleach section with a lesson number andclose to form a booklet.Begin with one sheet of 11” by 17” paper.120-159 CH05-875695.indd 120 6/29/06 9:53:18 AMGlencoe Algebra 2 121Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Vocabulary TermFoundon PageDefinitionDescription orExampleaxis of symmetrycompleting the squarecomplex conjugatescomplex numberconstant termdiscriminantdihs-KRIH-muh-nuhntimaginary unitlinear termmaximum valueminimum valueThis is an alphabetical list of the new vocabulary terms you will learn in Chapter 5.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.C H A P T E R5BUILD YOUR VOCABULARYChapter5(continued on the next page)120-159 CH05-875695.indd 121 6/29/06 9:53:40 AMThe Chapter Openercontains instructions andillustrations on how to makea Foldable that will help youto organize your notes.A Note-Taking Tipprovides a helpfulhint you can usewhen taking notes.The Build Your Vocabularytable allows you to writedefinitions and examplesof important vocabularyterms together in oneconvenient place.Within each chapter,Build Your Vocabularyboxes will remind youto fill in this table.vi Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 148 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.A function written in the form, y ϭ (x Ϫ h)2ϩ k, where(h, k) is the of the parabola and x ϭ h is its, is referred to as the vertex form.5–7• Analyze quadraticfunctions of the formy ϭ a(x Ϫ h)2ϩ k.• Write a quadraticfunction in the formy ϭ a(x Ϫ h)2ϩ k.MAIN IDEASAnalyzing Graphs of Quadratic FunctionsBUILD YOUR VOCABULARY (pages 121–122)Graph a Quadratic Function in Vertex FormAnalyze y ϭ (x Ϫ 3)2ϩ 2. Then draw its graph.The vertex is at (h, k) or and the axis of symmetry isx ϭ . The graph has the same shape as the graph ofy ϭ x2, but is translated 3 units right and 2 units up.Now use this information to draw the graph.Step 1 Plot the vertex, .Step 2 Draw the axis of symmetry,.Step 3 Find and plot two pointson one side of the axis ofsymmetry, such as (2, 3)and (1, 6).Step 4 Use symmetry to complete the graph.Check Your Progress Analyze y ϭ (x ϩ 2)2Ϫ 4. Thendraw its graph.xf(x)O(1, 6)(1, 6)(5, 6)(5, 6)(2, 3)(2, 3) (4, 3)(4, 3)(3, 2)(3, 2)f(x) ϭ (x Ϫ 3)2 ϩ 2On the page forLesson 5-7, sketch agraph of a parabola.Then sketch the graphof the parabola after avertical translation anda horizontal translation.ORGANIZE ITTEKS 2A.4 Thestudent connectsalgebraic andgeometric representationsof functions. (A) Identifyand sketch graphsof parent functions,including linear (f(x) ϭ x),quadratic (f(x) ϭ x2),exponential (f(x) ϭ ax),and logarithmic (f(x)ϭ loga x) functions,absolute value of x (f(x) ϭ|x|), square root of x (f(x)ϭ √x), and reciprocal ofx (f(x) ϭ 1_x). (B) Extendparent functions withparameters such as ain f(x) ϭ a_xand describethe effects of theparameter changeson the graph of parentfunctions.120-159 CH05-875695.indd 148 7/6/06 2:00:45 PMGlencoe Algebra 2 viiGlencoe Algebra 2 155Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.BRINGING IT ALL TOGETHERBUILD YOURVOCABULARYUse your Chapter 5 Foldable tohelp you study for your chaptertest.To make a crossword puzzle,word search, or jumblepuzzle of the vocabulary wordsin Chapter 5, go to:glencoe.comYou can use your completedVocabulary Builder(pages 121–122) to help yousolve the puzzle.VOCABULARYPUZZLEMAKERC H A P T E R5STUDY GUIDERefer to the graph at the right as you compe the followingsentences.1. The curve is called a .2. The line x ϭ Ϫ2 called the .3. The point (Ϫ2, 4) is called the .Determine whether each function has a maximum orminimum value. Then find the maximum or minimum valueof each function.4. ƒ(x) ϭ Ϫx2ϩ 2x ϩ 5 5. ƒ(x) ϭ 3x2Ϫ 4x Ϫ 2Solve each equation. If exact roots cannot be found, state theconsecutive integers between which the roots are located.6. x2Ϫ 2x ϭ 8 7. x2ϩ 5x Ϫ 7 ϭ 0xf(x)O(0, –1)(–2, 4)5-1Graphing Quadratic Functions5-2Solving Quadratic Equations by Graphing120-159 CH05-875695.indd 155 6/29/06 10:03:55 AMLessons cover the content ofthe lessons in your textbook.As your teacher discusses eachexample, follow along andcomplete the fill-in boxes.Take notes as appropriate.5–8154 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Page(s):Exercises:HOMEWORKASSIGNMENTSolve a Quadratic InequalitySolve x2ϩ x Յ 2 algebraically.First, solve the related equation x2ϩ x ϭ 2.x2ϩ x ϭ 2 Related quadratic equationϭ 0 Subtract 2 from each side.ϭ 0 Factor.ϭ 0 or ϭ 0 Zero Product Propertyx ϭ x ϭ Solve each equation.Plot the values on a number line. Use closed circles since thesesolutions are included. Note that the number line is separatedinto 3 intervals.4–3 –2 –1 0 1 2 3 5–5 –4x ≤ Ϫ2 x ≥ 1Ϫ2 ≤ x ≤ 1Test a value in each interval to see if it satisfies the originalinequality.x Յ Ϫ2 Ϫ2 Յ x Յ 1 x Ն 1Test x ϭ Ϫ3. Test x ϭ 0. Test x ϭ 2.x2ϩ x Յ 2 x2ϩ x Յ 2 x2ϩ x Յ 2(Ϫ3)2Ϫ 3 Յ?2 02ϩ 0 Յ?2 22ϩ 2 Յ?26 Յ 2 ϫ 0 Յ 2 ✔ 6 Յ 2 ϫThe solution set is .4–3 –2 –1 0 1 2 3 5–5 –4Check Your Progress Solve x2ϩ 5x Ͻ Ϫ6 algebraically.120-159 CH05-875695.indd 154 6/29/06 10:03:37 AMExamples parallel theexamples in your textbook.Foldables featurereminds you to takenotes in your Foldable.Check Your ProgressExercises allow you tosolve similar exercises onyour own.Bringing It All TogetherStudy Guide reviewsthe main ideas and keyconcepts from each lesson.Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Each lesson iscorrelated to theTexas EssentialKnowledge andSkills.
    • viii Glencoe Algebra 2Your notes are a reminder of what you learned in class. Taking good notescan help you succeed in mathematics. The following tips will help you takebetter classroom notes.• Before class, ask what your teacher will be discussing in class. Reviewmentally what you already know about the concept.• Be an active listener. Focus on what your teacher is saying. Listen forimportant concepts. Pay attention to words, examples, and/or diagramsyour teacher emphasizes.• Write your notes as clear and concise as possible. The following symbolsand abbreviations may be helpful in your note-taking.• Use a symbol such as a star (★) or an asterisk (*) to emphasize importantconcepts. Place a question mark (?) next to anything that you do notunderstand.• Ask questions and participate in class discussion.• Draw and label pictures or diagrams to help clarify a concept.• When working out an example, write what you are doing to solve theproblem next to each step. Be sure to use your own words.• Review your notes as soon as possible after class. During this time, organizeand summarize new concepts and clarify misunderstandings.Note-Taking Don’ts• Don’t write every word. Concentrate on the main ideas and concepts.• Don’t use someone else’s notes as they may not make sense.• Don’t doodle. It distracts you from listening actively.• Don’t lose focus or you will become lost in your note-taking.NOTE-TAKING TIPSWord or PhraseSymbol orAbbreviationWord or PhraseSymbol orAbbreviationfor example e.g. not equalsuch as i.e. approximately ≈with w/ therefore Іwithout w/o versus vsand ϩ angle ЄCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Solving Equations and InequalitiesUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.C H A P T E R1NOTE-TAKING TIP: When you take notes, it isoften a good idea to use symbols to emphasizeimportant concepts.Chapter1Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Glencoe Algebra 2 1Fold 2” tabs oneach of theshort sides.Then fold in half inboth directions andcut as shown.Refold along thewidth. Staple eachpocket. Label pockets asAlgebraic Expressions,Properties of RealNumbers, SolvingEquations andAbsolute ValueEquations, andSolve and GraphInequalities.Begin with one sheet of notebook paper.
    • 2 Glencoe Algebra 2C H A P T E R1BUILD YOUR VOCABULARYCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Vocabulary TermFoundon PageDefinitionDescription orExampleabsolute valuealgebraic expressioncoefficientKOH-uh-FIH-shuhntcompound inequalityconstantdegreeempty setequationformulaintersectionirrational numbersThis is an alphabetical list of new vocabulary terms you will learn in Chapter 1.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description on thesepages. Remember to add the textbook page numbering in the second columnfor reference when you study.
    • Chapter BUILD YOUR VOCABULARY1Glencoe Algebra 2 3Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Vocabulary TermFoundon PageDefinitionDescription orExamplelike termsmonomialopen sentenceorder of operationspolynomialpowerrational numbersreal numbersset-builder notationsolutiontermtrinomialunionvariable
    • 1–14 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.KEY CONCEPTExpressions and FormulasOrder of Operations1. Evaluate expressionsinside groupingsymbols, such asparentheses, ( ),brackets, [ ], braces, { },and fraction bars, as in5 ϩ 7_2.2. Evaluate all powers.3. Do all multiplicationsand/or divisions fromleft to right.4. Do all additions and/or subtractions fromleft to right.Write theorder of operations inthe Algebraic Expressionspocket of your Foldable.• Use the order ofoperations to evaluateexpressions.• Use formulas.MAIN IDEAS Expressions that contain at least one variable arecalled algebraic expressions.Evaluate Algebraic Expressionsa. Evaluate (x ؊ y)3؉ 3 if x ‫؍‬ 1 and y ‫؍‬ 4.(x Ϫ y)3ϩ 3 ϭ (1 Ϫ 4)3ϩ 3 x ϭ and y ϭ .ϭ ϩ 3 Find Ϫ33.ϭ Add Ϫ27 and 3.b. Evaluate s ؊ t(s2؊ t) if s ‫؍‬ 2 and t ‫؍‬ 3.4.s Ϫ t(s2Ϫ t) ϭ 2 Ϫ 3.4(22Ϫ 3.4) Replace s with 2 andt with 3.4.ϭ 2 Ϫ 3.4(4 Ϫ 3.4) Find 22.ϭ 2 Ϫ 3.4΂ ΃ Subtract 3.4 from 4.ϭ 2 Ϫ Multiply 3.4 and 0.6.ϭ Subtract 2.04 from 2.c. Evaluate8xy ϩ z3_y2 ϩ 5if x ‫؍‬ 5, y ‫؍‬ ؊2, and z ‫؍‬ ؊1.8xy ϩ z3_y2ϩ 5ϭ8(5)(Ϫ2) ϩ (Ϫ1)3__(Ϫ2)2ϩ 5x ϭ 5, y ϭ Ϫ2,and z ϭ Ϫ1ϭ Evaluate the numeratorand the denominatorseparately.ϭ Multiply 40 by Ϫ2.ϭ Ϫ81_9or Simplify the numeratorand the denominator.Then divide.BUILD YOUR VOCABULARY (page 2)(Ϫ2) ϩ (Ϫ1)ϩ 5Ϫ 14 ϩ 5Reinforcement of TEKS A.3 The student understands how algebra can be used toexpress generalizations and recognizes and uses the power of symbols to representsituations. (A) Use symbols to represent unknowns and variables.
    • 1–1Glencoe Algebra 2 5Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Check Your Progressa. Evaluate x – y2(x ϩ 5) if x ϭ 2 and y ϭ 4.b. Evaluate 3ab ϩ c2_b Ϫ cif a ϭ 5, b ϭ 2, and c ϭ 4.Use a FormulaFind the area of a trapezoid with base lengths of13 meters and 25 meters and a height of 8 meters.A ϭ 1_2h(b1 ϩ b2) Area of a trapezoidϭ 1_2(8)(13 ϩ 25) Replace h with 8, b1 with 13,and b2 with 25.ϭ Add 13 and 25.ϭ Multiply 8 by 1_2.ϭ Multiply 4 and 38.The area of the trapezoid is square meters.Check Your Progress The formula for the5 cm5 cm6 cmvolume V of a pyramid is V ϭ 1_3Bh, where B isthe area of the base and h is the height of thepyramid. Find the volume of the pyramid shown.Page(s):Exercises:HOMEWORKASSIGNMENTWhy is it importantto follow the orderof operations whenevaluating expressions?WRITE IT
    • 1–2 Properties of Real Numbers6 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Classify NumbersName the sets of numbers to which each numberbelongs.a. √6√6 lies between 2 and 3 so it is not a whole number.b. 5c. ؊2_3d. ؊43e. ؊23.3Check Your Progress Name the sets of numbers towhich each number belongs.a. 3_5b. –2.5ෆ2ෆc. √5 d. √121KEY CONCEPTSReal NumbersRational Numbers Arational number can beexpressed as a ratio m_n ,where m and n areintegers and n is notzero. The decimal formof a rational number iseither a terminating orrepeating decimal.Irrational Numbers Areal number that is notrational is irrational. Thedecimal form of anirrational numberneither terminates norrepeats.• Classify real numbers.• Use the propertiesof real numbers toevaluate expressions.MAIN IDEASReinforcementof TEKS A.4The studentunderstands theimportance of the skillsrequired to manipulatesymbols in order to solveproblems and uses thenecessary algebraicskills required to simplifyalgebraic expressionsand solve equations andinequalities in problemsituations. (B) Use thecommutative, associative,and distributive propertiesto simplify algebraicexpressions.
    • 1–2Glencoe Algebra 2 7Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Identify Properties of Real NumbersName the property illustrated by the equation(Ϫ8 ϩ 8) ϩ 15 ϭ 0 ϩ 15.The Property says that anumber plus its opposite is 0.Check Your Progress Name the property illustratedby each equation.a. 3 ϩ 0 ϭ 3b. 5 и 1_5ϭ 1Additive and Multiplicative InversesIdentify the additive and multiplicative inverse of ؊7.Since Ϫ7 ϩ 7 ϭ 0, the additive inverse is . Since(Ϫ7)(Ϫ1_7) ϭ 1, the multiplicative inverse is .Check Your Progress Identify the additive inverseand multiplicative inverse for each number.a. 5 b. Ϫ2_3Use the Distributive Property to Solve a ProblemPOSTAGE Audrey went to the post office and boughteight 39-cent stamps and eight 24-cent postcard stamps.How much did Audrey spend altogether on stamps?To find the total amount spent on stamps, multiply the price ofeach type of stamp by 8 and then add.S ϭ 8(0.39) ϩ 8(0.24)ϭ ϩ orSo, Audrey spent stamps.Real Number PropertiesFor any real numbers a,b, and c:Commuativea ϩ b ϭ b ϩ a anda и b ϭ b и aAssociative(a ϩ b) ϩ c ϭ a ϩ (b ϩ c)and (a и b) и c ϭ a и (b и c)Identitya ϩ 0 ϭ a ϭ 0 ϩ a anda и 1 ϭ a ϭ 1 и aInversea ϩ (Ϫa) ϭ 0 ϭ (Ϫa) ϩ aIf a 0, then a и 1_a ϭ 1ϭ 1_a и a.Distributivea(b ϩ c) ϭ ab ϩ ac and(b ϩ c)a ϭ ba ϩ caKEY CONCEPT
    • Page(s):Exercises:HOMEWORKASSIGNMENT1–28 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Check Your Progress Joel went to the grocery storeand bought 3 plain chocolate candy bars for $0.69 each and 3chocolate-peanut butter candy bars for $0.79 each. How muchdid Joel spend altogether on candy bars?Simplify an ExpressionSimplify 4(3a ؊ b) ؉ 2(b ؉ 3a).4(3a Ϫ b) ϩ 2(b ϩ 3a)ϭ 4( )Ϫ 4( )ϩ 2( )ϩ 2( ) DistributivePropertyϭ 12a Ϫ 4b ϩ 2b ϩ 6a Multiply.ϭ 12a ϩ 6a Ϫ 4b ϩ 2bProperty (؉)ϭ ( )a ϩ ( )b Distributive Propertyϭ 18a Ϫ 2b Simplify.Check Your Progress Simplify 2(3x Ϫ y) ϩ 4(2x ϩ 3y).
    • 1–3 Solving EquationsGlencoe Algebra 2 9Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.• Translate verbalexpressions intoalgebraic expressionsand equations, andvice versa.• Solve equations usingthe properties ofequality.MAIN IDEASVerbal to Algebraic ExpressionWrite an algebraic expression to representa. 7 less than a numberb. the square of a number decreased by the product of 5and the numberCheck Your Progress Write an algebraic expressionto represent each verbal expression.a. 2 less than the cube of a numberb. 10 decreased by the product of a number and 2Algebraic to Verbal SentenceWrite a verbal sentence to represent each equation.a. 6 ϭ Ϫ5 ϩ x is equal to plus a number.b. 7y Ϫ 2 ϭ 19 times a number minusisCheck Your Progress Write a verbal sentence torepresent each equation.a. 5 ϭ 2 ϩ xb. 3a ϩ 2 ϭ 11A mathematical sentence containing one or moreis called an open sentence.BUILD YOUR VOCABULARY (page 3)Reinforcementof TEKS A.7The studentformulates equationsand inequalities basedon linear functions, usesa variety of methods tosolve them, and analyzesthe solutions in terms ofthe situation. (A) Analyzesituations involving linearfunctions and formulatelinear equations orinequalities to solveproblems. (B) Investigatemethods for solving linearequations and inequalitiesusing concrete models,the properties of equality,select a method, andsolve the equations andinequalities.
    • 1–310 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Identify Properties of EqualityName the property illustrated by each statement.a. a Ϫ 2.03 ϭ a Ϫ 2.03b. If 9 ϭ x, then x ϭ 9.Check Your Progress Name the property illustratedby each statement.a. If x ϩ 4 ϭ 3, then 3 ϭ x ϩ 4.b. If 3 ϭ x and x ϭ y, then 3 ϭ y.Solve One-Step EquationsSolve each equation.a. s Ϫ 5.48 ϭ 0.02s Ϫ 5.48 ϭ 0.02 Original equations Ϫ 5.48 ϩ ϭ 0.02 ϩ Add 5.48 toeach side.s ϭ Simplify.b. 18 ϭ 1_2t18 ϭ 1_2t Original equation18 ϭ 1_2t Multiply each side by themultiplicative inverse of 1_2.ϭ t Simplify.Check Your Progress Solve each equation.a. x ϩ 5 ϭ 3 b. 2_3x ϭ 10Properties of EqualityReflexive For any realnumber a, a ϭ a.Symmetric For all realnumbers a and b, ifa ϭ b, then b ϭ a.Transitive For all realnumbers a, b, and c,if a ϭ b and b ϭ c,then a ϭ c.Substitution If a ϭ b,then a may be replacedby b and b may bereplaced by a.KEY CONCEPT
    • 1–3Glencoe Algebra 2 11Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Solve a Multi-Step EquationSolve 53 ϭ 3(y Ϫ 2) Ϫ 2(3y Ϫ 1).53 ϭ 3(y Ϫ 2) Ϫ 2(3y Ϫ 1) Original equation53 ϭ 3y Ϫ 6 Ϫ 6y ϩ 2 Distributive and SubstitutionProperties53 ϭ Commutative, Distributive,and Substitution Propertiesϭ Addition and SubstitutionPropertiesϭ y Division and SubstitutionPropertiesCheck Your Progress Solve 25 ϭ 3(2x ϩ 2) Ϫ 5(2x ϩ 1).Solve for a VariableGEOMETRY The formula for the area of a trapezoid isA ϭ1_2(b1 ϩ b2)h, where A is the area, b1 is the length ofthe base, b2 is the length of the other base, and h is theheight of the trapezoid. Solve the formula for h.A ϭ 1_2(b1 ϩ b2)h Area of a trapezoidA ϭ и 1_2(b1 ϩ b2)h Multiply each side by 2.2A ϭ (b1 ϩ b2)h Simplify.2Aϭ(b1 ϩ b2)hDivide each side by(b1 ϩ b2).ϭ h Simplify.Apply Properties of EqualityTEST EXAMPLE If 4g ϩ 5 ϭ4_9, what is the value of4g Ϫ 2?A Ϫ41_36B Ϫ59_9C Ϫ41_9D Ϫ67_7Write a multi-stepequation in theEquations column ofyour Foldable. Thensolve the equation,justifying each step withone of the propertiesyou have learned.ORGANIZE IT
    • 1–312 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.You are asked to find the value of the expression .Your first thought might be to find the value of andthen evaluate the expression using this value. Notice thatyou are not required to find the value of g. Instead, you canuse the .Solve the Test Item.4g ϩ 5 ϭ 4_9Original equation4g ϩ 5 Ϫ ϭ 4_9Ϫ Subtract from each side.4g Ϫ 2 ϭ Ϫ 59_94_9Ϫ 7 ϭ 4_9Ϫ 63_9ϭ Ϫ59_9The answer is .Write an EquationHOME IMPROVEMENT Carl wants to replace 5 windowsin his home. His neighbor Will is a carpenter and hehas agreed to help install them for $250. If Carl hasbudgeted $1000 for the total cost, what is the maximumamount he can spend on each window?Let c represent the cost of each window. Write and solve anequation to find the value of c.cost ofNumber of each totalwindows times window plus installation equals cost.5 и c ϩ 250 = 10005c ϩ 250 ϭ 1000 Original equation5c ϩ 250 Ϫ ϭ 1000 Ϫ Subtractfrom each side.5c ϭ 750 Simplify.c ϭ Divide each sideby 5.Carl can afford to spend on each window.Check Your Progress Kelly wants to repair the sidingon her house. Her contractor will charge her $300 plus $1.50per square foot of siding. How much siding can she repairfor $1500?Page(s):Exercises:HOMEWORKASSIGNMENT
    • What is the differencebetween an algebraicexpression and anequation?(Lessons 1-1, 1-3)Evaluate an Expression with Absolute ValueEvaluate 2.7 ϩ |6 Ϫ 2x| if x ϭ 4.2.7 ϩ |6 Ϫ 2x| ϭ 2.7 ϩ |6 Ϫ 2(4)| Replace x with 4.ϭ Simplify Ϫ2(4) first.ϭ Subtract 8 from 6.ϭ |Ϫ2| ϭ 2ϭ Add.Check Your Progress Evaluate 2.3 Ϫ |3y Ϫ 10|if y ϭ Ϫ2.Solve an Absolute Value EquationSolve |y ϩ 3| ϭ 8.Case 1 a ϭ by ϩ 3 ϭ 8y ϩ 3 Ϫ ϭ 8 Ϫy ϭThe solutions are .Thus, the solution set is .The absolute value of a number is its distance fromon the number line.The solution set for an equation that has no solution is theempty set, symbolized by or .1–4Glencoe Algebra 2 13Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Solving Absolute Value Equations• Evaluate expressionsinvolving absolutevalues.• Solve absolute valueequations.MAIN IDEASBUILD YOUR VOCABULARY (page 2)Case 2 a ϭ Ϫby ϩ 3 ϭ Ϫ8y ϩ 3 Ϫ ϭ Ϫ8 Ϫy ϭREVIEW ITReinforcementof TEKS A.7The studentformulates equationsand inequalities basedon linear functions, usesa variety of methods tosolve them, and analyzesthe solutions in terms ofthe situation. (A) Analyzesituations involving linearfunctions and formulatelinear equations orinequalities to solveproblems. (B) Investigatemethods for solving linearequations and inequalitiesusing concrete models,the properties of equality,select a method, andsolve the equations andinequalities.
    • 1–4Page(s):Exercises:HOMEWORKASSIGNMENT14 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.No SolutionSolve |6 Ϫ 4t| ϩ 5 ϭ 0.|6 Ϫ 4t| ϩ 5 ϭ 0 Original equation|6 Ϫ 4t| ϭ Subtract from each side.This sentence is never true. So, the solution set is .One SolutionSolve |8 ϩ y| ϭ 2y Ϫ 3. Check your solutions.Case 1 a ϭ b8 ϩ y ϭ 2y Ϫ 3ϭϭ yThere appear to be two solutions.Check:|8 ϩ y| ϭ 2y Ϫ 3|8 ϩ | ϭ 2( )Ϫ 3՘ 1919 ϭSince 19_3Ϫ19_3, the only solution is .Check Your Progress Solve each equation. Checkyour solutions.a.|2x ϩ 5| ϭ 15 b. 5x Ϫ 2_3 ϩ 7 ϭ 0c. |3x Ϫ 5| ϭ Ϫ4x ϩ 37Case 2 a ϭ Ϫb8 ϩ y ϭ Ϫ(2y Ϫ 3)8 ϩ y ϭ Ϫ2y ϩ 3ϭ 3ϭy ϭ|8 ϩ y| ϭ 2y Ϫ 38 ϩ՘ 2( )Ϫ 319_13՘ Ϫ 319_13ϭREMEMBER ITDetermine if yoursolutions are reasonablebefore checking them.In Example 3, there is noneed to consider the twocases because absolutevalue is never negative.
    • Solve an Inequality Using Addition or SubtractionSolve 4y Ϫ 3 Ͻ 5y ϩ 2. Graph the solution set on anumber line.4y Ϫ 3 Ͻ 5y ϩ 2 Original inequality4y Ϫ 3 Ϫ Ͻ 5y ϩ 2 Ϫ Subtract 4y fromeach side.Ͻ Simplify.Ͻ Subtract from each side.Ͻ y or y Ͼ Simplify.Any real number greaterthan Ϫ5 is a solution ofthis inequality.Check Your Progress Solve 6x Ϫ 2 Ͻ 5x ϩ 7. Graph thesolution on a number line.1–5Glencoe Algebra 2 15Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Solving Inequalities• Solve inequalities.• Solve real-worldproblems involvinginequalities.MAIN IDEASProperties of InequalityAddition Property ofInequality For any realnumbers a, b, and c:If a Ͼ b, thena ϩ c Ͼ b ϩ c.If a Ͻ b, thena ϩ c Ͻ b ϩ c.Subtraction Property ofInequality For any realnumbers a, b, and c:If a Ͼ b, thena Ϫ c Ͼ b Ϫ c.If a Ͻ b, thena Ϫ c Ͻ b Ϫ c.KEY CONCEPTSThe Trichotomy Property says that, for any two realnumbers, a and b, exactly of the followingstatements is true.a Ͻ b a ϭ b a Ͼ bThe of an inequality can be expressedby using set-builder notation, for example, {x x Ͼ 9}.BUILD YOUR VOCABULARY (page 3)A circle means that thispoint is not included inthe solution set.Reinforcementof TEKS A.7The studentformulates equationsand inequalities basedon linear functions, usesa variety of methods tosolve them, and analyzesthe solutions in terms ofthe situation. (A) Analyzesituations involving linearfunctions and formulatelinear equations orinequalities to solveproblems. (B) Investigatemethods for solving linearequations and inequalitiesusing concrete models,the properties of equality,select a method, andsolve the equations andinequalities.3 4 5 6 7 8 10 11 129
    • 1–516 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Solve an Inequality Using Multiplication orDivisionSolve 12 Ն Ϫ0.3p. Graph the solution set on anumber line.12 Ն Ϫ0.3p Original inequalityՅ Divide each side by ,reversing the inequality symbol.Յ p Simplify.p Ն Rewrite with p first.The solution set is {p|p Ն Ϫ40].Solve a Multi-Step InequalitySolve Ϫx Ͼx Ϫ 7_2. Graph the solution set on anumber line.Ϫx Ͼ x Ϫ 7_2Original inequalityϪ2x Ͼ x Ϫ 7 Multiply each side by 2.Ͼ Add Ϫx to each side.x Ͻ Divide each side by Ϫ3, reversingthe inequality symbol.The solution set is (Ϫϱ, 7_3) and is graphed below.3 427310Check Your Progress Solve each inequality. Grapheach solution on a number line.a. Ϫ3x Ն 21Properties of InequalityMultiplication Propertyof Inequality For anyreal numbers a, b, and c,where c is positive:if a Ͼ b, then ac Ͼ bc.if a Ͻ b, then ac Ͻ bc.c is negative:if a Ͼ b, then ac Ͻ bc.if a Ͻ b, then ac Ͼ bc.Division Property ofInequality For any realnumbers a, b, and c,where c is positive:if a Ͼ b, then a_cϾ b_c.if a Ͻ b, then a_cϽ b_c.c is negative:if a Ͼ b, then a_cϽ b_c.if a Ͻ b, then a_cϾ b_c.KEY CONCEPTS12 Ϫ0.3pϪ38Ϫ39Ϫ40Ϫ41Ϫ42 Ϫ37 Ϫ36A dot means that thispoint is included in thesolution set.REMEMBER ITThe symbol ϱrepresents infinity.Ϫ12 Ϫ11 Ϫ10 Ϫ9 Ϫ8 Ϫ7 Ϫ5 Ϫ4Ϫ6
    • 1–5Glencoe Algebra 2 17Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.b. Ϫ2x Ͼ x ϩ 5_3Write an InequalityCONSUMER COSTS Alida has at most $15.00 to spendat a convenience store. She buys a bag of potato chipsand a can of soda for $1.59. If gasoline at this store costs$2.89 per gallon, how many gallons of gasoline can Alidabuy for her car, to the nearest tenth of a gallon?Let g ϭ the gallons of gasoline Alida can buy for her car.Write and solve an inequality.1.59 ϩ 2.89g Յ 15.00 Originalinequality1.59 ϩ 2.89g ϩ Յ 15.00 Ϫ Subtract fromeach side.Յ Simplify.Յ Divide.g Յ Simplify.Alida can buy up to gallons of gasoline for her car.Check Your Progress Jeb wants to rent a car for hisvacation. Value Cars rents cars for $25 per day plus $0.25 permile. How far can he drive for one day if he wants to spend nomore than $200 on car rental?Page(s):Exercises:HOMEWORKASSIGNMENTORGANIZE ITWrite the Properties ofInequality in your notes.Ϫ2 Ϫ1 057Ϫ
    • 1–618 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Solving Compound and AbsoluteValue InequalitiesSolve an “and” Compound InequalitySolve 10 Յ 3y Ϫ 2 Ͻ 19. Graph the solution set.METHOD 1 Write the compound inequality using the wordand. Then solve each inequality.10 Յ 3y Ϫ 2 and 3y Ϫ 2 Ͻ 19Յ 3y 3y ϽՅ y y ϽՅ y ϽMETHOD 2 Solve both parts at the same time by adding 2 toeach part. Then divide each part by 3.Յ 3y Ϫ 2 ϽՅ 3y ϽՅ y ϽGraph each solution set. Then find their intersection.1 2 3 4 5 6 8 9 1071 2 3 4 5 6 8 9 1071 2 3 4 5 6 8 9 107y Նy ϽՅ y ϽThe solution set is {y|4 Յ y Ͻ 7}.• Solve compoundinequalities.• Solve absolute valueinequalities.MAIN IDEAS A compound inequality consists of two inequalities joinedby the word or the word .The graph of a compound inequality containing and is theintersection of the solution sets of the two inequalities.BUILD YOUR VOCABULARY (page 2)“And” CompoundInequalitiesA compound inequalitycontaining the word andis true if and only if bothinequalities are true.KEY CONCEPTWrite thisconcept in your notes.Reinforcementof TEKS A.7The studentformulates equationsand inequalities basedon linear functions, usesa variety of methods tosolve them, and analyzesthe solutions in terms ofthe situation. (A) Analyzesituations involving linearfunctions and formulatelinear equations orinequalities to solveproblems. (B) Investigatemethods for solving linearequations and inequalitiesusing concrete models,the properties of equality,select a method, andsolve the equations andinequalities.
    • Solve an “or” Compound InequalitySolve x ϩ 3 Ͻ 2 or Ϫx Յ Ϫ4. Graph the solution set.Solve each inequality separately.x ϩ 3 Ͻ 2 or Ϫx Յ Ϫ4x Ͻ x ՆϪ4Ϫ3Ϫ2Ϫ1 0 1 2 3 4 5 6 7 8Ϫ4Ϫ3Ϫ2Ϫ1 0 1 2 3 4 5 6 7 8Ϫ4Ϫ3Ϫ2Ϫ1 0 1 2 3 4 5 6 7 8x Ͻx Նx Ͻ or x ՆThe solution set is {x|x Ͻ Ϫ 1 or x Ն 4}.Check Your Progress Solve x ϩ 5 Ͻ 1 or Ϫ2x Յ Ϫ6.Graph the solution set.Solve an Absolute Value Inequality (Ͻ)Solve 3 Ͼ |d|. Graph the solution set on a number line.You can interpret 3 Ͼ |d| to mean that the distance betweend and 0 on a number line is less than units. To make1–6Glencoe Algebra 2 19Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Check Your Progress Solve 11 Յ 2x ϩ 5 Ͻ 17. Graph thesolution set.The graph of a compound inequality containingis the union of the solution sets of the two inequalities.BUILD YOUR VOCABULARY (page 3)REMEMBER ITCompoundinequalities containingand are conjunctions.Compound inequalitiescontaining or aredisjunctions.“Or” CompoundInequalitiesA compound inequalitycontaining the word oris true if one or more ofthe inequalities is true.KEY CONCEPTWrite thisconcept in your notes.1 2 3 4 5 6 8 9 107Ϫ6Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1 0 1 2 3 4 5 6
    • 1–6Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.3 Ͼ |d| true, you must substitute numbers for d that arefewer than units from .10Ϫ1Ϫ2Ϫ3Ϫ4 2 3 4All of the numbers not at or between Ϫ3 and 3 are less thanunits from 0. The solution set is .Check Your Progress Solve ԽxԽ Ͻ 5. Graph the solution.Solve a Multi-Step Absolute Value InequalitySolve |2x Ϫ 2| Ն 4. Graph the solution set.|2x Ϫ 2| Ն 4 is equivalent to 2x Ϫ 2 Ն 4 or 2x Ϫ 2 Յ Ϫ4.Solve each inequality.2x Ϫ 2 Ն 4 or 2x Ϫ 2 Յ Ϫ4Ն Յx Ն x ՅThe solution set is .Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5Check Your Progress Solve |3x Ϫ 3| Ͼ 9. Graph thesolution set.Notice that the graph of3 Ͼ |d| is the same asthe graph of d Ͼand d Ͻ .20 Glencoe Algebra 2Ϫ6Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1 0 1 2 3 4 5 6Ϫ4 Ϫ3Ϫ2Ϫ1 0 1 2 3 4 5 6
    • Write an Absolute Value InequalityHOUSING According to a recent survey, the averagemonthly rent for a one-bedroom apartment in one cityis $750. However, the actual rent for any given one-bedroom apartment might vary as much as $250 fromthe average.a. Write an absolute value inequality to describe thissituation.Let r ϭ the actual monthly rentThe rent for an apartment candiffer from the average by as much as $250.b. Solve the inequality 750 Ϫ r Յ 250 to find therange of monthly rent. Rewrite the absolute valueinequality as a compound inequality. Then solvefor r.Ϫ250 Յ 750 Ϫ r Յ 250Ϫ250 Յ Ϫr Յ 250Ϫ1000 Յ Ϫr Յ Ϫ500Ն r ՆThe solution set is . The actualrent falls between and , inclusive.Check Your Progress The average birth weight of anewborn baby is 7 pounds. However, this weight canvary by as much as 4.5 pounds.a. What is an absolute value inequality to describe thissituation?b. What is the range of birth weights for newborn babies?1–6Page(s):Exercises:HOMEWORKASSIGNMENTGlencoe Algebra 2 21Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • BRINGING IT ALL TOGETHERBUILD YOURVOCABULARYVOCABULARYPUZZLEMAKERC H A P T E R1STUDY GUIDE22 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.1. Find the value of 30 Ϫ 42Ϭ 2 и 4.2. Evaluate 2x2Ϫ 3xy if x ϭ Ϫ4 and y ϭ 5.3. Why is it important for everyone to use the same order ofoperations for evaluating expressions?4. Name the sets of numbers to which Ϫ7_8belongs.5. Write the Associative Property of Addition in symbols. Thenillustrate this property by finding the sum 12 ϩ 18 ϩ 45.Complete each sentence.6. The Property of Addition says that adding 0to any number does not change its value.7. The numbers can be written as ratios of twointegers, with the integer in the denominator not being 0.Use your Chapter 1 Foldable tohelp you study for your chaptertest.To make a crossword puzzle,word search, or jumblepuzzle of the vocabulary wordsin Chapter 1, go to:glencoe.comYou can use your completedVocabulary Builder(pages 2–3) to help you solvethe puzzle.1-1Expressions and Formulas1-2Properties of Real Numbers
    • Chapter BRINGING IT ALL TOGETHER1Glencoe Algebra 2 23Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Solve each equation. Check your solution.8. 3 Ϫ 5y ϭ 4y ϩ 6 9. 2_3Ϫ 1_2x ϭ 1_610. Solve A ϭ 1_2h(a ϩ b) for h.Read the following problem and then write an equation that youcould use to solve it. Do not actually solve the equation. In yourequation, let m be the number of miles driven.11. When Louisa rented a moving truck, she agreed to pay $28 perday plus $0.42 per mile. If she kept the truck for 3 days andthe rental charges (without tax) were $153.72, how manymiles did Louisa drive the truck?12. Evaluate |m Ϫ 5n| if m ϭ Ϫ3 and n ϭ 2.Solve each equation.13. Ϫ2|4x Ϫ 5| ϭ Ϫ46 14. |7 ϩ 3x| ϭ x Ϫ 515. Explain why the absolute value of a number can never benegative.1-3Solving Equations1-4Solving Absolute Value Equations
    • Chapter BRINGING IT ALL TOGETHER124 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.There are several different ways to write or show solution sets ofinequalities. Write each of the following in interval notation.16. {x|x Ͻ Ϫ3}17. Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5Solve each inequality. Graph the solution set.18. 5y ϩ 9 Ͼ 3419. Ϫ 1 Ϫ 5x Յ 4(x ϩ 2)Complete each sentence.20. Two inequalities combined by the word and or the word orform a .21. The graph of a compound inequality containing the word andis the of the graphs of the two separateinequalities.Solve each inequality. Graph the solution set.22. Ϫ11 Ͻ 3m Ϫ 2 Ͻ 2223. |x ϩ 3| Ն 11-5Solving Inequalities1-6Solving Compound and Absolute Value Inequalities0Ϫ1 1 2 3 4 5 7 8 9 106Ϫ3Ϫ4 Ϫ2 Ϫ1 0 1 2 4 5 6 7301Ϫ3Ϫ4 Ϫ2 20Ϫ1 1 3 4 5 7 8 9 106Ϫ7Ϫ8 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 0 1 2 3
    • C H A P T E R1ChecklistStudent Signature Parent/Guardian SignatureTeacher SignatureVisit glencoe.com toaccess your textbook,more examples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 1.Glencoe Algebra 2 25Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Check the one that applies. Suggestions to help you study aregiven with each item.ARE YOU READY FORTHE CHAPTER TEST?I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.• You may want to take the Chapter 1 Practice Test on page 53of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 1 Study Guide and Reviewon pages 49–52 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 1 Practice Test onpage 53 of your textbook.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 1 Foldable.• Then complete the Chapter 1 Study Guide and Review onpages 49–52 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 1 Practice Test onpage 53 of your textbook.
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.26 Glencoe Algebra 2Linear Relations and FunctionsUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin this Interactive StudyNotebook to help you in taking notes.C H A P T E R2NOTE-TAKING TIP: When taking notes, makeannotations. Annotations are usually notes takenin the margins of books you own to organize thetext for review or study.Fold in half along thewidth and staple alongthe fold.Turn the fold to theleft and write the titleof the chapter on thefront. On each left-handpage of the booklet,write the title of alesson from the chapter.Begin with two sheets of grid paper.
    • Vocabulary TermFoundon PageDefinitionDescription orExampleabsolute value functionboundaryCartesian coordinateplaneconstant functionfamily of graphsfunctiongreatest integer func-tionidentity functionlinear functionline of fitThis is an alphabetical list of new vocabulary terms you will learn in Chapter 2.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.C H A P T E R2BUILD YOUR VOCABULARYGlencoe Algebra 2 27Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Chapter2(continued on the next page)
    • 28 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Chapter BUILDING YOUR VOCABULARY2Vocabulary TermFoundon PageDefinitionDescription orExampleparent graphpiecewise functionPEES-WYZpoint-slope formpositive correlationprediction equationpree-DIHK-shuhnrelationscatter plotslopeslope-intercept formIHN-tuhr-SEHPTstandard formstep function
    • A relation is a set of .A function is a special type of relation in which eachelement of the domain is paired withelement of the range.A function where each element of the is pairedwith exactly one element of the is called aone-to-one function.Glencoe Algebra 2 29Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Domain and RangeState the domain and range of the relation shown in thegraph. Is the relation a function?The relation is {(1, 2), (3, 3), (0, Ϫ2),(0, –2)(–3, 1)(–4, 0)(1, 2)(3, 3)xyO(Ϫ4, 0), (Ϫ3, 1)}.The domain is .The range is .Each member of the domain is paired with exactly onemember of the range, so this relation is a .Check Your Progress State the domain and range of therelation shown in the graph. Is the relation a function?yxO(–3, –1)(3, 1)(2, 0)(0, –2)2–1• Analyze and graphrelations.• Find functional values.MAIN IDEASBUILD YOUR VOCABULARY (pages 27–28)Relations and FunctionsTEKS 2A.1 The student uses properties and attributes of functions and appliesfunctions to problem situations. (A) Identify the mathematical domains and rangesof functions and determine reasonable domain and range values for continuousand discrete situations.
    • Vertical Line TestTRANSPORTATION The table shows the average fuelefficiency in miles per gallon for light trucks for severalyears. Graph this information and determine whetherit represents a function. Is this relation discrete orcontinuous?The year values correspond to Year Fuel Efficiency (mi/gal)1995 20.51996 20.81997 20.61998 20.91999 20.52000 20.52001 20.4the values, and the FuelEfficiency values correspond tothe values.Source: U.S. Environmental Protection AgencyGraph the ordered pairs on acoordinate grid.Use the vertical line test.Notice that no vertical linecan be drawn that containsmore than one data pointThis relation is a function,and it is .Check Your Progress HEALTH The table shows theaverage weight of a baby for several months during thefirst year. Graph this information and determine whether itrepresents a function.Age(months)Weight(pounds)1 12.52 164 226 249 2512 262–130 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.On the page labeledRelations and Functions,define function andrelation. Then draw agraph of a function anda relation.ORGANIZE IT
    • 2–1Glencoe Algebra 2 31Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Graph a Relationa. Graph the relation represented by y ‫؍‬ 3x ؊ 1.Make a table of values to find ordered pairs that satisfy theequation. Choose values for x and find the correspondingvalues for y. Then graph the ordered pairs.x yϪ1012xyO(–1, –4)(0, –1)(1, 2)(2, 5)b. Find the domain and range.The domain and range are both all numbers.c. Determine whether the equation is a function andstate whether it is discrete or continuous.This graph passes the vertical line test. For each-value, there is exactly one -value.The equation represents a .All (x, y) values that satisfy the equation lie on a ,so the relation is .Check Your Progressa. Graph y ϭ 2x ϩ 5.b. Find the domain and range.c. Determine whether the equation is a function and statewhether it is.Vertical Line Test If novertical line intersectsa graph in more thanone point, the graphrepresents a function.If some vertical lineintersects a graph intwo or more points,the graph does notrepresent a function.KEY CONCEPT
    • Evaluate a FunctionGiven ƒ(x) ‫؍‬ x3؊ 3, find each value.a. ƒ(؊2)ƒ(x) ϭ x3Ϫ 3 Original functionƒ(Ϫ2) ϭ Substitute.ϭ or Simplify.b. ƒ(2t)ƒ(x) ϭ x3Ϫ 3 Original functionƒ(2t) ϭ Substitute.ϭ (ab)2ϭ a2b2Check Your Progress Given ƒ(x) ϭ x2ϩ 5 andh(x) ϭ 0.5x2ϩ 2x ϩ 2.5, find each value.a. ƒ(Ϫ1)b. h(1.5)c. ƒ(3a)REVIEW IT2–1Page(s):Exercises:HOMEWORKASSIGNMENT32 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Explain what it means toevaluate an expression.(Lesson 1-1)
    • An equation such as x ϩ y ϭ 4 is called a linear equation.A linear equation has no operations other than, , andof a variable by a constant.Glencoe Algebra 2 33Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Identify Linear FunctionsState whether each function is a linear function.Explain.a. g(x) ‫؍‬ 2x ؊ 5This is a function because it is in the formg(x) ϭ mx ϩ b. m ϭ , b ϭb. p(x) ‫؍‬ x3؉ 2This is not a linear function because x has an exponentother than .c. t(x) ‫؍‬ 4 ؉ 7xThis is a function because it is in the formt(x) ϭ mx ϩ b. m ϭ , b ϭCheck Your Progress State whether each function isa linear function. Explain.a. h(x) ϭ 3x Ϫ 2 b. g(x, y) ϭ 3xy2–2• Identify linear equationsand functions.• Write linear equationsin standard form andgraph them.MAIN IDEASGive an example ofa linear function anda nonlinear function.Explain how you can tellthe difference betweenthe two functions.WRITE ITBUILD YOUR VOCABULARY (page 27)Linear EquationsTEKS 2A.1 The student uses properties and attributes of functions and appliesfunctions to problem situations. (A) Identify the mathematical domains and rangesof functions and determine reasonable domain and range values for continuousand discrete situations.
    • Evaluate a Linear FunctionThe linear function f(C) ϭ 1.8C ϩ 32 can be used to findthe number of degrees Fahrenheit f that are equivalentto a given number of degrees Celsius C.a. On the Celsius scale, normal body temperature is37ЊC. What is it in degrees Fahrenheit?f(C) ϭ 1.8C ϩ 32 Original functionf ΂ ΃ ϭ 1.8΂ ΃ ϩ 32 Substitute.ϭ Simplify.b. There are 100 Celsius degrees between the freezingand boiling points of water and 180 Fahrenheitdegrees between these two points. How manyFahrenheit degrees equal 1 Celsius degree?100ЊC ϭ ЊF Given100ЊC ϭ ЊF Divide each side by.1ЊC ϭ Simplify.Check Your Progress The linear function d(s) ϭ 1_5scan be used to find the distance d in miles from a storm,based on the number of seconds s that it takes to hearthunder after seeing lightning.a. If you hear thunder 10 seconds after seeing lightning,how far away is the storm?b. If the storm is 3 miles away, how long will it take to hearthunder after seeing lightning?2–234 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Standard Form of aLinear Equation Thestandard form of alinear equation isAx ϩ By ϭ C, whereA Ն 0, A and B are notboth zero.KEY CONCEPT
    • 2–2Glencoe Algebra 2 35Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Standard FormWrite each equation in standard form. Identify A, B,and C.a. y ϭ 3x Ϫ 9y ϭ 3x Ϫ 9 Original equationϭ Subtract 3x from each side.ϭ Multiply each side by Ϫ1 sothat A Ն 0.So, A ϭ , B ϭ , and C ϭ .b. Ϫ2_3x ϭ 2y Ϫ 1Ϫ2_3x ϭ 2y – 1 Original equationϭ Subtract 2y from each side.ϭ Multiply each side by Ϫ3 so thatthe coefficients are all integers.So, A ϭ , B ϭ , and C ϭ .Check Your Progress Write each equation in standardform. Identify A, B, and C.a. y ϭ Ϫ2x ϩ 5b. 3_5x ϭ Ϫ3y ϩ 2c. 3x Ϫ 9y ϩ 6 ϭ 0
    • Page(s):Exercises:HOMEWORKASSIGNMENT2–236 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Use Intercepts to Graph a LineFind the x-intercept and the y-intercept of the graph of؊2x ؉ y ؊ 4 ‫؍‬ 0. Then graph the equation.The x-intercept is the value of x when y ϭ 0.Ϫ2x ϩ y Ϫ 4 ϭ 0 Original equationϪ2x ϩ Ϫ 4 ϭ 0 Substitute 0 for y.ϭ Add 4 to each side.x ϭ Divide each side by Ϫ2.The x-intercept is Ϫ2. The graph crosses the x-axis at .Likewise, the y-intercept is the value of y when x ϭ 0.Ϫ2x ϩ y Ϫ 4 ϭ 0 Original equationϪ2΂ ΃ ϩ y Ϫ 4 ϭ 0 Substitute 0 for x.y ϭ Add 4 to each side.The y-intercept is 4. The graph crosses the y-axis at .Use the ordered pairs to graph this equation.The x-intercept is ,and the y-intercept is .Check Your Progress Find the x-intercept and they-intercept of the graph of 3x Ϫ y ϩ 6 ϭ 0. Then graph theequation.REMEMBER ITAn equation suchas x ϭ 3 represents avertical line, and onlyhas an x-intercept.An equation such asy ϭ 5 represents ahorizontal line, and onlyhas a y-intercept.(–2, 0)(0, 4)xxyyOO
    • Glencoe Algebra 2 37Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Find Slope and Use Slope to GraphFind the slope of the line that passes through (1, 3) and(؊2, ؊3). Then graph the line.m ϭy2 Ϫ y1_x2 Ϫ x1Slope formulaϭ (x1, y1) ϭ (1, 3), (x2, y2) ϭ (Ϫ2, Ϫ3)ϭ or Simplify.Graph the two ordered pairs anddraw the line. Use the slopeto check your graph by selectingany on the line. Thengo up units and rightunit or go 2 unitsand left 1 unit. This point shouldalso be on the line.Check Your Progress Find the slope of the line thatpasses through (2, 3) and (Ϫ1, 5). Then graph the line.• Find and use the slopeof a line.• Graph parallel andperpendicular lines.MAIN IDEASSlope of a Line Theslope of a line is theratio of the change iny-coordinates to thechange in x-coordinates.KEY CONCEPT2–3 Slope(–2, –3)(–2, –3)(1, 3)(1, 3)xyOTEKS 2A.4 Thestudent connectsalgebraic andgeometric representationsof functions. (A) Identifyand sketch graphsof parent functions,including linear(f(x) ϭ x), quadratic(f(x) ϭ x2), exponential(f(x) ϭ ax), and logarithmic(f(x) ϭ loga x), functions,absolute value of x(f(x) ϭ x), square rootof x (f(x) ϭ √x), andreciprocal of x (f(x) ϭ 1_x).
    • 2–338 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Find Rate of ChangeBUSINESS Refer to the graphat the right, which shows dataon the fastest-growingrestaurant chain in the U.S.during the time period of thegraph. Find the rate of changeof the number of stores from1999 to 2004.m ϭy2 Ϫ y1_x2 Ϫ x1Slope formulaϭϪ___ϪSubstitute.ϭ 5.4 Simplify.Between 1999 and 2004, the number of stores in the U.S.increases at an average rate of orstores per yearCheck Your Progress Refer to the graph in Example 2.Find the rate of change of the number of stores from 2001 to2004.Parallel LinesGraph the line through (1, ؊2) that is parallel to theline with the equation x ؊ y ‫؍‬ ؊2.The x-intercept is , and they-intercept is .Use the intercepts to graphx Ϫ y ϭ Ϫ2.The line rises 1 unit for every1 unit it moves to the right,so the slope is .Now, use the slope and the point at (1, Ϫ2) to graph the lineparallel to x Ϫ y ϭ Ϫ2.Parallel Lines In a plane,nonvertical lines withthe same slope areparallel. All vertical linesare parallel.KEY CONCEPT
    • 2–3Glencoe Algebra 2 39Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Page(s):Exercises:HOMEWORKASSIGNMENTCheck Your Progress Graph theline through (2, 3) that is parallel tothe line with the equation 3x ϩ y ϭ 6.Perpendicular LinesGraph the line through (2, 1) that is perpendicular tothe line with the equation 2x ؊ 3y ‫؍‬ 3.The x-intercept is or , and the y-intercept is. Use the intercepts to graph 2x Ϫ 3y ϭ 3.The line rises 1 unit for every 1.5 units it moves to the right,so the slope is or .The slope of the line perpendicularis the opposite reciprocal ofor . Start at (2, 1) and godown units and right units.Use this point and (2, 1) to graph the line.Check Your Progress Graph the line through (Ϫ3, 1)that is perpendicular to the line with the equation5x Ϫ 10y ϭ Ϫ20.Perpendicular Lines In aplane, two oblique linesare perpendicular if andonly if the product oftheir slopes is Ϫ1.On the pageopposite the pagelabeled Slope, graph twolines that are parallel.Then graph two linesthat are perpendicular.KEY CONCEPT
    • Write an Equation Given Slope and a PointWrite an equation in slope-intercept form for the linethat has a slope of ؊3_5and passes through (5, ؊2).y ϭ mx ϩ b Slope-intercept formϪ2 ϭ Ϫ3_5(5) ϩ b (x, y) ϭ (5, Ϫ2), m ϭ Ϫ3_5ϭ Simplify.ϭ Add 3 to each side.The y-intercept is . So the equation in slope-interceptform is .Check Your Progress Write an equation in slope-interceptform for the line that has a slope of 2_3and passes through(Ϫ3, Ϫ1).Write an Equation Given Two PointsWhat is an equation of the line through (2, ؊3)and (؊3, 7)?A y ϭ Ϫ2x Ϫ 1 C y ϭ 1_2x ϩ 1B y ϭ Ϫ 1_2x ϩ 1 D y ϭ Ϫ2x ϩ 1First, find the slope of the line.m ϭy2 Ϫ y1_x2 Ϫ x1Slope formulaϭϪ ΂ ΃__Ϫ(x1, y1) ϭ (2, ؊3), (x2, y2) ϭ (Ϫ7)ϭ or Simplify.40 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.2–4• Write an equation of aline given the slope anda point on the line.• Write an equationof a line parallel orperpendicular to agiven line.MAIN IDEASSlope-Intercept Form ofa Linear Equation Theslope-intercept form ofthe equation of a line isy ϭ mx ϩ b, where m isthe slope and b is they-intercept.Point-Slope Form of aLinear Equation Thepoint-slope form ofthe equation of a lineis y – y1 ϭ m(x Ϫ x1),where (x1, y1) are thecoordinates of a pointon the line and m is theslope of the line.KEY CONCEPTSWriting Linear EquationsReinforcementof TEKS A.6The studentunderstands the meaningof the slope and interceptsof the graphs of linearfunctions and zeros oflinear functions andinterprets and describesthe effects of changesin parameters of linearfunctions in real-world andmathematical situations.(D) Graph and writeequations of lines givencharacteristics such astwo-points, a point anda slope, or a slope andy-intercept.
    • The slope is Ϫ2. That eliminates choices B and C.Now use the point-slope formula to find an equation.y Ϫ y1 ϭ m(x Ϫ x1) Point-slope formy Ϫ (Ϫ3) ϭ Ϫ2(x Ϫ 2) m ϭ Ϫ2; you can useeither point for (x1, y1).ϭ Distributive Propertyy ϭ Subtract 3 from each side.The answer is .Check Your Progress What is an equation of the linethrough (2, 5) and (Ϫ1, 3)?A y ϭ 2_3x ϩ 11_3C y ϭ Ϫ2_3x ϩ 19_3B y ϭ 3_2x ϩ 9_2D y ϭ Ϫ3_2x ϩ 8Write an Equation of a Perpendicular LineWrite an equation for the line that passes through(3, ؊2) and is perpendicular to the line whose equationis y ‫؍‬ ؊5x ؉ 1.The slope of the given line is . Since the slopes ofperpendicular lines are opposite reciprocals, the slope of theperpendicular line is .Use the point-slope form and the ordered pair towrite the equation.y Ϫ y1 ϭ m(x Ϫ x1) Point-slope formy Ϫ ΂ ΃ ϭ ΂x Ϫ ΃ (x1, y1) ϭ (3, Ϫ2), m ϭ 1_5ϭ Distributive Propertyy ϭ Subtract 2 from each side.An equation of the line is .2–4Glencoe Algebra 2 41Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.REVIEW ITName the property youwould use to simplify8(z ϩ 4). (Lesson 1-2)
    • Page(s):Exercises:HOMEWORKASSIGNMENT2–442 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Check Your Progress Write an equation for the line thatpasses through (3, 5) and is perpendicular to the line withequation y ϭ 3x Ϫ 2.
    • 2–5Glencoe Algebra 2 43Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Find and use a Prediction EquationEDUCATION The table below shows the approximatepercent of students who sent applications to twocolleges in various years since 1985.a. Draw a scatterplot and a line of fit for the data.Graph the data as ordered pairs, with the number of yearssince 1985 on the axis and the percentageon the axis.b. Find a prediction equation. What do the slope andy-intercept indicate?Find an equation of the line through (3, 18) and (15, 13).Begin by finding the slope.m ϭy2 Ϫ y1_x2 Ϫ x1Slope formulam ϭ 13 Ϫ 18_15 Ϫ 3Substitute.ϭ Ϸ Simplify.• Draw scatter plots.• Find and use predictionequations.MAIN IDEASYears Since 1985 0 3 6 9 12 15Percent 20 18 15 15 14 13Source: U.S. News & World ReportStatistics: Using Scatter Plots(continued on the next page)TEKS 2A.1 The student uses properties and attributes of functions and appliesfunctions to problem situations. (B) Collect and organize data, make and interpretscatterplots, fit the graph of a function to the data, interpret the results, andproceed to model, predict, and make decisions and crtical judgments
    • 2–544 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.y Ϫ y1 ϭ m(x Ϫ x1) Point-slope formy Ϫ ϭ ΂x Ϫ ΃ m ϭ Ϫ0.42,(x1, y1) ϭ (3, 18)ϭ Distributive Propertyy ϭ Add 18 to each side.The slope indicates that the percent of students sendingapplications to two colleges is falling at about eachyear. The y-intercept indicates that the percent in 1985should have been about .c. Predict the percent in 2010.The year 2010 is 25 years after 1985, so use x ϭ 25 to find y.y ϭ Ϫ0.42x ϩ 19.26 Prediction equationy ϭ Ϫ0.42΂ ΃ ϩ 19.26 x ϭy ϭ Simplify.In 2010 the percent should be about .d. How accurate is the prediction?The fit is only , so the prediction maynot be very accurate.Check Your Progress The table below shows theapproximate percent of drivers who wear seat belts invarious years since 1994.a.Make a scatter plot of the data and draw a line of best fit.An outlier is a datapoint that does notappear to belong withthe rest of the set.Should you include anoutlier when finding aline of fit? Explain.WRITE ITYears Since 1994 0 1 2 3 4 5 6 7Percent 57 58 61 64 69 68 71 73Source: National Highway Traffic Safety Administration
    • 2–5Page(s):Exercises:HOMEWORKASSIGNMENTGlencoe Algebra 2 45Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.b. Find a prediction equation. What do the slope andy-intercept indicate?c. Predict the percent of drivers who will be wearing seat beltsin 2005.
    • A function whose graph is a series of line segments iscalled a step function.The greatest integer function, written ƒ(x) ϭ [[x]], is anexample of a step function.ƒ(x) ϭ b is called a constant function.When a function does not change the value,ƒ(x) ϭ x is called the identity function.Another special function is the absolute value function,ƒ(x) ϭ ΗxΗ.A that is written using two or moreexpressions is called a piecewise function.2–646 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Step FunctionPSYCHOLOGY One psychologist charges for counselingsessions at the rate of $85 per hour or any fractionthereof. Draw a graph that represents this situation.Use the pattern of times and costs to make a table, where x isthe number of hours of the session and C(x) is the total cost.Then draw the graph.x C(x)0 Ͻ x Յ 11 Ͻ x Յ 22 Ͻ x Յ 33 Ͻ x Յ 44 Ͻ x Յ 5xC(x)510425340255170851 2 3 4 50 6• Identify and graph step,constant, and identityfunctions.• Identify and graphabsolute value andpiecewise functions.MAIN IDEASBUILD YOUR VOCABULARY (pages 27–28)Special FunctionsTEKS 2A.1 Thestudent usespropertiesand attributes offunctions and appliesfunctions to problemsituations. (A) Identifythe mathematicaldomains and ranges offunctions and determinereasonable domainand range values forcontinuous and discretesituations.2A.4 The studentconnects algebraic andgeometric representationsof functions. (A) Identifyand sketch graphsof parent functions,including linear(f(x) ϭ x), quadratic(f(x) ϭ x2), exponential(f(x) ϭ ax), and logarithmic(f(x) ϭ loga x), functions,absolute value of x(f(x) ϭ x), square rootof x (f(x) ϭ √x), andreciprocal of x (f(x) ϭ 1_x).
    • Absolute Value FunctionsGraph ƒ(x) ‫؍‬ ͦx ؊ 3ͦ and g(x) ‫؍‬ ͦx ؉ 2ͦ on the samecoordinate plane. Determine the similarities anddifferences in the two graphs.Find several ordered pairs for each function.x ͦx ؊ 3ͦ0 31234x ͦx ؉ 2ͦϪ4 2Ϫ3Ϫ2Ϫ10Graph the points and connect them.xyO• The domain of both graphsis all numbers.• The range of both graphs is ΆyΗy Ն ·.• The graphs have the same shapebut different .• The graph of g(x) is the graph of ƒ(x) translated left units.Check Your Progressa. Graph ƒ(x) ϭ ΗxΗ Ϫ 2 and g(x) ϭ ΗxΗ ϩ 3 on the samecoordinate plane. Determine the similarities and differencesin the two graphs.2–6Glencoe Algebra 2 47Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Consider the generalform of an absolutevalue function, ƒ(x) ϭ ΗxΗ.Explain why the rangeof the function onlyincludes nonnegativenumbers.WRITE ITUnder the tab forGraphing LinearFunctions, graph anexample of each of thefollowing functions:step, constant, absolutevalue, and piecewise.ORGANIZE IT
    • 2–648 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.b. Graph the piecewise function ƒ(x) ϭ2x ϩ 1 if x Ͼ Ϫ1Ϫ3 if x Յ Ϫ1.Identify the domain and range.Piecewise FunctionGraph f(x) ϭx Ϫ 1 if x Յ 3Ϫ1 if x Ͼ 3. Identify the domain andrange.First graph the linear functionf(x) ϭ x Ϫ 1 for x Յ 3. Since 3satisfies this inequality, beginwith a closed circle at .Then graph the function f(x) ϭ Ϫ1for x Ͼ 3. Since 3 does not satisfythis inequality, begin with ancirlce at .The is all real numbers. The is{yͦy Յ 2}.Check Your Progress Graph ƒ(x) ϭ2x ϩ 1 if x Ͼ Ϫ1Ϫ3 if x Յ Ϫ1.Identify the domain and range.
    • 2–6Page(s):Exercises:HOMEWORKASSIGNMENTIndentify FunctionsDetermine whether each graph represents a stepfunction, a constant function, an absolute valuefunction, or a piecewise function.a.xOy Since this graph consists of differentrays and segments, it is afunction.b.xOy Since this graph is V-shaped, it is anfunction.Check Your Progress Determine whether each graphrepresents a step function, a constant function, anabsolute value function, or a piecewise function.a. b.Glencoe Algebra 2 49Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 50 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.When graphing inequalities, the graph of the line is theboundary of each region.2–7Dashed BoundaryGraph x ؊ 2y Ͻ 4.The boundary is the graph of x Ϫ 2y ϭ 4. Since the inequalitysymbol is Ͻ, the boundary will be dashed. Use the slope-intercept form, y ϭ 1_2x Ϫ 2. Test (0, 0).x Ϫ 2y Ͻ 4 OriginalxyOinequality(x, y) ϭ (0, 0)0 Ͻ 4 trueShade the region that contains (0, 0)Solid BoundaryEDUCATION One tutoring company advertises that itspecializes in helping students who have a combinedSAT verbal and math score of 900 or less.a. Write an inequality to describe the combined scoresof students who are prospective tutoring clients. Letx represent the verbal score and y the math score.verbal score and math scoreare atmost nine hundredx ϩ y Յ 900The inequality is .b. Does a student with a verbal score of 480 and a mathscore of 410 fit the tutoring company’s guidelines?x ϩ y Յ 900 Original inequality480 ϩ 410 Յ 900 (x, y) ϭ (480, 410)890 Յ 900 SimplifySo, a student with a verbal score of 480 and a math scoreof 410 the tutoring company’s guidelines.BUILD YOUR VOCABULARY (page 27)Graphing Inequalities• Graph linearinequalities.• Graph absolute valueinequalities.MAIN IDEASReinforcementof TEKS A.7The studentformulates equationsand inequalities basedon linear functions, usesa variety of methods tosolve them, and analyzesthe solutions in terms ofthe situation. (A) Analyzesituations involving linearfunctions and formulatelinear equations orinequalities to solveproblems. (B) Investigatemethods for solving linearequations and inequalitiesusing concrete models,the properties of equality,select a method, andsolve the equations andinequalities.
    • 2–7Page(s):Exercises:HOMEWORKASSIGNMENTGlencoe Algebra 2 51Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Absolute Value InequalityGraph y § ͦxͦ ؊ 2.Since the inequality symbol is Ն, the graph of the relatedequation y ϭ ΗxΗ Ϫ 2 is solid. Graph the equation. Test (0, 0).y Ն ΗxΗ Ϫ 2xyO0 2 Η0Η Ϫ 20 Ն 0 Ϫ 2 or ՆShade the region that contains .Check Your Progress Graph each inequality.a. 2x ϩ 5y Ͼ 10 b. y Ͻ ΗxΗ ϩ 3c. CLASS TRIP Two social studies classes are going ona field trip. The teachers have asked for parentvolunteers to go on the trip as chaperones. However,there is only enough seating for 60 people on the bus.Write an inequality to describe the number ofstudents and chaperones that can ride on the bus.Can 45 students and 10 chaperones go on the trip?
    • BRINGING IT ALL TOGETHERBUILD YOURVOCABULARYUse your Chapter 2 Foldable tohelp you study for your chaptertest.To make a crossword puzzle,word search, or jumblepuzzle of the vocabulary wordsin Chapter 2, go to:glencoe.comYou can use your completedVocabulary Builder(pages 27–28) to help you solvethe puzzle.VOCABULARYPUZZLEMAKERC H A P T E R2STUDY GUIDE52 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.2-1Relations and FunctionsFor Exercises 1 and 2, refer to the graph shown at the right.1. Write the domain and range of the relation.D:(0, 4)(3, 1)(3, –4)(–1, –5)(–2, 0)(–3, 2)xyOR:2. Is this relation a function? Explain.3. Write x Ϫ 2 ϭ 1_6y in standard form. Identify A, B, and C.Write yes or no to tell whether each linear equation is instandard form. If it is not, explain why it is not.4. Ϫx ϩ 2y ϭ 5 5. 5x Ϫ 7y ϭ 32-2Linear Equations
    • Chapter BRINGING IT ALL TOGETHER2Glencoe Algebra 2 53Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.6. How are the slopes of two nonvertical parallel lines related?7. How are the slopes of two oblique perpendicular lines related?Find the slope of the line that passes through each pair of points.8. (3, 7), (8, Ϫ1) 9.΂8, Ϫ1_4΃,΂0, Ϫ1_4΃Write the equation in slope-intercept form for the line thatsatisfies each set of conditions.10. slope 4, passes through (0, 3) 11. passes through (5, Ϫ6) and (3, 2)12. Write an equation for the line that passes through (8, Ϫ5) andis perpendicular to the line whose equation is y ϭ 1_2x Ϫ 8.13. Draw a scatter plot for the data. Then state which ofthe data points is an outlier.x 2 6 10 14 20 24y 15 20 30 16 40 502-4Writing Linear Equations2-5Statistics: Using Scatter Plots2-3Slope
    • 2-6Write the letter of the term that best describes each function.14. ƒ(x) ϭ Η4x ϩ 3Η15. ƒ(x) ϭ [[x]] ϩ 116. ƒ(x) ϭ 617. ƒ(x) ϭ Άx ϩ 3 if x Ͻ 02 Ϫ x if x Ն 018. When graphing a linear inequality in two variables, how doyou know whether to make the boundary a solid line or adashed line?18. Graph the inequality 10 Ϫ 5y Ͻ 2x.2-7Graphing InequalitiesChapter BRINGING IT ALL TOGETHER254 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Special Functionsa. constant functionb. absolute value functionc. piecewise functiond. step function
    • C H A P T E R2ChecklistCheck the one that applies. Suggestions to help you study aregiven with each item.I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.• You may want to take the Chapter 2 Practice Test onpage 111 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 2 Study Guide and Reviewon pages 106–110 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 2 Practice Test onpage 111 of your textbook.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 2 Foldable.• Then complete the Chapter 2 Study Guide and Review onpages 106–110 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 2 Practice Test onpage 111 of your textbook.Student Signature Parent/Guardian SignatureTeacher SignatureVisit glencoe.com toaccess your textbook,more examples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 2.Glencoe Algebra 2 55Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc. ARE YOU READY FORTHE CHAPTER TEST?
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.56 Glencoe Algebra 2C H A P T E R3 Systems of Equations and InequalitiesUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.NOTE-TAKING TIP: When taking notes, summarizethe main ideas presented in the lesson. Summariesare useful for condensing data and realizing whatis important.Fold the short sidesof the 11” ϫ 17” paperto meet in the middle.Cut each table in halfas shown.Cut 4 sheets of gridpaper in half and foldthe half-sheets in half.Insert two foldedhalfsheets under each ofthe four tabs and staplealong the fold. Labeleach tab as shown.Begin with one sheet of 11” ϫ 17” paper and foursheets of grid paper.
    • Chapter3Chapter4Glencoe Algebra 2 57Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Vocabulary TermFoundon PageDefinitionDescription orExampleboundedconsistentconstraintskuhn-STRAYNTSdependentelimination methodfeasible regionFEE-zuh-buhlinconsistentihn-kuhn-SIHS-tuhntindependentThis is an alphabetical list of new vocabulary terms you will learn in Chapter 3.As you complete the study notes for the chapter, you will see Build Your Vocabularyreminders to complete each term’s definition or description on these pages.Remember to add the textbook page numbering in the second column for referencewhen you study.C H A P T E R3BUILD YOUR VOCABULARY(continued on the next page)
    • 58 Glencoe Algebra 2Chapter BRINGING IT ALL TOGETHER3Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Vocabulary TermFoundon PageDefinitionDescription orExamplelinear programmingordered triplesubstitution methodsystem of equationssystem of inequalitiesunboundedvertex
    • 3–1A system of equations is or more equationswith the same variables.A system of equations is consistent if it has at leastsolution and inconsistent if it has solutions.A consistent system is independent if it has exactlysolution or dependent if it has an numberof solutions.Glencoe Algebra 2 59Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Solving Systems of Equations by GraphingSolve the System of Equations byCompleting a TableSolve the system of equations by completing a table.x ؉ y ‫؍‬ 3؊2x ؉ y ‫؍‬ ؊6Write each equation in slope-intercept form.x ϩ y ϭ 3Ϫ2x ϩ y ϭ Ϫ6Use a table to find the solution that satisfies both equations.x y1ϭ Ϫx ϩ 3 y1 y2 ϭ 2x Ϫ 6 y2 (x1,y1) (x2,y2)0 y1 ϭ (0) ϩ 3 3 y2 ϭ 2(0) Ϫ 6 Ϫ6 (0, 3) (0, Ϫ6)1 y1 ϭ Ϫ(1) ϩ 3 2 y2 ϭ 2(0) Ϫ 6 Ϫ4 (1, 2) (1, Ϫ4)2 y1 ϭ Ϫ(2) ϩ 3 1 y2 ϭ 2(0) Ϫ 6 Ϫ2 (2, 1) (2, Ϫ2)3 y1 ϭ Ϫ(3) ϩ 3 0 y2 ϭ 2(0) Ϫ 6 0 (3, 0) (3, 0)The solution of the system of equations is .• Solve systems of linearequations by graphing.• Determine whethera system of linearequations is consistentand independent,consistent anddependent, orinconsistent.MAIN IDEASBUILD YOUR VOCABULARY (pages 57–58)TEKS 2A.3 Thestudent formulatessystems ofequations and inequalitiesfrom problem situations,uses a variety of methodsto solve them, andanalyzes the solutions interms of the situations. (A)Analyze situations andformulate systems ofequations in two or moreunknowns or inequalitiesin two unknowns to solveproblems. (B) Usealgebraic methods,graphs, tables, ormatrices, to solvesystems of equationsor inequalities. (C)Interpret and determinethe reasonableness ofsolutions to systems ofequations or inequalitiesfor given contexts.
    • Solve by GraphingSolve the system of equations by graphing.x ؊ 2y ‫؍‬ 0x ؉ y ‫؍‬ 6Write each equation in slope-intercept form.x Ϫ 2y ϭ 0x ϩ y ϭ 6The graphs appear to intersectat .Check: Substitute the coordinates into each equation.x Ϫ 2y ϭ 0 x ϩ y ϭ 6 Original equations.4 Ϫ 2(2) ՘ 4 ϩ 2 ՘ 6 Replace x and y.ϭ 0 ϭ 6 Simplify.So, the solution of the system is .a. Solve the system by completing a table.x ϩ y ϭ 2x Ϫ 3y ϭ Ϫ6b. Solve the system by graphing.x ϩ 3y ϭ 7x Ϫ y ϭ 33–160 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Check Your ProgressxyOx ϩ y ϭ 6x Ϫ 2y ϭ 0REMEMBER ITWhen solving asystem of equations bygraphing, you shouldalways check theordered pair in each ofthe original equations.
    • 3–1Break-Even Point AnalysisSALES A service club is selling copies of its holidaycookbook to raise funds for a project. The printer’s set-up charge is $200, and each book costs $2 to print. Thecookbooks will sell for $6 each. How much must theclub sell before it makes a profit?Income frombooksisprice perbooktimesnumber ofbooks.y ϭ 6 и xThe graphs interect at .This is the break-even point. If thegroup sells less than 50 books, theywill money. If the groupsells more than 50 books, they willmake a .Check Your Progress The student government is sellingcandy bars. It costs $1 for each candy bar plus a $60 set-upfee. The group will sell the candy bars for $2.50 each. Howmany do they need to sell to break even?Same LineGraph the system of equations and describe it asconsistent and independent, consistent and dependent,or inconsistent.9x ؊ 6y ‫؍‬ ؊66x ؊ 4y ‫؍‬ ؊49x Ϫ 6y ϭ Ϫ66x Ϫ 4y ϭ Ϫ4Since the equations are equivalent, their graphs are theline. There are many solutions.This system is and .Glencoe Algebra 2 61Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.xyO6x Ϫ 4y ϭ Ϫ49x Ϫ 6y ϭ Ϫ6Explain the steps youwould use to write2x ϩ 5y ϭ10 in slope-intercept form.(Lesson 1-3)REVIEW IT
    • Check Your Progress Graph each systemof equations and describe it as consistent andindependent, consistent and dependent, or inconsistent.a. x ϩ y ϭ 5 b. x ϩ y ϭ 32x ϭ y Ϫ 11 2x ϭ Ϫ2y ϩ 6Parallel LinesGraph the system of equations and describe it asconsistent and independent, consistent and dependent,or inconsistent.15x Ϫ 6y ϭ 05x Ϫ 2y ϭ 1015x Ϫ 6y ϭ 05x Ϫ 2y ϭ 10The lines do not intersect. Their graphs are parallel lines. So,there are no solutions that satisfy both equations. This systemis .Check Your Progress Graph the system of equationsy ϭ 3x ϩ 2 and Ϫ6x ϩ 2y ϭ 10 and describe it as consistentand independent, consistent and dependent, or inconsistent.3–1Page(s):Exercises:HOMEWORKASSIGNMENTxyO15x Ϫ 6y ϭ 05x Ϫ 2y ϭ 1062 Glencoe Algebra 2Under the Systems ofEquations tab, graphthe three possiblerelationships between asystem of equations andthe number of solutions.Write the number ofsolutions below eachgraph.ORGANIZE ITCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 3–2Using the substitution method, one equation is solved forone in terms of . Then, thisexpression is substituted for the variable in the otherequation. Using the elimination method, you eliminate oneof the variables by or theequations.BUILD YOUR VOCABULARYGlencoe Algebra 2 63Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Solve by Using SubstitutionUse substitution to solve x ϩ 4y ϭ 26 andx Ϫ 5y ϭ Ϫ10.Solve the first equation for x in terms of y.x ϩ 4y ϭ 26 First equationx ϭ Subtract 4y from each side.Substitute 26 Ϫ 4y for x in the second equation and solve for y.x Ϫ 5y ϭ Ϫ10 Second equationϪ 5y ϭ Ϫ10 Substitute for x.ϭ Subtract from each side.y ϭ Divide each side.Now substitute the value for y in either of the originalequations and solve for x.x ϩ 4y ϭ 26 First equationx ϩ 4( )ϭ 26 Replace y with 4.x ϩ ϭ 26 Simplify.x ϭ Subtract from each side.The solution of the system is .• Solve systems of linearequations by usingsubstitution.• Solve systems of linearequations by usingelimination.MAIN IDEASREMEMBER ITIn Example 1, youcan substitute the valuefor y in either of theoriginal equations.Choose the equationthat is easiest to solve.(pages 57–58)Solving Systems of Equations AlgebraicallyTEKS 2A.3 Thestudent formulatessystems ofequations and inequalitiesfrom problem situations,uses a variety of methodsto solve them, andanalyzes the solutions interms of the situations. (A)Analyze situations andformulate systems ofequations in two or moreunknowns or inequalitiesin two unknowns tosolve problems. (B)Use algebraic methods,graphs, tables, ormatrices, to solvesystems of equationsor inequalities. (C)Interpret and determinethe reasonableness ofsolutions to systems ofequations or inequalitiesfor given contexts.
    • 3–264 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Check Your Progress Use Substitution to solve thesystem of equations.x Ϫ 3y ϭ 2x ϩ 7y ϭ 12Solve by SubstitutionFURNITURE Lancaster Woodworkers FurnitureStore builds two types of wooden outdoor chairs. Arocking chair sells for $265 and an Adirondack chairwith footstool sells for $320. The books show thatlast month, the business earned $13,930 for the 48outdoor chairs sold. How many chairs were sold?Read the Test ItemYou are asked to find the number of each type of chair sold.Solve the Test Item.Step 1 Define variables and write the system of equations.Let x represent the number of rocking chairs soldand y represent the number of Adirondack chairs.x ϩ y ϭ The total number of chairs sold265x ϩ 320y ϭ The total earned.Step 2 Solve one of the equations for one of the variablesin terms of the other. Since the coefficient of x is, solve the first equation for x in terms of y.x ϩ y ϭ 48 First equationx ϭ 48 Ϫ y Subtract from each side.Step 3 Substitute 48 Ϫ y for x in the second equation.265x ϩ 320y ϭ 13,930 Second equation265( )ϩ 320y ϭ 13,930 Substitutefor x.Ϫ ϩ 320y ϭ 13,930 DistributiveProperty55y ϭ 1210 Simplify.y ϭ Divide each sideby .
    • Step 4 Now find the value of x. Substitute the valuefor y into either equation.x ϩ y ϭ 48 First equationx ϩ ϭ 48 Replace y with .x ϭ 26 Subtract from each side.They sold rocking chairs and Adirondack chairsCheck Your Progress At Amy’s Amusement Park, ticketssell for $24.50 for adults and $16.50 for children. On one daythe amusement park took in $6405 from selling 330 tickets.How many of each kind of ticket was sold?Multiply, Then Use EliminationUse the elimination method to solve 2x ؉ 3y ‫؍‬ 12 and5x ؊ 2y ‫؍‬ 11.Multiply the first equation by 2 and the second equation by 3.Then add the equations to eliminate one of the variables.2x ϩ 3y ϭ 12 ϩ ϭ5x Ϫ 2y ϭ 11 (ϩ) Ϫ ϭ19x ϭ 57x ϭReplace x and solve for y.2x ϩ 3y ϭ 12 First equation2( ) ϩ 3y ϭ 12 Replace x with .ϩ 3y ϭ 12 Multiply.3y ϭ 6 Subtract 6 from each side.y ϭ 2 Divide each side by 3.The solution is .3–2Glencoe Algebra 2 65Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.When solving a systemof equations, how doyou choose whetherto use the substitutionmethod or theelimination method?WRITE ITMultiply by 2.Multiply by 3.
    • 3–2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Inconsistent SystemUse the elimination method to solve ؊3x ؉ 5y ‫؍‬ 12 and6x ؊ 10y ‫؍‬ ؊21.Use multiplication to eliminate x.Ϫ3x ϩ 5y ϭ 12 ϩ ϭ6x Ϫ 10y ϭ (ϩ) Ϫ ϭ0 ϭSince there are no values of x and y that will make the equationϭ true, there are no solutions for the systemof equations.Check Your Progress Use the elimination method tosolve each system of equations.a. x ϩ 3y ϭ 7 d. 2x ϩ 3y ϭ 112x ϩ 5y ϭ 10 Ϫ4x Ϫ 6y ϭ 20Page(s):Exercises:HOMEWORKASSIGNMENTMultiply by 2.66 Glencoe Algebra 2Under the Systems ofEquations tab, writehow you recognize aninconsistent system ofequations. Then writehow you recognize aconsistent system ofequations.ORGANIZE IT
    • 3–3Intersecting RegionsSolve each system of inequalities by graphing.a. y Ն 2x Ϫ 3y Ͻ Ϫx ϩ 2Graph each inequality.solution of y Ն 2x Ϫ 3 Regionssolution of y Ͻ Ϫx ϩ 2 RegionsThe intersection of these regions is Region , whichis the solution of the system of inequalities. Notice that thesolution is a region containing an number ofordered pairs.b. y Յ Ϫx ϩ 1x ϩ 1 Ͻ 3The inequality x ϩ 1 Ͻ 3 can be written as x ϩ 1 3and x ϩ 1 Ϫ3Graph all of the inequalities onthe same coordinate planeand shade the region orregions that are common to all.Glencoe Algebra 2 67Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Solving Systems of Inequalities by Graphing• Solve systems ofinequalities bygraphing.• Determine thecoordinates of thevertices of a regionformed by the graph ofa system of inequalities.MAIN IDEAS To solve a system of inequalities, find the pairsthat satisfy of the inequalities in the system.BUILD YOUR VOCABULARY (page 57)y ϭ 2x Ϫ 3y ϭ Ϫx ϩ 2Region 1Region 2Region 3TEKS 2A.3 Thestudent formulatessystems ofequations and inequalitiesfrom problem situations,uses a variety of methodsto solve them, andanalyzes the solutions interms of the situations. (A)Analyze situations andformulate systems ofequations in two or moreunknowns or inequalitiesin two unknownsto solve problems.(B) Use algebraicmethods, graphs, tables,or matrices, to solvesystems of equationsor inequalities. (C)Interpret and determinethe reasonableness ofsolutions to systems ofequations or inequalitiesfor given contexts.
    • Check Your Progress Solve each system ofinequalities by graphing.a. y Ն 3x Ϫ 3 b. y Ն Ϫ2x Ϫ 3y Ͼ x ϩ 1 x ϩ 2 Ͻ 1Separate RegionsSolve the system of inequalities by graphing.y § ؊3_4x ؉ 1y ∞ Ϫ3_4x ؊ 2Graph both inequalities.The graphs do not overlap, so thesolutions have points in common.The solution set is .Check Your Progress Solve the system of inequalitiesy Ͻ 1_2x ϩ 2 and y Ͼ 1_2x ϩ 4 by graphing.3–368 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.REMEMBER ITYou can indicate thata system of inequalitieshas no solutions in thefollowing ways: emptyset, null set, л, or {}.xyOy ϭ Ϫ34x ϩ 1y ϭ Ϫ34x Ϫ 2Under the tab forSystems of Inequalities,solve the following bygraphing.y Յ 3x Ϫ 6y Ͼ Ϫ4x ϩ 2ORGANIZE IT
    • Write and Use a System of InequalitiesMEDICINE Medical Professionals recommend thatpatients have a cholesterol level below 200 milligramsper deciliter (mg/dL) of blood and a triglyceride levelsbelow 150 mg/dL. Write and graph a system ofinequalities that represents the range of cholesterollevel and triglyceride levels for patients.Source: American Heart Association.Let c represent the cholesterol levels in mg/dL. It must be lessthan mg/dL. Since cholesterol levels cannot benegative, we can write this as 0 Յ c Ͻ .Let t represent the triglyceride levels in mg/dL. It must be lessthan mg/dL. Since triglyceride levels also cannot benegative, we can write this as 0 Յ t Ͻ .Graph both inequalities. Anyordered pair in the intersectionof the graphs is aof the system.Check Your Progress The speed limits while drivingon the highway are different for trucks and cars. Cars mustdrive between 45 and 65 miles per hour, inclusive. Trucks arerequired to drive between 40 and 55 miles per hour, inclusive.Let c represent the speed range for cars and t representthe speed range for trucks. Write and graph a system ofinequalities to represent this situation.3–3Glencoe Algebra 2 69Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Find VerticesFind the coordinates of the vertices of the figureformed by 2x ؊ y § ؊1, x ؉ y ∞ 4, and x ؉ 4y § 4.Graph each inequality.(0, 1)(1, 3)(4, 0)xyO2x Ϫ y ϭ Ϫ1x ϩ y ϭ 4x ϩ 4y ϭ 4The of the graphsforms a triangular region.The vertices of the triangle are at, , and .Check Your Progress Find the coordinates of thevertices of the figure formed by x ϩ 2y Ն 1, x ϩ y Յ 3,and Ϫ2x ϩ y Յ 3.3–3Page(s):Exercises:HOMEWORKASSIGNMENT70 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 3–4Bounded RegionGraph the following system of inequalities. Name thecoordinates of the vertices of the feasible region.Find the maximum and minimum values of the functionƒ(x, y) ‫؍‬ 3x ؊ 2y for this region.x ∞ 5y ∞ 4x ؉ y § 2First, find the vertices of the region.Graph the inequalities.The polygon formed is a trianglewith vertices at, , and .(5, –3)(–2, 4)(5, 4)xyOx ϩ y ϭ 2x ϭ 5y ϭ 4In a graph of a system of inequalities, theare called the constraints.The intersection of the graphs is called the feasible region.When the graph of a system of constraints is a polygonalregion, we say that the region is bounded.The maximum or minimum value of a related functionoccurs at one of the vertices of thefeasible region.When a system of inequalities forms a region that is, the region is said to be unbounded.The process of finding orvalues of a function for a region defined by inequalities iscalled linear programming.(pages 57–58)BUILD YOUR VOCABULARYGlencoe Algebra 2 71Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Linear Programming• Find the maximum andminimum values of afunction over a region.• Solve real-worldproblems using linearprogramming.MAIN IDEASTEKS 2A.3 Thestudent formulatessystems ofequations and inequalitiesfrom problem situations,uses a variety of methodsto solve them, andanalyzes the solutions interms of the situations. (A)Analyze situations andformulate systems ofequations in two or moreunknowns or inequalitiesin two unknownsto solve problems.(B) Use algebraicmethods, graphs, tables,or matrices, to solvesystems of equationsor inequalities. (C)Interpret and determinethe reasonableness ofsolutions to systems ofequations or inequalitiesfor given contexts.
    • Next, use a table to find the maximum and minimum values ofƒ(x, y). Substitute the coordinates of the vertices into thefunction.The vertices of the feasible region are , ,and . The maximum value is at .The minimum value is at .Unbounded RegionGraph the following system of inequalities. Name thecoordinates of the vertices of the feasible region. Findthe maximum and minimum values of the functionf(x, y) ϭ 2x ϩ 3y for this region.Ϫx ϩ 2y Յ 2x Ϫ 2y Յ 4x ϩ y Ն Ϫ2Graph the system ofinequalities. there areonly two points ofintersection,and .(x, y) 2x ϩ 3y f(x, y)(Ϫ2, 0)(0, Ϫ2)(x, y) 3x Ϫ 2y ƒ(x, y)(Ϫ2, 4)(5, Ϫ3)(5, 4)3–472 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.minimummaximumExplain how yourecognize theunbounded region of asystem of inequalities.WRITE ITUnder the LinearProgramming tab,sketch a graph of asystem of inequalitieslike the one shown inExample 1. Then labelthe constraints, feasibleregion, and vertices ofthe graph.ORGANIZE IT
    • 3–4The minimum value is at (0, Ϫ2). Although f(Ϫ2, 0)is , it is not the maximum value since there are otherpoints that produce greater values. For example, f(2, 1) isand f(3, 1) is . It appears that because the regionis unbounded, f(x, y) has no maximum value.Check Your Progress Graph each system ofinequalities. Name the coordinates of the vertices of thefeasible region. Find the maximum and minimum valuesof the given function for this region.a. x Յ 4y Յ 5x ϩ y Ն 6ƒ(x, y) ϭ 4x Ϫ 3yb. x ϩ 3y Յ 6Ϫx Ϫ 3y Յ 92y Ϫ x Ն Ϫ6ƒ(x, y) ϭ x ϩ 2yGlencoe Algebra 2 73Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Linear ProgrammingLANDSCAPING A landscaping company has crewswho mow lawns and prune shrubbery. The companyschedules 1 hour for mowing jobs and 3 hours forpruning jobs. Each crew is scheduled for no more than2 pruning jobs per day. Each crew’s schedule is set upfor a maximum of 9 hours per day. On the average, thecharge for mowing a lawn is $40 and the charge forpruning shrubbery is $120. Find a combination of mowinglawns and pruning shrubs that will maximize the incomethe company receives per day from one of its crews.Step 1 Define the variables.m ϭ the number of mowing jobsp ϭ the number of pruning jobsStep 2 Write a system of inequalities.Since the number of jobs cannot be negative,m and p must be nonnegative numbers.m Ն , p ՆMowing jobs take hour. Pruning jobs takehours. There are hours to do the jobs.ϩ Յ 9There are no more than 2 pruning jobs a day. Յ 2Step 3 Graph the system pmO(3, 2)(0, 2)(0, 0) (9,0)of inequalities.Step 4 Find the coordinates ofthe vertices of thefeasible region.From the graph, thevertices are at ,, , and .Step 5 Write the function to be maximized.The function that describes the income isƒ(m, p) ϭ 40m ϩ 120p. We want to find thevalue for this function.3–474 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Linear ProgrammingProcedureStep 1 Define thevariables.Step 2 Write a systemof inequalities.Step 3 Graph thesystem ofinequalities.Step 4 Find thecoordinates ofthe vertices ofthe feasibleregion.Step 5 Write a functionto be maximizedor minimized.Step 6 Substitute thecoordinates ofthe vertices intothe function.Step 7 Select thegreatest or leastresult. Answerthe problem.KEY CONCEPT
    • 3–4Glencoe Algebra 2 75Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc. Step 6 Substitute the coordinates of the vertices into thefunction.Step 7 Select the amount. Answer theproblem.The maximum values are at andat . This means that the company receives the mostmoney with mows and prunings or mowsand prunings.Check Your Progress A landscaping company has crewswho rake leaves and mulch. The company schedules 2 hoursfor mulching jobs and 4 hours for raking jobs. Each crew isscheduled for no more than 2 raking jobs per day. Each crew’sschedule is set up for a maximum of 8 hours per day. On theaverage, the charge for raking a lawn is $50 and the chargefor mulching is $30. Find a combination of raking leaves andmulching that will maximize the income the company receivesper day from one of its crews.Page(s):Exercises:HOMEWORKASSIGNMENT(m, p) 40m ϩ 120p ƒ(m, p)(0, 2)(3, 2)(9, 0)(0, 0)
    • 3–576 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Solving Systems of Equations inThree VariablesOne SolutionSolve the system of equations.5x ϩ 3y ϩ 2z ϭ 22x ϩ y Ϫ z ϭ 5x ϩ 4y ϩ 2z ϭ 16Use elimination to make a system of two equations in twovariables. First, eliminate z in the first and second equations.5x ϩ 3y ϩ 2z ϭ 2 5x ϩ 3y ϩ 2z ϭ 22x ϩ y Ϫ z ϭ 5 (ϩ) 4x ϩ 2y Ϫ 2z ϭ 10ϭEliminate z in the first and third equations.5x ϩ 3y ϩ 2z ϭ 2(Ϫ)x ϩ 4y ϩ 2z ϭ 16ϭSolve the system of two equations. Eliminate y.9x ϩ 5y ϭ 12 9x ϩ 5y ϭ 124x Ϫ y ϭ Ϫ14(ϫ 5)(ϩ) ϭϭx ϭSubstitute for x in one of the two equationswith two variables and solve for y.4x Ϫ y ϭ Ϫ14 Equation with two variables4( )Ϫ y ϭ Ϫ14 Replace x.Ϫ y ϭ Ϫ14 Multiplyy ϭ Simplify.System of Equations inThree VariablesOne solution• planes intersect in onepointInfinite Solutions• planes intersect in aline• planes intersect in thesame planeNo solution• planes have no pointin commonKEY CONCEPTThe solution of a system of equations in three variables,x, y, and z is called an ordered triple and is written as (x, y, z).BUILD YOUR VOCABULARY (page 57)(ϫ 2)• Solve systems of linearequations in threevariables.• Solve real-worldproblems using systemsof linear equations inthree variables.MAIN IDEASTEKS 2A.3 Thestudent formulatessystems ofequations and inequalitiesfrom problem situations,uses a variety of methodsto solve them, andanalyzes the solutions interms of the situations. (A)Analyze situations andformulate systems ofequations in two or moreunknowns or inequalitiesin two unknowns tosolve problems. (B)Use algebraic methods,graphs, tables, ormatrices, to solvesystems of equationsor inequalities. (C)Interpret and determinethe reasonableness ofsolutions to systems ofequations or inequalitiesfor given contexts.
    • Substitute for x and y in one of the originalequations with three variables.2x ϩ y Ϫ z ϭ 5 Equation with three variables2( )ϩ 6 Ϫ z ϭ 5 Replace x and y.ϩ 6 Ϫ z ϭ 5 Multiply.z ϭ Simplify.The solution is . You can check this solutionin the other two original equations.Infinite SolutionsSolve the system of equations.2x ؉ y ؊ 3z ‫؍‬ 5x ؉ 2y ؊ 4z ‫؍‬ 76x ؉ 3y ؊ 9z ‫؍‬ 15Eliminate y in the first and third equations.2x ϩ y Ϫ 3z ϭ 5(ϫ 3)6x ϩ 3y Ϫ 9z ϭ 156x ϩ 3y Ϫ 9z ϭ 15 (Ϫ)6x ϩ 3y Ϫ 9z ϭ 15ϭThe equation 0 ϭ 0 is always . This indicates thatthe first and third equations represent the same plane. Checkto see if this plane intersects the second planex ϩ 2y Ϫ 4z ϭ 7(ϫ 6)6x ϩ 12y Ϫ 24z ϭ 426x ϩ 3y Ϫ 9z ϭ 15 (Ϫ)6x ϩ 3y Ϫ 9z ϭ 15ϭϭThe planes intersect in a . So, there are anof solutions.Glencoe Algebra 2 77Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.3–5Under the Systems ofEquations in ThreeVariables tab, sketch agraph of a system with(a) one solution,(b) infinite soultions,(c) no solutions.ORGANIZE IT
    • 3–578 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.No SolutionSolve the system of equations.3x ؊ y ؊ 2z ‫؍‬ 46x ؉ 4y ؉ 8z ‫؍‬ 119x ؉ 6y ؉ 12z ‫؍‬ ؊3Eliminate x in the second two equations.6x ϩ 4y ϩ 8z ϭ 11(ϫ 3)ϩ 12y ϩ ϭ9x ϩ 6y ϩ 12z ϭ Ϫ3(ϫ 2)(Ϫ)18x ϩ ϩ 24z ϭϭThe equation ϭ is never true. So, there isno solution of this system.Check Your Progress Solve each system of equations.a. 2x ϩ 3y Ϫ 3z ϭ 16x ϩ y ϩ z ϭ Ϫ3x Ϫ 2y Ϫ z ϭ Ϫ1b. x ϩ y Ϫ 2z ϭ 3Ϫ3x Ϫ 3y ϩ 6z ϭ Ϫ92x ϩ y Ϫ z ϭ 6c. 3x ϩ y Ϫ z ϭ 5Ϫ15x Ϫ 5y ϩ 5z ϭ 11x ϩ y ϩ z ϭ 2
    • 3–5Write and Solve a System of EquationsSPORTS There are 49,000 seats in a sports stadium.Tickets for the seats in the upper level sell for $25, theones in the middle level cost $30, and the ones in thebottom level are $35 each. The number of seats in themiddle and bottom levels together equals the number ofseats in the upper level. When all of the seats are soldfor an event, the total revenue is $1,419,500. How manyseats are there in each level?Explore Read the problem and define the .u ϭ number of seats in the upper levelm ϭ number of seats in the middle levelb ϭ number of seats in the bottom levelPlan There are seats.u ϩ m ϩ b ϭ 49,000When all the seats are sold, the revenue is.Seats cost , , and .25u ϩ 30m ϩ 35b ϭ 1,419,500The number of seats in the middle and bottom levelstogether equal the number of seats in the upperlevel.m ϩ b ϭSolve Substitute m ϩ b for u in each of the first twoequations.u ϩ m ϩ b ϭ 49,000( )ϩ m ϩ b ϭ 49,000 Replace u with .ϩ ϭ 49,000 Simplify.m ϩ b ϭ 24,500 Divide by 2.25u ϩ 30m ϩ 35b ϭ 1,419,50025( )ϩ 30m ϩ 35b ϭ 1,419,500 Replace uwith m ϩ b.Ϫ 30m Ϫ 35b ϭ 1,419,500 DistributivePropertyϩ ϭ 1,419,500 Simplify.Glencoe Algebra 2 79Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Now, solve the system of two equations in two variablesm ϩ b ϭ 24,500(ϫ 55)ϩ ϭ55m ϩ 60b ϭ 1,419,500 (Ϫ)55m ϩ 60b ϭ 1,419,500ϭb ϭ 14,400Substitute 14,400 for b in the one of the equations with twovariables and solve for m.m ϩ b ϭ 24,500 Equation with two variablesm ϩ ϭ 24,500 b ϭ 14,400m ϭ 10,100 Subtract from each side.Substitute 14,400 for b and 10,100 for m in one of the originalequation with three variables.m ϩ b ϭ u Equation with three variablesϩ ϭ u m ϭ 10,100, b ϭ 14,400ϭ u Add.There are upper level, middlelevel, and bottom level seats.Check Check to see if all the criteria are met.There are seats in the stadium.24,500 ϩ 10,100 ϩ 14,400 ϭ 49,000The number of seats in the middle and bottom levelsequals the number of seats in the upper level.10,100 ϩ 14,400 ϭ 24,500When all of the seats are sold, the revenue is.24,500($25) ϩ 10,100($30) ϩ 14,400($35)ϭ 1,419,500Check Your Progress The school store sells pens, pencils,and paper. The pens are $1.25 each, the pencils are $0.50each, and the paper is $2 per pack. Yesterday the store sold 25items and earned $32. The number of pens sold equaled thenumber of pencils sold plus the number of packs of paper soldminus 5. How many of each item did the store sell?3–5Page(s):Exercises:HOMEWORKASSIGNMENT80 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • C H A P T E R3Under each system graphed below, write all of the following wordsthat apply: consistent, inconsistent, dependent, and independent.1.xyO2.xyO3.xyO4. Solve the system x ϩ y ϭ 3and 3x Ϫ y ϭ 1 by graphing.5. Solve x ϭ 4y ϩ 3 and x Ϫ 2y ϭ 9by using substitution.6. Solve Ϫ2x ϩ 5y ϭ 7 and 2x ϩ 4y ϭ 11by using elimination.BRINGING IT ALL TOGETHERBUILD YOURVOCABULARYUse your Chapter 3 Foldable tohelp you study for your chaptertest.To make a crossword puzzle,word search, or jumblepuzzle of the vocabulary wordsin Chapter 3, go to:glencoe.comYou can use your completedVocabulary Builder(pages 57–58) to help you solvethe puzzle.VOCABULARYPUZZLEMAKERSTUDY GUIDEGlencoe Algebra 2 81Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.3-2Solving Systems of Equations Algebraically3-1Solving Systems of Equations by Graphing
    • Chapter BRINGING IT ALL TOGETHER382 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.7. Which system of inequalities matches the graph shownat the right?a. x Ϫ y Յ Ϫ2 b. x Ϫ y Ն Ϫ2x Ϫ y Ͼ 2 x Ϫ y Ͻ 2c. x ϩ y Յ Ϫ2 d. x Ϫ y Ͼ Ϫ2x ϩ y Ͼ 2 x Ϫ y Յ 2Find the coordinates of the vertices of the figure formed byeach system of inequalities.8. x ϩ y Ն 8 9. x ϩ y Ն 6y Ն 5 x Յ 8x Ն 0 y Յ 510. A polygonal feasible region has vertices at (4, Ϫ1), (3, 0), (1, Ϫ6),and (Ϫ2, 2). Find the maximum and minimum of the functionƒ(x, y) ϭ Ϫ2x ϩ y over this region.Solve each system of equations.11. 5x ϩ y ϩ 4z ϭ 9 12. 2x ϩ y Ϫ 4z ϭ Ϫ3Ϫ5x ϩ 3y ϩ z ϭ Ϫ15 Ϫ6x Ϫ 3y ϩ 12z ϭ 715x ϩ 5y ϩ 7z ϭ 9 x Ϫ 9y ϩ 4z ϭ 113. The sum of three numbersis 22. The sum of the first andsecond numbers is 19, and thefirst number is 5 times thethird number. Find the numbers.3-5Solving Systems of Equations in Three Variables3-3Solving Systems of Inequalities by Graphing3-4Linear ProgrammingxyO
    • C H A P T E R3ChecklistCheck the one that applies. Suggestions to help you study aregiven with each item.I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.• You may want to take the Chapter 3 Practice Test onpage 157 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 3 Study Guide and Reviewon pages 153–156 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 3 Practice Test onpage 157 of your textbook.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 3 Foldable.• Then complete the Chapter 3 Study Guide and Review onpages 153–156 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 3 Practice Test onpage 157 of your textbook.Student Signature Parent/Guardian SignatureTeacher SignatureVisit glencoe.com toaccess your textbook,more examples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 3.Glencoe Algebra 2 83Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc. ARE YOU READY FORTHE CHAPTER TEST?
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.84 Glencoe Algebra 2MatricesC H A P T E R4Use the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.NOTE-TAKING TIP: When you take notes, writedescriptive paragraphs about your learningexperiences.Fold lengthwise tothe holes. Cut eighttabs in the top sheet.Label each tab with alesson number and title.Begin with one sheet of notebook paper.
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Vocabulary TermFoundon PageDefinitionDescription orExampleCramer’s Rule[KRAY-muhrs]determinantdilation[dy-LAY-shuhn]dimensionelementequal matricesexpansion byminorsidentity matriximageinverseisometryeye-SAH-muh-treeThis is an alphabetical list of new vocabulary terms you will learn in Chapter 4.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description on thesepages. Remember to add the textbook page numbering in the second column forreference when you study.C H A P T E R4BUILD YOUR VOCABULARY(continued on the next page)Glencoe Algebra 2 85Chapter4
    • 86 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Chapter BUILDING YOUR VOCABULARY4Vocabulary TermFoundon PageDefinitionDescription orExamplematrixMAY-trihksmatrix equationminorpreimagereflectionrotationscalar multiplicationSKAY-luhrsquare matrixtransformationtranslationvertex matrixzero matrix
    • A matrix is a rectangular array of variables or constants inand vertical columns, usuallyenclosed in brackets.Each in the matrix is called an element.A matrix that has the same number of rows andis called a square matrix.Another special type of matrix is the zero matrix, in whichevery element is .Glencoe Algebra 2 87Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Organize Data into a MatrixCOLLEGE Kaitlin wants to attend one of three Iowauniversities next year. She has gathered informationabout tuition (T), room and board (R/B), and enrollment(E) for the universities. Use a matrix to organize theinformation. Which university’s total cost is lowest?Iowa State University:T - $5426 R/B - $5958 E - 26,380University of Iowa:T - $5612 R/B - $6560 E - 28,442University of Northern Iowa:T - $5387 R/B - $5261 E - 12,927Organize the data into labeled columns and rows.T R/B EISUUIUNIThe University of has the lowest total cost.4–1• Organize data inmatrices.• Solve equationsinvolving matrices.MAIN IDEASREMEMBER ITAn element ofa matrix can berepresented by thenotation aij. This refersto the element in row i,column j.BUILD YOUR VOCABULARY (pages 85–86)Introduction to MatricesTEKS 2A.1 The student uses properties and attributes of functions and appliesfunctions to problem situations. (B) Collect and organize data, make and interpretscatterplots, fit the graph of a function to the data, interpret the results, and proceed tomodel, predict, and make decisions and critical judgements.
    • 4–188 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Check Your Progress Justin is going out for lunch.The information he has gathered from the two fast-foodrestaurants is listed below. Use a matrix to organize theinformation. When is each restaurant’s total cost lessexpensive?Burger Complex Lunch ExpressHamburger Meal $3.39 Hamburger Meal $3.49Cheeseburger Meal $3.59 Cheeseburger Meal $3.79Chicken Sandwich Chicken SandwichMeal $4.99 Meal $4.89Dimensions of a MatrixState the dimensions of matrix G ifG ϭ΄21Ϫ150Ϫ33Ϫ1΅.G ϭ΄21Ϫ150Ϫ33Ϫ1΅ rowscolumnsSince matrix G has rows and columns, thedimensions of matrix G are .Check Your Progress State the dimensions of matrix Gif G ϭ ΄20Ϫ1344΅.Under the tab forLesson 4-1, tell howto find the dimensionsof a matrix.ORGANIZE IT
    • 4–1Page(s):Exercises:HOMEWORKASSIGNMENTGlencoe Algebra 2 89Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Solve an Equation Involving MatricesSolve ΄y3΅ϭ΄3x Ϫ 22y ϩ x΅for x and y.Since the matrices are equal, the corresponding elements areequal. When you write the sentences to solve this equation,two linear equations are formed.y ϭ 3x Ϫ 23 ϭ 2y ϩ xThis system can be solved using substitution.3 ϭ 2y ϩ x Second equationϭ 2΂ ΃ ϩ x Substitute for y.ϭ ϩ x Distributive Propertyϭ Add 4 to each side.ϭ x Divide each side by 7.To find the value for y, substitute 1 for x in either equation.y ϭ 3x Ϫ 2 First equationy ϭ 3΂ ΃ Ϫ 2 Substitute for x.y ϭ Simplify.The solution is .Check Your Progress Solvey2xϭ3x Ϫ 1y Ϫ 2for x and y.
    • 4–290 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Add Matricesa. Find A ϩ B if A ϭ ΄ 6Ϫ140΅and B ϭ΄Ϫ3013΅.A ϩ B ϭ΄ 6Ϫ140΅ϩ΄Ϫ3013΅ Definition ofmatrix additionAddϭ΄ ΅ correspondingelements.ϭ΄ ΅ Simplify.b. Find A ϩ B if A ϭ΄41Ϫ250Ϫ1΅and B ϭ΄Ϫ6Ϫ973΅.Since their dimensions are , these matricesbe added.Subtract MatricesFind A Ϫ B if A ϭ ΄ 3Ϫ120΅and B ϭ΄Ϫ201Ϫ1΅.A Ϫ B ϭ΄ 3Ϫ120΅Ϫ΄Ϫ201Ϫ1΅ Definition ofmatrix subtractionSubtractϭ΄ ΅ correspondingelements.ϭ΄ ΅ Simplify.Check Your Progressa. Find A ϩ B if A ϭ ΄Ϫ213Ϫ1΅and B ϭ΄6350΅.• Add and subtractmatrices.• Multiply by a matrixscalar.MAIN IDEASOperations with MatricesAddition of MatricesIf A and B are twom ϫ n matrices, thenA ϩ B is an m ϫ nmatrix in which eachelement is the sum ofthe correspondingelements of A and B.Subtraction of MatricesIf A and B are twom ϫ n matrices, thenA Ϫ B is an m ϫ n matrixin which each elementis the difference of thecorresponding elementsof A and B.KEY CONCEPTSPreparation for TEKS 2A.3 The student formulates systems of equations andinequalities from problem situations, uses a variety of methods to solve them, andanalyzes the solutions in terms of the situations. (B) Use algebraic methods, graphs,tables, or matrices, to solve systems of equations or inequalities.
    • b. Find A Ϫ B if A ϭ΄ 3Ϫ120΅and B ϭ΄ 5Ϫ1Ϫ23 ΅.Solve ProblemsSCHOOL ATHLETES The table below shows the totalnumber of student athletes and the number of femaleathletes in three high schools. Use matrices to find thenumber of male athletes in each school.School Total Number of Athletes Female AthletesJefferson 863 281South 472 186Ferguson 1023 346The data in the table can be organized in matrices.Find the difference of the matrix that represents total numberof and the matrix that represents the number ofathletes.Total Female MaleϪϭSubtractcorrespondingelementsϭThere are male athletes at Jefferson, maleathletes at South, and male athletes at Ferguson.4–2Glencoe Algebra 2 91Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • You can multiply any matrix by a constant called a. This operation is called scalar multiplication.4–292 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Multiply a Matrix by a ScalarIf A ϭ2Ϫ10135, find 2A.2A ϭ 22Ϫ10135Substitutionϭ1(2)Ϫ1(2)5(2)Multiply each element by 2.ϭSimplify.BUILD YOUR VOCABULARY (page 85–86)Under thetab for Lesson 4-2,write your own examplethat involves scalarmultiplication.Then perform themultiplication.KEY CONCEPTScalar MultiplicationThe product of a scalar kand an m ϫ n matrix isthe matrix in which eachelement equals k timesthe correspondingelements of the originalmatrix.Check Your Progress The table shows the percent ofstudents at Clark High School who passed the 9th and 10thgrade proficiency tests in 2005 and 2006. Use matrices to findhow the percent of passing students changed from 2005to 2006.Proficiency Tests Passing PercentagesYear 2001 2002Grade 9 10 9 10Math 86.2 87.3 88.4 89.6Reading 83.5 90.1 81.9 91.2Science 79.6 89.7 85.0 89.9Citizenship 86.1 87.4 86.4 85.7
    • 4–2Page(s):Exercises:HOMEWORKASSIGNMENTCombination of Matrix OperationsIf A ϭ2Ϫ130and B ϭϪ201Ϫ1, find 4A Ϫ 3B.Perform the scalar multiplication first. Then subtract thematrices.4A Ϫ 3Bϭ 42Ϫ130Ϫ 3Ϫ201Ϫ1Substitutionϭ4(2)4(Ϫ1)4(3)4(0)Ϫ3(Ϫ2)3(0)3(1)3(Ϫ1)Multiply each element inthe first matrix by 4 andmultipy each element inthe second matrix by 3.ϭϪSimplify.ϭSubtract correspondingelements.ϭ14Ϫ493Simplify.Check Your Progressa. If A ϭ2Ϫ13Ϫ506find 4A.b. If A ϭ32Ϫ15and B ϭϪ2Ϫ134, find 5A Ϫ 2B.Glencoe Algebra 2 93Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 4–394 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Dimensions of Matrix Productsa. Determine if the product of A3 ϫ 4 and B4 ϫ 2 isdefined. If so, state the dimensions of the product.A и B ϭ AB3 ϫ 4 4 ϫ 2The inner dimensions are so the matrix product is. The dimensions of the product are .b. Determine whether the product of A3 ϫ 2 and B4 ϫ 3 isdefined. If so, state the dimensions of the product.A и B3 ϫ 2 4 ϫ 3The inner dimensions are , so the matrixproduct is .Check Your Progress Determine if the product of A2 ϫ 3and B2 ϫ 3 is defined. If so, state the dimensions of the product.Multiply Square MatricesFind RS if R ϭ3Ϫ120and S ϭϪ211Ϫ1.RS ϭ3Ϫ120иϪ211Ϫ1ϭϩϪ1(Ϫ2) ϩ 0(1)3(1) ϩ 2(Ϫ1)ϩϭ• Multiply matrices.• Use the properties ofmatrix multiplication.MAIN IDEASMultiplying MatricesThe element aij of AB isthe sum of the productsof the correspondingelements in row i of Aand column j of B.Under the tabfor Lesson 4-3, write anexample of multiplyingsquare matrices.KEY CONCEPTMultiplying MatricesPreparation for TEKS 2A.3 The student formulates systems of equations andinequalities from problem situations, uses a variety of methods to solve them, andanalyzes the solutions in terms of the situations. (B) Use algebraic methods, graphs,tables, or matrices, to solve systems of equations or inequalities.
    • 4–3Glencoe Algebra 2 95Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Check Your Progress Find RS if R ϭ31Ϫ10andS ϭ2Ϫ21Ϫ3.Multiply Matrices with Different DimensionsCHESS Three teams competed in the final round of theChess Club’s championships. For each win, a team wasawarded 3 points and for each draw a team received1 point. Which team won the tournament?Team Wins DrawsBlue 5 4Red 6 3Green 4 5RP ϭ564435и31. Write an equation.ϭϩϩϩMultiply columns by rows.ϭSimplify.The labels for the product matrix Total Pointsare shown. The red team won thechampionship with a total ofBlueRedGreenpoints.Is multiplication ofmatrices commutative?Explain.WRITE IT
    • Check Your Progress Three players made the pointslisted below. They scored 1 point for the free-throws, 2 pointsfor the 2-point shots, and 3 points for the 3-point shots. Howmany points did each player score and who scored the mostpoints?Player Free-throws 2-point 3-pointWarton 2 3 2Bryant 5 1 0Chris 2 4 5Commutative PropertyFind each product if K ϭϪ3Ϫ12Ϫ220and L ϭ140Ϫ23Ϫ1.a. KL ϭϪ3Ϫ12Ϫ220и140Ϫ23Ϫ1SubstitutionϭϪ3 ϩ 8 ϩ 02 Ϫ 6 ϩ 0Multiply.ϭSimplify.b. LK ϭ140Ϫ23Ϫ1ϭϪ3Ϫ12Ϫ220SubstitutionϭϪ3 ϩ 2 2 ϩ 48 Ϫ 6 8 Ϫ 00 ϩ 1 0 Ϫ 0Multiply.ϭSimplify.4–396 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 4–3Page(s):Exercises:HOMEWORKASSIGNMENTCheck Your Progress Find each product if A ϭ0314and B ϭϪ2310.a. ABb. BADistributive PropertyFind each product if A ϭϪ1021, B ϭ130Ϫ2, andC ϭϪ3Ϫ110.a. A(B ϩ C) ϭϪ1021и ΂130Ϫ2ϩϪ3Ϫ110΃ϭϪ1021ϩϪ221Ϫ2Add.ϭϪ1 Ϫ 40 Ϫ 2or62Multiply.b. AB ϩ AC ϭϪ1021и130Ϫ2ϩϪ1021иϪ3Ϫ110ϭ53Ϫ4Ϫ2ϩ1Ϫ1Ϫ10Multiply.ϭ5 ϩ 13 Ϫ 1or62Add.Check Your Progress Find AB ϩ AC if A ϭ31Ϫ20,B ϭ210Ϫ5and C ϭ0Ϫ313.Glencoe Algebra 2 97Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Transformations are functions that map points of a preimageonto its image.A translation occurs when a figure is moved from onelocation to another without changing its ,, or orientation.4–498 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Translate a FigureFind the coordinates of the vertices of the image ofquadrilateral ABCD with A(Ϫ5, Ϫ1), B(Ϫ2, Ϫ1), C(Ϫ1, Ϫ4),and D(Ϫ3, Ϫ5) if it is moved 3 units to the right and4 units up.Write the vertex matrix for quadrilateral ABCD.Ϫ1Ϫ1 Ϫ1 Ϫ5To translate the quadrilateral 3 units to the right, add 3 toeach x-coordinate. To translate the figure 4 units up, add 4 toeach y-coordinate. This can be done by adding the translationmatrix to the vertex matrix.Vertex Matrix Translationof ABCD MatrixϪ5Ϫ1Ϫ2Ϫ1Ϫ1Ϫ4Ϫ3Ϫ5ϩϭThe coordinates of AЈBЈCЈDЈ are BЈCЈDЈA BCDAЈxyOAЈ , BЈ ,CЈ , DЈ .By graphing the preimage and theimage, you find that the coordinatesof AЈBЈCЈDЈ are correct.• Use matrices todetermine thecoordinates of atranslated or dilatedfigure.• Use matrixmultiplication to findthe coordinates of areflected or rotatedfigure.MAIN IDEASBUILD YOUR VOCABULARY (pages 85–86)Preparation for TEKS 2A.3 The student formulates systems of equations andinequalities from problem situations, uses a variety of methods to solve them, andanalyzes the solutions in terms of the situations. (B) Use algebraic methods, graphs,tables, or matrices, to solve systems of equations or inequalities.Transformations with MatricesUnder the tab forLesson 4-4, write eachnew Vocabulary Builderword. Then give anexample of each word.ORGANIZE IT
    • A reflection occurs when every point of a figure is mappedto a corresponding image acrossusing a reflection matrix.A rotation occurs when a figure is moved around a, usually the .4–4Glencoe Algebra 2 99Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.BUILD YOUR VOCABULARY (pages 85–86)Check Your Progress Find the coordinates of the verticesof the image of quadrilateral HIJK with H(2, 3), I(3, Ϫ1),J(Ϫ1, Ϫ3), and K(Ϫ2, 5) if it is moved 2 units to the left and2 units up.Find a Translation MatrixTEST EXAMPLE Rectangle EЈFЈGЈHЈ is the result of atranslation of rectangle EFGH. A table of the vertices ofeach rectangle is shown. Find the coordinates of GЈ.RectangleEFGHRectangleEЈFЈGЈHЈE(Ϫ2, 2) EЈ(Ϫ5, 0)F(4, 2) FЈ(1, 0)G(4, Ϫ2) GЈ( , )H(Ϫ2, Ϫ2) HЈ(Ϫ5, Ϫ4)A (1, 0) C (7, 0)B (1, Ϫ4) D (7, Ϫ4)Read the Test ItemYou are given the coordinates of the preimage and image ofpoints E, F, and H. Use this information to find the translationmatrix. Then you can use the translation matrix to find thecoordinates of G.
    • Solve the Test ItemStep 1 Write a matrix equation. Let (c, d) represent thecoordinates of G.Ϫ22424Ϫ2Ϫ2Ϫ2ϩxyxyxyxyϭϪ5010cdϪ5Ϫ4ϭϪ5010cdϪ5Ϫ4Step 2 The matrices are equal, so corresponding elementsare equal.Ϫ2 ϩ x ϭ Ϫ5 Solve for x. 2 ϩ y ϭ 0 Solve for y.x ϭ y ϭStep 3 Use the values for x and y to find the values forGЈ(c, d).4 ϩ (Ϫ3) ϭ c Ϫ2 ϩ (Ϫ2) ϭ dϭ c ϭ dSo, the coordinates for G are , and the answeris .DilationDilate XYZ with X(1, 2), Y(3, Ϫ1), and Z(Ϫ1, Ϫ2) so thatits perimeter is twice the original perimeter. Find thecoordinates of the vertices of XЈYЈZЈ.If the perimeter of a figure is twice the original perimeter, thenthe lengths of the sides of the figure will be themeasure of the original lengths. Multiply the vertex matrix bythe scale factor of .2ϭThe coordinates of the vertices of XYZ are XЈ ,YЈ , and ZЈ .4–4100 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Reflection MatricesFor a counterclockwiserotation about theorigin of:90Њ 01Ϫ10180Њ Ϫ100Ϫ1270Њ 0Ϫ110KEY CONCEPT4–4Check Your Progressa. Rectangle AЈBЈCЈDЈ is the translation of rectangle ABCDwith vertices A(Ϫ4, 5), B(Ϫ1, 5), C(Ϫ1, 0), and D(Ϫ4, 0).What are the coordinates of AЈ?A (Ϫ13, 10) B (5, 10) C (5, 0) D (Ϫ13, 0)b. ABC has vertices A(2, 1), B(Ϫ3, Ϫ2), and C(1, 4). If ABCis dilated so its perimeter is 4 times the original perimeter,what are the coordinates of the vertices of AЈBЈCЈ?ReflectionFind the coordinates of the vertices of the imageof pentagon PENTA with P(Ϫ3, 1), E(0, Ϫ1), N(Ϫ1, Ϫ3),T(Ϫ3, Ϫ4), and A(Ϫ4, Ϫ1) after a reflection across thex-axis.Write the ordered pairs as a vertex matrix. Thenthe vertex matrix by the matrix for the x-axis.иϪ310Ϫ1Ϫ1Ϫ3Ϫ3Ϫ4Ϫ4Ϫ1ϭϪ3 Ϫ11 4 1The coordinates of the vertices yxONEPTANEPTAof PЈEЈNЈTЈAЈare PЈ(Ϫ3, Ϫ1), EЈ(0, 1),NЈ(Ϫ1, 3), TЈ(Ϫ3, 4), and AЈ(Ϫ4, 1). Thegraph of the preimage and image showsthat the coordinates of PЈEЈNЈTЈAЈ arecorrect.Check Your Progress Find the coordinates of the verticesof the image of pentagon PENTA with P(Ϫ5, 0), E(Ϫ3, 3),N(1, 2), T(1, Ϫ1), and A(Ϫ4, Ϫ2) after a reflection across theline y ϭ x.Glencoe Algebra 2 101Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Reflection Matricesx-axis100Ϫ1y-axisϪ1001line 0110y ϭ xKEY CONCEPT
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.4–4Page(s):Exercises:HOMEWORKASSIGNMENT102 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.RotationFind the coordinates of the vertices of the image of᭝DEF with D(4, 3), E(1, 1), and F(2, 5) after it is rotated90° counterclockwise about the origin.Write the ordered pairs in a vertex matrix. Then multiply thevertex matrix by the rotation matrix.и431125ϭϪ54 1The coordinates of the vertices of yxOFEDFEDtriangle DЈEЈFЈ are DЈ ,EЈ , FЈ .The graph of the preimage and imageshow that the coordinates of DЈEЈFЈare correct.Check Your Progress Find the coordinates of thevertices of the image of TRI with T(Ϫ1, 2), R(Ϫ3, 0) andI(Ϫ2, Ϫ2) after it is rotated 180° counterclockwise about theorigin.
    • 4–5Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.• Evaluate thedeterminant of a 2 ϫ 2matrix.• Evaluate thedeterminant of a 3 ϫ 3matrix.MAIN IDEASSecond-OrderDeterminant The valueof a second-orderdeterminant is foundby calculating thedifference of theproducts of the twodiagonals.KEY CONCEPTSecond-Order DeterminantFind the value of the determinant  6Ϫ140. 6Ϫ140ϭ Ϫ Definition ofdeterminantϭ ϩ Multiply.ϭ Simplify.Expansion by MinorsEvaluate1240Ϫ1Ϫ2Ϫ13Ϫ3using expansion by minors.Decide which row of elements to use for the expansion. Forthis example, let’s use the first row.1240Ϫ1Ϫ2Ϫ13Ϫ3ϭ 1Ϫ3Ϫ3Ϫ 0243Ϫ3ϩ (Ϫ1)Ϫ14ϭ 1΂3 Ϫ ΂ ΃΃ Ϫ 0΂ Ϫ 12΃ Ϫ 1΂Ϫ4 Ϫ ΂ ΃΃ϭ 1΂ ΃ Ϫ Ϫ 1΂ ΃ϭPreparation for TEKS 2A.3 The student formulates systems of equations andinequalities from problem situations, uses a variety of methods to solve them, andanalyzes the solutions in terms of the situations. (B) Use algebraic methods, graphs,tables, or matrices, to solve systems of equations or inequalities.DeterminantsGlencoe Algebra 2 103Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 4–5104 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Use DiagonalsEvaluate321Ϫ2Ϫ12Ϫ10Ϫ3using diagonals.Step 1 Rewrite the first 2 columns to the right of thedeterminant.321Ϫ2Ϫ12Ϫ10Ϫ3Step 2 Find the product of the elements of the diagonals.321Ϫ2Ϫ12Ϫ10Ϫ3321Ϫ2Ϫ12321Ϫ2Ϫ12Ϫ10Ϫ3321Ϫ2Ϫ12Step 3 Add the bottom products and subtract the top products.ϭThe value of the determinant is .Check Your Progress Evaluate each determinant.a. 3120 b.Ϫ312243016c.251Ϫ302Ϫ1Ϫ25Under the tab forLesson 4-5, write yourown 3 ϫ 3 matrix. Thenevaluate your matrix.Include the steps shownin Example 3.ORGANIZE IT
    • SURVEYING A survey crew located three points on amap that formed the vertices of a triangular area. Acoordinate grid in which one unit equals 10 inches isplaced over the map so that the vertices are located at(0, Ϫ1), (Ϫ2, Ϫ6), and (3, Ϫ2). Use a determinant to findthe area of the triangle.A ϭ1_2acebdf111 Area Formula(a, b) ϭ ,ϭ1_20Ϫ23Ϫ1Ϫ6Ϫ2111 (c, d) ϭ ,(e, f) ϭϭ1_20Ϫ61Ϫ (Ϫ1)Ϫ2311ϩ 1Ϫ63Expansion by minorsϭ1_20΂Ϫ6 Ϫ ΂ ΃΃Ϫ (Ϫ1)΂Ϫ2 Ϫ ΃ϩ 1΂4 Ϫ ΂ ΃΃Evaluate 2 ϫ 2 determinants.ϭ1_2΂ Ϫ ϩ ΃ Multiply.ϭ1_2΂ ΃or Simplify.Remember that 1 unit equals 10 inches, so 1 square unit ϭ10 ϫ 10 or 100 square inches. Thus, the area is ϫ 100or square inches.Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.4-5Page(s):Exercises:HOMEWORKASSIGNMENTGlencoe Algebra 2 105Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Cramer’s Rule uses determinants to solve systems ofequations.• Solve systems of twolinear equations byusing Cramer´s Rule.• Solve systems of threelinear equations byusing Cramer´s Rule.MAIN IDEASCramer’s Rule for TwoVariables The solutionof the system of linearequations ax ϩ by ϭ eand cx ϩ dy ϭ f is (x, y),wherex ϭefbd_acbd, y ϭacef_acbd,and acbd 0.Under thetab for Lesson 4-6, writethis rule.KEY CONCEPTBUILD YOUR VOCABULARY (page 85)System of Two EquationsUse Cramer’s Rule to solve the system of equations5x ϩ 4y ϭ 28 and 3x Ϫ 2y ϭ 8.x ϭefbd_acbdCramer’s Rule y ϭacef_acbdϭ a ϭ 5, b ϭ 4,c ϭ 3, d ϭ Ϫ2,ϭ e ϭ 28, and ϭf ϭ 8ϭ Evaluate. ϭ5(8) Ϫ 3(28)_5(Ϫ2) Ϫ 3(4)ϭ or Simplify. ϭ orThe solution is .Check Your Progress Use Cramer’s Rule to solve thesystem of equations 3x ϩ 2y ϭ 1 and 2x Ϫ 5y ϭ Ϫ12.5(Ϫ2) Ϫ 3(4)Ϫ8435 435283TEKS 2A.3 Thestudent formulatessystems ofequations and inequalitiesfrom problem situations,uses a variety of methodsto solve them, andanalyzes the solutions interms of the situations. (A)Analyze situations andformulate systems ofequations in two or moreunknowns or inequalitiesin two unknowns tosolve problems. (B)Use algebraic methods,graphs, tables, ormatrices, to solvesystems of equations orinequalities.4–6 Cramer’s Rule106 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.4–6Page(s):Exercises:HOMEWORKASSIGNMENTGlencoe Algebra 2 107Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.System of Three EquationsUse Cramer’s Rule to solve the system of equations.2x Ϫ 3y ϩ z ϭ 5x ϩ 2y ϩ z ϭ Ϫ1x Ϫ 3y ϩ 2z ϭ 1x ϭjkbehcfi_adgbehcfiϭ5Ϫ11Ϫ32Ϫ3112__211Ϫ32Ϫ3112y ϭadgjkcfi_adgbehcfiϭ2115Ϫ11112_211Ϫ32Ϫ3112z ϭadgbehjk_adgbehcfiϭ211Ϫ32Ϫ35Ϫ11__211Ϫ32Ϫ3112Use a calculator to evaluate each determinant.x ϭ or 9_4y ϭ Ϫ9_12or Ϫ z ϭ or Ϫ7_4The solution is .Check Your Progress Use Cramer’s Rule to solve thesystem of equations.2x ϩ y ϩ z ϭ Ϫ3Ϫ3x ϩ 2y Ϫ z ϭ 5x Ϫ y ϩ 3z ϭ 1The solution of thesystem whose equationsareax ϩ by ϩ cz ϭ jdx ϩ ey ϩ fz ϭ kgx ϩ hy ϩ iz ϭ lis (x, y, z), wherex ϭjklbehcfi_adgbehcfi,y ϭadgjklcfi_adgbehcfi,z ϭadgbehjkl_adgbehcfi, andandadgbehcfi 0.KEY CONCEPT
    • The identity matrix is a square matrix that, when multipliedby another matrix, equals that same matrix.Two n ϫ n matrices are inverses of each other if theiris the .4–7108 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.• Determine whether twomatrices are inverses.• Find the inverse of a2 ϫ 2 matrix.MAIN IDEASBUILD YOUR VOCABULARY (page 85)Verify Inverse MatricesDetermine whether each pair of matrices are inverses.X ϭ3Ϫ1Ϫ21and Y ϭ1123Find X и Y.X и Y ϭ3Ϫ1Ϫ21и1123ϭ6 Ϫ 6Ϫ1 ϩ 1orFind Y и X.Y и X ϭ1123и3Ϫ1Ϫ21ϭ3 Ϫ 23 Ϫ 3orSince X и Y ϭ Y и X ϭ I, X and Y are inverses.Check Your Progress Determine whether each pairof matrices are inverses.a. A ϭϪ2130and B ϭϪ2121b. C ϭ3211and D ϭ1Ϫ2Ϫ13Identity Matrix forMultiplication Theidentity matrix formultiplication I is asquare matrix with 1for every element ofthe main diagonal,from upper left to lowerright, and 0 in all otherpositions. For any squarematrix A of the samedimensions as I,A и I ϭ I и A ϭ A.Inverse of a 2 ϫ 2 matrixThe inverse of Matrix Aϭ acbd is AϪ1 ϭ1_ad Ϫ bc dϪcϪba, wheread Ϫ bc 0.KEY CONCEPTSIdentify and Inverse MatricesPreparation for TEKS 2A.3 The student formulates systems of equations andinequalities from problem situations, uses a variety of methods to solve them, andanalyzes the solutions in terms of the situations. (B) Use algebraic methods, graphs,tables, or matrices, to solve systems of equations or inequalities.
    • 4–7Glencoe Algebra 2 109Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Find the Inverse of a MatrixFind the inverse of each matrix, if it exists.a. S ϭϪ180Ϫ2Find the value of the determinant.Ϫ180Ϫ2ϭ 2 Ϫ ϭSince the determinant is not equal to 0, SϪ1exists.SϪ1ϭ 1_ad Ϫ bcdϪcϪbaϭ 1__Ϫ1(Ϫ2) Ϫ (0)Ϫ1Ϫ8ϭ 1_ Ϫ2Ϫ80Ϫ1orϪ1_2b. T ϭϪ4Ϫ263Find the value of the determinant.Ϫ4Ϫ263ϭ ϩ ϭSince the determinant equals 0, TϪ1does not exist.Check Your Progress Find the inverse of each matrixif it exists.a. A ϭ2613b. B ϭ67Ϫ2Ϫ2Under the tab forLesson 4-7, write yourown 2 ϫ 2 matrix. Thenfind the inverse of thematrix, if it exists.ORGANIZE IT
    • a. CRYPTOGRAPHY Use the table at the beginning ofthe lesson to assign a number to each letter in themessage ALWAYS_SMILE. Then code the messagewith the matrix A ϭ1123. Convert the message tonumbers using the table.A L W A Y S _ S M I L E1 12 23 1 25 19 0 19 13 9 12 5Write the message in matrix form. Arrange the numbers ina matrix with 2 columns and as many rows as are needed.Then multiply the message matrix B by the coding matrix A.BA ϭ1232501312121191995иWrite an equation.ϭMultiply the matrices.ϭSimplify.The coded message is.4–7110 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 4–7b. Use the inverse matrix AϪ1to decode the message inpart a. First find the inverse matrix of A ϭ1123.AϪ1ϭ 1_ad Ϫ bcdϪcϪbaDefinition of inverseϭ 1_1(3) Ϫ 2(1)a ϭ , b ϭ ,c ϭ , d ϭϭ3Ϫ1Ϫ21Simplify.ϭSimplify.Next, decode the message by multiplying the coded matrixC by AϪ1.CAϪ1ϭ1324441922173849107575339и3Ϫ1Ϫ21Write an equation.ϭMultiply thematrices.ϭWrite an equation.Use the table again to convert the numbers to letters. Youcan now read the message.A L W A Y S _ S M I L E1 12 23 1 25 19 0 19 13 9 12 5Glencoe Algebra 2 111Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Check Your Progressa. Use the table to assign a Code_ 0 I 9 R 18A 1 J 10 S 19B 2 K 11 T 20C 3 L 12 U 21D 4 M 13 V 22E 5 N 14 W 23F 6 O 15 X 24G 7 P 16 Y 25H 8 Q 17 Z 26number to each letter in themessage FUN_MATH. Thencode the matrix A ϭϪ2013.b. Use the inverse matrix ofA ϭϪ2013to decode themessage 12 63 28 14 26 1640 44.4–7Page(s):Exercises:HOMEWORKASSIGNMENT112 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Glencoe Algebra 2 113Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.A system of equations can be written withexpressing the system of equations as a matrix equation.Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Two-Variable Matrix EquationWrite a matrix equation for the system of equations.x ϩ 3y ϭ 3x ϩ 2y ϭ 7Determine the coefficient, variable, and constant matrices.x ϩ 3y ϭ 3x ϩ 2y ϭ 7Write the matrix equation.A и X ϭ BиϭProblem Solve with Matrix EquationsFABRICS The table below shows the composition ofthree types of fabric.Type Wool Silk Cotton CostR 10% 20% 70% $7S 20% 30% 50% $8T 20% 50% 30% $10a. Write a system of equations that represents the totalcost for each of the three fabric components.Let w represent the cost of wool.Let s represent the cost of silk.Let c represent the cost of cotton.BUILD YOUR VOCABULARY (page 86)• Write matrix equationsfor systems ofequations.• Solve systems ofequations using matrixequations.MAIN IDEASTEKS 2A.3 Thestudent formulatessystems ofequations and inequalitiesfrom problem situations,uses a variety of methodsto solve them, andanalyzes the solutions interms of the situations. (A)Analyze situations andformulate systems ofequations in two or moreunknowns or inequalitiesin two unknowns tosolve problems. (B)Use algebraic methods,graphs, tables, ormatrices, to solvesystems of equations orinequalities.4–8 Using Matrices to Solve Systems of EquationsUnder the tab forLesson 4-8, write amatrix equation for thissystem of equations.x ϩ 2y ϭ 53x ϩ 5y ϭ 14ORGANIZE IT
    • 4–8114 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.0.1w ϩ 0.2s ϩ 0.7c ϭ 70.2w ϩ 0.3s ϩ 0.5c ϭ 80.2w ϩ 0.5s ϩ 0.3c ϭ 10b. Write a matrix equation for the system of equations.Determine the coefficient, variable, and constantmatrices. Then write the matrix equation.0.1w ϩ 0.2s ϩ 0.7c ϭ 70.2w ϩ 0.3s ϩ 0.5c ϭ 8иwsc0.2w ϩ 0.5s ϩ 0.3c ϭ 10ϭSolve a System of EquationsUse a matrix equation to solve the system of equations.a. 5x ϩ 3y ϭ 134x ϩ 7y ϭ Ϫ8The matrix equation iswhen A ϭ, X ϭ, and B ϭ.Step 1 Find the inverse of the coefficient matrix.AϪ1ϭϪ7Ϫ4Ϫ35or7Ϫ4Ϫ35Step 2 Multiply each side of the matrix equation by theinverse matrix.1_237Ϫ4Ϫ35и5437иxyϭ 1_237Ϫ4Ϫ35и13Ϫ8иϭ 1_23ϭThe solution is .
    • 4–8Page(s):Exercises:HOMEWORKASSIGNMENTb. 10x ϩ 5y ϭ 156x ϩ 3y ϭ Ϫ6The matrix equation is10653иxyϭ15Ϫ6when A ϭ10653, X ϭxy, and B ϭ15Ϫ6.Find the inverse of the coefficient matrix.AϪ1ϭ 1_30 Ϫ 30or3Ϫ6Ϫ510The determinant of the coefficient matrix3Ϫ6Ϫ510is, so AϪ1does not exist. There is no uniqueof this system.Check Your Progressa. Write a matrix equation for the system of equations.x Ϫ 2y ϭ 6 and 3x ϩ 4y ϭ 7.b. Use a matrix equation to solve the system of equations.3x ϩ 4y ϭ0 and x Ϫ 2y ϭ 10.Glencoe Algebra 2 115Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Match each matrix with its dimensions.a. 3 ϫ 2b. 2 ϫ 3c. 2 ϫ 2d. 1 ϫ 21.3Ϫ120562. [30 Ϫ84]3.013Ϫ24.4Ϫ160215. Write a system of equations that you could use to solve thefollowing matrix equation for x, y, z.3xx ϩ yy Ϫ zϭϪ9566. Use M ϭ320Ϫ124and N ϭϪ2351Ϫ40to find 2M ϩ 3N.BRINGING IT ALL TOGETHERBUILD YOURVOCABULARYUse your Chapter 4 Foldable tohelp you study for your chaptertest.To make a crossword puzzle,word search, or jumblepuzzle of the vocabulary wordsin Chapter 4, go to:glencoe.comYou can use your completedVocabulary Builder(pages 85–86) to help you solvethe puzzle.VOCABULARYPUZZLEMAKERC H A P T E R4STUDY GUIDE116 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.4-1Introduction to Matrices4-2Operations with Matrices
    • Chapter BRINGING IT ALL TOGETHER4Glencoe Algebra 2 117Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Determine whether each indicated matrix product isdefined. If so, state the dimensions of the product. If not,write undefined.7. M3 ϫ 2 and N2 ϫ 3 MN:8. M1 ϫ 2 and N1 ϫ 2 MN:9. M4 ϫ 1 and N1 ϫ 4 MN:10. Find the product, if possible.210Ϫ135и15Ϫ3320Refer to quadrilateral ABCD shown.xyOABCD11. Write the vertex matrix for thequadrilateral ABCD.12. Write the vertex matrix that represents the position of thequadrilateral AЈBЈCЈDЈ that results when quadrilateralABCD is translated 3 units to the right 2 units down.13. Find the value of 823Ϫ1.14. Evaluate310Ϫ51296Ϫ10Ϫ2using expansion by minors.4-3Multiplying Matrices4-4Transformations with Matrices4-5Determinants
    • Chapter BRINGING IT ALL TOGETHER4118 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.15. The two sides of an angle are contained in the lines whoseequations are 3x ϩ y ϭ 5 and 2x ϩ 3y ϭ 8. Find thecoordinates of the vertex of the angle.16. Use Cramer’s Rule to solve the system of equations.2x ϩ 5y ϩ 3z ϭ 103x Ϫ y ϩ 4z ϭ 85x Ϫ 2y ϩ 7z ϭ 12Indicate whether each of the following statements is trueor false.17. Every element of an identity matrix is 1.18. There is a 3 ϫ 2 identity matrix.19. If M is a matrix, MϪ1represents the reciprocal of M.20. Every square matrix has an inverse.21. Determine whether A ϭ1Ϫ3Ϫ27andB ϭ7321are inverses.22. Write a matrix equation for the following system equations.3x ϩ 5y ϭ 102x Ϫ 4y ϭ Ϫ723. Solve the system of equations 4x Ϫ 5y ϭ 4 and 2x Ϫ y ϭ 8using inverse matrices.4-6Cramer´s Rule4-7Identity and Inverse Matrices4-8Using Matrices to Solve Systems of Equations
    • C H A P T E R4ChecklistGlencoe Algebra 2 119Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc. ARE YOU READY FORTHE CHAPTER TEST?Check the one that applies. Suggestions to help you study aregiven with each item.I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.• You may want to take the Chapter 4 Practice Test onpage 229 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 4 Study Guide and Reviewon pages 224–228 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 4 Practice Test onpage 229.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 4 Foldable.• Then complete the Chapter 4 Study Guide and Review onpages 224–228 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 4 Practice Test onpage 229 of your textbook.Visit glencoe.com toaccess your textbook,more examples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 4.Student Signature Parent/Guardian SignatureTeacher Signature
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.120 Glencoe Algebra 2C H A P T E R5 Quadratic Functions and InequalitiesUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.NOTE-TAKING TIP: When you take notes, you maywish to use a highlighting marker to emphasizeimportant concepts.Fold in half lengthwise. Then fold infourths crosswise. Cut along the middlefold from the edge to the last crease asshown.Staple along the lengthwise fold andstaple the uncut section at the top. Labeleach section with a lesson number andclose to form a booklet.Begin with one sheet of 11” by 17” paper.
    • Glencoe Algebra 2 121Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Vocabulary TermFoundon PageDefinitionDescription orExampleaxis of symmetrycompleting the squarecomplex conjugatescomplex numberconstant termdiscriminantdihs-KRIH-muh-nuhntimaginary unitlinear termmaximum valueminimum valueThis is an alphabetical list of the new vocabulary terms you will learn in Chapter 5.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.C H A P T E R5BUILD YOUR VOCABULARYChapter5(continued on the next page)
    • 122 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Chapter BUILDING YOUR VOCABULARY5Vocabulary TermFoundon PageDefinitionDescription orExampleparabolapuh-RA-buh-luhpure imaginary numberquadratic equationkwah-DRA-tihkquadratic functionquadratic inequalityquadratic termrootsquare rootvertexvertex formzero
    • 5–1• Graph quadraticfunctions.• Find and interpret themaximum and minimumvalues of a quadraticfunction.MAIN IDEASGlencoe Algebra 2 123Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Graph a Quadratic FunctionGraph ƒ(x) ϭ x2ϩ 3x Ϫ 1 by making a table of values.First, choose integer values for x. Then, evaluate the function foreach x value. Graph the function.x x2ϩ 3x Ϫ 1 ƒ(x) (x, y)Ϫ3 (Ϫ3)2ϩ 3(Ϫ3) Ϫ 1Ϫ2 (Ϫ2)2ϩ 3(Ϫ2) Ϫ 1Ϫ1 (Ϫ1)2ϩ 3(Ϫ1) Ϫ 10 (0)2ϩ 3(0) Ϫ 11 (1)2ϩ 3(1) Ϫ 1Check Your Progress Graph ƒ(x) ϭ 2x2ϩ 3x ϩ 2.Axis of Symmetry, y-intercept, and VertexConsider the quadratic function ƒ(x) ϭ 2 Ϫ 4x ϩ x2.a. Find the y-intercept, the equation of the axis ofsymmetry, and the x-coordinate of the vertex.Begin by rearranging the terms of the function. Thenidentify a, b, and c.ƒ(x) ϭ ax2ϩ bx ϩ cƒ(x) ϭ 2 Ϫ 4x ϩ x2ƒ(x) ϭ 1x2Ϫ 4x ϩ 2So, a ϭ , b ϭ , and c ϭ .xf(x)Of(x) ϭ x2 ϩ 3x Ϫ 1Graphing Quadratic FunctionsTEKS 2A.6The studentunderstands thatquadratic functions can berepresented in differentways and translatesamong their variousrepresentations.(A) Determine thereasonable domainand range values ofquadratic functions,as well as interpretand determine thereasonableness ofsolutions to quadraticequations and inequalities.TEKS 2A.8 The studentformulates equations andinequalities based onquadratic functions, usesa variety of methods tosolve them, and analyzesthe solutions in terms ofthe situation. (A) Analyzesituations involvingquadratic functions andformulate quadraticequations or inequalitiesto solve problems. Alsoaddresses TEKS 2A.1(A)and 2A.6(B).
    • 5–1124 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.The y-intercept is 2. You can find the equation of theaxis of symmetry by using a and b.x ϭ Ϫ b_2aEquation of the axis of symmetryx ϭ a ϭ 1, b ϭ Ϫ4x ϭ Simplify.The y-intercept is . The equation of the axis ofsymmetry is x ϭ . Therefore, the x-coordinate of thevertex is .b. Make a table of values that includes the vertex.Choose some values for x that are less than 2 and some thatare greater than 2.x x2Ϫ 4x ϩ 2 ƒ(x) (x, ƒ(x))0 02Ϫ 4(0) ϩ 21 12Ϫ 4(1) ϩ 22 22Ϫ 4(2) ϩ 23 32Ϫ 4(3) ϩ 24 42Ϫ 4(4) ϩ 2c. Use this information to graph thexf(x)Ox ϭ 2(4, 2)(0, 2)(3, –1)(2, –2)(1, –1)function.Graph the vertex and y-intercept.Then graph the points from your tableand the y-intercept, connecting themwith a smooth curve.As a check, draw the axis of symmetry,x ϭ 2, as a dashed line.The graph of the function should be symmetricabout this line.Check Your Progress Consider the quadraticfunction ƒ(x) ϭ 3 Ϫ 6x ϩ x2.a. Find the y-intercept, the equation of the axis of symmetryand the x-coordinate of the vertex.Why is it helpfulto know the axisof symmetry whengraphing a quadraticfunction?WRITE ITGraph of a QuadraticFunction Consider thegraph ofy ϭ ax2ϩ bx ϩc,where a 0.• The y-intercept isa(0)2ϩ b(0) ϩ c or c.• The equation of theaxis of symmetry isx ϭ Ϫ b_2a.• The x-coordinate ofthe vertex is Ϫ b_2a.KEY CONCEPT
    • b. Make a table of values that includes the vertex.c. Use this informationto graph the function.5–1Glencoe Algebra 2 125Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Maximum or Minimum ValueConsider the function ƒ(x) ϭ Ϫx2ϩ 2x ϩ 3.a. Determine whether the function has a maximum or aminimum value.For this function, a ϭ , b ϭ , and c ϭ .Since a 0, the graph opens and thefunction has a .b. State the maximum or minimum value of the function.The maximum value of this function is the y-coordinate ofthe vertex.The x-coordinate of the vertex is or .Find the y-coordinate of the vertex by evaluating thefunction for x ϭ 1.ƒ(x) ϭ Ϫx2ϩ 2x ϩ 3 Original functionƒ(1) ϭ or x ϭ 1The maximum value of the function is .c. State the domain and range of the function.The domain is all real numbers. The range is all reals lessthan or equal to the value. That is.Maximum and MinimumValue The graph ofƒ(x) ϭ ax2ϩ bx ϩ c,where a 0,• opens up and has aminimum value whena Ͼ 0, and• opens down and has amaximum value whena Ͻ 0.On the pagefor Lesson 5-1, sketch aparabola. Then label theaxis of symmetry, vertex,maximum value, andminimum value.KEY CONCEPTS
    • Check Your Progress Consider the functionƒ(x) ϭ x2ϩ 4x Ϫ 1.a. Determine whether the function has a maximum or aminimum value.b. State the maximum or minimum value of the function.c. State the domain and range of the function.ECONOMICS A souvenir shop sells about 200 coffeemugs each month for $6 each. The shop ownerestimates that for each $0.50 increase in the price, hewill sell about 10 fewer coffee mugs per month.a. How much should the owner charge for each mugin order to maximize the monthly income from theirsales?Words The income is the number of mugs multiplied bythe cost per mug.Variables Let x = the number of $0.50 price increases.Then 6 ϩ 0.50x ϭ the price per mug and200 Ϫ 10x ϭ the number of mugs sold.Let I(x) ϭ income as a function of x.The income is the number of mugs multiplied by theper mug.EquationI(x) ϭ (200 Ϫ 10x) и (6 ϩ 0.50x)ϭ 200(6) ϩ 200(0.50x) Ϫ 10x(6) Ϫ 10x(0.50x)ϭ 1200 ϩ 100x Ϫ 60x Ϫ 5x2Multiply.ϭ ϩ Ϫ Simplify.ϭ Ϫ5x2ϩ 40x ϩ 1200 Rewrite inax2ϩ bx ϩ c form.5–1126 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Glencoe Algebra 2 127Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.I(x) is a quadratic function with a ϭ Ϫ5, b ϭ 40, andc ϭ 1200. Since a Ͻ 0, the function has a maximum value atthe of the graph. Use the formula to find thex-coordinate of the vertex.x-coordinate of the vertex ϭ Ϫ b_2aFormula for thex-coordinate of thevertex.ϭ Ϫ 40_2(Ϫ5)a ϭ , b ϭϭ Simplify.This means the souvenir shop should make 4 price increasesof $0.50 to maximize their income. Thus, the price of a mugshould be or .b. What is the maximum monthly income the owner canexpect to make from the mugs?To determine maximum income, find the maximum value ofthe function by evaluating I(x) for x ϭ 4.I(x) ϭ Ϫ5x2ϩ 40x ϩ 1200 Income functionI΂ ΃ ϭ Ϫ5΂ ΃2ϩ 40΂ ΃ ϩ 1200 x ϭϭ Use a calculator.Thus, the maximum income the souvenir shop can expectis .Check Your Progress ECONOMICS A sports teamsells about 100 coupon books for $30 each during theirannual fund-raiser. They estimate that for each $0.50decrease in the price, they will sell about 10 morecoupon books.a. How much should they charge for each book in order tomaximize the income from their sales?b. What is the maximum monthly income the team can expectto make from these items?Page(s):Exercises:HOMEWORKASSIGNMENT5–1
    • 128 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.A quadratic equation can be written in the formax2ϩ bx ϩ c ϭ 0, where a 0.The of a quadratic equation are called theroots of the equation. One method for finding the roots ofa quadratic is to find the zeros of the relatedquadratic .5–2• Solve quadraticequations by graphing.• Estimate solutions ofquadratic equations bygraphing.MAIN IDEASSolving Quadratic Equations by GraphingTwo Real SolutionsSolve x2Ϫ 3x Ϫ 4 ϭ 0 by graphing.Graph the related quadratic functionƒ(x) ϭ x2Ϫ 3x Ϫ 4. The equation ofthe axis of symmetry is x ϭ Ϫ Ϫ3_2(1)or 3_2. Make a table using x-valuesaround 3_2. Then graph each point.x Ϫ1 0 1 2 3 4ƒ(x)From the table and the graph, we can see that the zeros of thefunction are Ϫ1 and 4. The solutions of the equation areand .Check Your Progress Solve x2ϩ 2x Ϫ 3 ϭ 0 by graphing.xf(x)Of(x) ϭ x2 Ϫ 3x Ϫ 4BUILD YOUR VOCABULARY (pages 121–122)TEKS 2A.8The studentformulatesequations and inequalitiesbased on quadraticfunctions, uses a varietyof methods to solvethem, and analyzes thesolutions in terms of thesituation. (C) Compareand translate betweenalgebraic and graphicalsolutions of quadraticequations. (D) Solvequadratic equationsand inequalities usinggraphs, tables, andalgebraic methods. Alsoaddresses TEKS 2A.6(B)and 2A.8(A).
    • 5–2Glencoe Algebra 2 129Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.One Real SolutionSolve x2Ϫ 4x ϭ Ϫ4 by graphing.Write the equation in ax2ϩ bx ϩ c ϭ 0 form.x2Ϫ 4x ϭ Ϫ4 ϭ 0 Add 4 to each side.Graph the related quadratic functionƒ(x) ϭ x2Ϫ 4x ϩ 4.Notice that the graph has onlyone x-intercept, 2.Check Your Progress Solve x2Ϫ 6x ϭ Ϫ9 by graphing.No Real SolutionNUMBER THEORY Find two real numbers whose sum is4 and whose product is 5 or show that no such numbersexist.Let x ϭ one of the numbers. Then 4 Ϫ x ϭ the other number.Since the product is 5, you know that x(4 Ϫ x) ϭ 5 orϪx2ϩ 4x Ϫ 5 ϭ 0.You can solve x2Ϫ 4x ϩ 5 ϭ 0by graphing the related functionƒ(x) ϭ x2Ϫ 4x ϩ 5.Notice that the graph has no x-intercepts.This means that the original equation hasno real solution.Solutions of a QuadraticEquation A quadraticequation can have onereal solution, two realsolutions, or no realsolution.KEY CONCEPTx 0 1 2 3 4ƒ(x)x 0 1 2 3 4ƒ(x)xf(x)Of(x) ϭ x2 Ϫ 4x ϩ 5xf(x)Of(x) ϭ x2 Ϫ 4x ϩ 4On the page for Lesson5-2, sketch a graph of aparabola that has onereal solution, two realsolutions, and no realsolution.ORGANIZE IT
    • 5–2130 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Check Your Progress Find two real numbers whosesum is 7 and whose product is 14 or show that no suchnumbers exist.Estimate RootsSolve x2Ϫ 6x ϩ 3 ϭ 0 by graphing. If exact roots cannotbe found, state the consecutive integers between whichthe roots are located.The equation of the axis of symmetryxf(x)Of(x) ϭ x2 Ϫ 6x ϩ 3of the related function isx ϭ .The x-intercepts of the graph are between andbetween .Check Your Progress Solve x2Ϫ 4x ϩ 2 ϭ 0 by graphing.If exact roots cannot be found, state the consecutive integersbetween which the roots are located.x 0 1 2 3 4 5 6ƒ(x)
    • 5–2Page(s):Exercises:HOMEWORKASSIGNMENTGlencoe Algebra 2 131Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.ROYAL GORGE BRIDGE The highest bridge in the U.S.is the Royal Gorge Bridge in Colorado. The deck is 1053feet above the river. Suppose a marble is dropped overthe railing from a height of 3 feet above the bridge deck.How long will it take the marble to reach the surface ofthe water, assuming there is no air resistance? Use theformula h(t) ϭ Ϫ16t2ϩ h0, where t is the time in secondsand h0 is the initial height above the water in feet.We need to find t when h0 ϭ and h(t) ϭ . Solve0 ϭ Ϫ16t2ϩ 1056.Graph the related function y ϭ Ϫ16t2ϩ 1056 on a graphingcalculator.Use the ZERO feature, 2nd [CALC], to find the positive zeroof the function, since time cannot be . Use thearrow keys to locate a left bound and press ENTER .Then, locate a right bound and press ENTER twice.The positive zero of the functionis approximately .The marble would fallfor about seconds.Check Your Progress One of the larger dams in theUnited States is the Hoover Dam on the Colorado River, whichwas built during the Great Depression. The dam is 726.4feet tall. Suppose a marble is dropped over the railing from aheight of 6 feet above the top of the dam. How long will it takethe marble to reach the surface of the water, assuming there isno air resistance? Use the formula from Example 5.
    • 5–3132 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.• Write quadraticequations in interceptform.• Solve quadraticequations by factoringMAIN IDEASSolving Quadratic Equations by FactoringWrite an Equation Given RootsWrite a quadratic equation with Ϫ2_3and 6 as itsroots. Write the equation in the form ax2ϩ bx ϩ c ϭ 0,where a, b, and c are integers.(x Ϫ p)(x Ϫ q) ϭ 0 Write the pattern.΄x Ϫ (Ϫ2_3)΅(x Ϫ 6) ϭ 0 Replace p with Ϫ2_3and q with 6.ϭ 0 Simplify.ϭ 0 Use FOIL.ϭ 0 Multiply each side by 3 so thatb is an integer.Check Your Progress Write a quadratic equation with Ϫ3_4and 5 as its roots. Write the equation in the formax2ϩ bx ϩ c = 0, where a, b, and c are integers.Two or Three TermsFactor each polynomial.a. 2y2Ϫ 3y Ϫ 5The coefficient of the y terms must be and since2(Ϫ5) ϭ Ϫ10 and 2 ϩ (Ϫ5) ϭ Ϫ3. Rewrite the expressionusing Ϫ5y and 2y in place of Ϫ3y and factor by grouping.2y2Ϫ 3y Ϫ 5 ϭ 2y2Ϫ 5y ϩ 2y Ϫ 5 Replace Ϫ3y.ϭ (2y2Ϫ 5y) ϩ (2y Ϫ 5) AssociativePropertyϭ (2y Ϫ 5) ϩ (2y Ϫ 5) Factor.ϭ (2y Ϫ 5) DistributivePropertyTEKS 2A.2The studentunderstands theimportance of the skillsrequired to manipulatesymbols in order to solveproblems and uses thenecessary algebraicskills required to simplifyalgebraic expressionsand solve equations andinequalities in problemsituations. (A) Use toolsincluding factoring andproperties of exponentsto simplify expressionsand to transform andsolve equations.TEKS 2A.6 Thestudent understands thatquadratic functions canbe represented in differentways and translatesamong their variousrepresentations. (C)Determine a quadraticfunction from its roots ora graph. Also addressesTEKS 2A.6(B), 2A.8(A),2A.8(C), and 2A.8(D).
    • 5–3Glencoe Algebra 2 133Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.b. 8x3Ϫ y38x3Ϫ y3ϭ3Ϫ y3This is thedifferenceof two cubes.ϭ [(2x)2ϩ (2x)y ϩ y2] Factor.ϭ (4x2ϩ 2xy ϩ y2) Simplify.Check Your Progress Factor each polynomial.a. 2x2ϩ x Ϫ 3b. 3x3Ϫ 12xc. a3b3Ϫ 27Two RootsSolve each equation by factoring.a. x2ϭ Ϫ4xx2ϭ Ϫ4x Original equationϭ 0 Add 4x to each side.ϭ 0 Factor the binomial.ϭ 0 or ϭ 0 Zero Product Propertyx ϭSolve the secondequation.The solution set is .Check Substitute and for x in the originalequation.x2ϭ Ϫ4x x2ϭ Ϫ4x2Ϫ42ϭ Ϫ4ϭ ✔ ϭ ✔Zero Product PropertyFor any real numbersa and b, if ab ϭ 0, theneither a ϭ 0, b ϭ 0, orboth a and b equal zero.KEY CONCEPTOn the page forLesson 5-3, writeyour own roots to aquadratic equation.Then find a quadraticequation with thoseroots. Check your resultby graphing the relatedfunction.ORGANIZE IT
    • 5–3Page(s):Exercises:HOMEWORKASSIGNMENT134 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.b. 3x2ϭ 5x ϩ 23x2ϭ 5x ϩ 2 Original equationϭ 0Subtract 5x and 2from each side.ϭ 0Factor thetrinomial.ϭ 0 or ϭ 0Zero ProductPropertyϭ x ϭSolve eachequation.x ϭThe solution set is .Check Your Progress Solve each equation byfactoring.a. x2ϭ 3x b. 6x2ϩ 11x ϭ Ϫ4Double RootSolve x2Ϫ 6x ϭ Ϫ9 by factoring.x2Ϫ 6x ϭ Ϫ9 Original equationϭ 0 Add 9 to each side.ϭ 0 Factor.ϭ 0 or ϭ 0 Zero Product Propertyx ϭ x ϭ Solve each equation.The solution set is .Check Your Progress Solve x2ϩ 10x ϭ Ϫ25 by factoring.
    • Glencoe Algebra 2 135Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.i is called the imaginary unit (i ϭ √Ϫ1 ).Pure imaginary numbers are square roots ofreal numbers.A complex number, such as the expression 5 ϩ 2i, is acomplex number since it has a real number (5) and apure imaginary number (2i).Two complex numbers of the form a ϩ bi andare called complex conjugates.5–4• Find square roots andperform operationswith pure imaginarynumbers.• Perform operationswith complex numbersMAIN IDEASComplex NumbersREMEMBER ITYou can write i tothe left or right of theradical symbol.However, i is usuallywritten to the left so itis clear that it is notunder the radical.BUILD YOUR VOCABULARY (pages 121–122)Properties of Square RootsSimplify.a. √18 b.10_81ϭ √9 и 2ϭ √9 и √2 ϭ√10_√81ϭϭCheck Your Progress Simplifya. √75 b. √ 7_36Square Roots of Negative NumbersSimplify √Ϫ32y3.√Ϫ32y3ϭ √Ϫ1 и 42и 2 и y2и yϭ √Ϫ1 и √42и √2 и √y2и √yϭ и 4 y √2 иϭTEKS 2A.2The studentunderstands theimportance of the skillsrequired to manipulatesymbols in order to solveproblems and uses thenecessary algebraicskills required to simplifyalgebraic expressionsand solve equations andinequalities in problemsituations. (B) Usecomplex numbers todescribe the solutions ofquadratic equations.
    • 5–4136 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Check Your Progress Simplify.a. √Ϫ32 b. √Ϫ50x5Multiply Pure Imaginary NumbersSimplify.a. ؊3i ؒ 2iϪ3i и 2i ϭ Ϫ6i2ϭ Ϫ6΂ ΃ i2ϭϭb. √؊12 ؒ √Ϫ2√Ϫ12 и √Ϫ2 ϭ иϭ i2√24ϭ orc. i35i35ϭ i и i34Multiplying powersϭ i и17Power of a Powerϭ i и17i2ϭϭ i и or (Ϫ1)17ϭCheck Your Progress Simplify.a. 3i и 5ib. √Ϫ2 и √Ϫ6c. i57
    • 5–4Glencoe Algebra 2 137Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Equation with Pure Imaginary SolutionsSolve 5y2ϩ 20 ϭ 0.5y2ϩ 20 ϭ 0 Original equation5y2ϭ Ϫ Subtract from each side.y2ϭ Ϫ4 Divide each side by .y ϭ Ϯ √Ϫ4 Square Root Propertyy ϭ √Ϫ4 ϭ √4 и √Ϫ1Check Your Progress Solve 2x2ϩ 50 ϭ 0.Equate Complex NumbersFind the values of x and y that make the equation2x ϩ yi ϭ Ϫ14 Ϫ 3i true.Set the real parts equal to each other and the imaginary partsequal to each other.2x ϭ Ϫ14 Real partsx ϭ Ϫ7 Divide each side by . y ϭ Imaginary partsx ϭ , y ϭCheck Your Progress Find the value of x and y thatmake the equation 3x Ϫ yi ϭ 15 ϩ 2i true.Add and Subtract Complex NumbersSimplify.a. (3 ϩ 5i) ϩ (2 Ϫ 4i)(3 ϩ 5i) ϩ (2 Ϫ 4i)ϭ (3 ϩ 2) ϩ (5 Ϫ 4)i Commutative and AssociativePropertiesϭ Simplify.Complex Numbers Acomplex number is anynumber that can bewritten in the forma ϩ bi, where a and bare real numbers and iis the imaginary unit. ais called the real part,and b is called theimaginary part.KEY CONCEPT
    • 5–4138 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.b. (4 Ϫ 6i) Ϫ (3 Ϫ 7i)(4 Ϫ 6i) Ϫ (3 Ϫ 7i)ϭ (4 Ϫ 3) ϩ [Ϫ6 Ϫ (Ϫ7)]i Commutative and AssociativePropertiesϭ Simplify.Check Your Progress Simplify.a. (2 ϩ 6i) ϩ (3 ϩ 4i) b. (3 ϩ 2i) Ϫ (Ϫ2 ϩ 5i)Multiply Complex NumbersELECTRICITY In an AC circuit, the voltage E, current I,and impedance Z are related to the formula E ϭ I и Z.Find the voltage in a circuit with current 1 ϩ 4 j ampsand impedance 3 Ϫ 6 j ohms.E ϭ I и Z Electricity formulaϭ (1 ϩ 4j) (3 Ϫ 6j) I ϭ 1 ϩ 4j, Z ϭ 3 Ϫ 6jϭ 1(3) ϩ 1(Ϫ6j) ϩ 4j(3) ϩ 4j(Ϫ6j) FOILϭ 3 Ϫ ϩ 12j Ϫ Multiplyϭ j2ϭ Ϫ1ϭ volts Add.Check Your Progress Refer to Example 5. Find thevoltage in a circuit with current 1 Ϫ 3j amps and impedance3 ϩ 2j ohms.
    • 5–4Page(s):Exercises:HOMEWORKASSIGNMENTDivide Complex NumbersSimplify.a. 5i_3 ϩ 2i5i_3 ϩ 2iϭ 5i_3 ϩ 2iи 3 Ϫ 2i and 3 ϩ 2i areconjugates.ϭ 15i ϩ 10i2_9 Ϫ 4i2Multiplyϭ15i ϩ 10i2ϭ Ϫ1ϭ ϩ Standard formb. 4 Ϫ i_5i4 Ϫ i_5iϭ 4 Ϫ i_5iи i_iϭ 4i Ϫ i2_Ϫ5i2Multiplyϭ4i ϩ 1i2ϭ Ϫ1ϭ Standard formCheck Your Progress Simplify.a. 3i_1 ϩ ib. 3 ϩ 2i_2iGlencoe Algebra 2 139Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.On the page for Lesson5-4, write your ownexample of complexconjugates. Then explainwhy the product ofcomplex conjugates isalways a real number.ORGANIZE IT
    • 140 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Equation with Rational RootsSolve x2ϩ 14x ϩ 49 ϭ 64 by using the Square RootProperty.x2ϩ 14x ϩ 49 ϭ 64 Original equationϭ 64 Factor the trinomial.ϭ Ϯ Square Root Propertyϭ √64 ϭ 8x ϭ Ϫ7 Ϯ 8 Subtract 7 from each side.x ϭ Ϫ7 ϩ 8 or x ϭ Ϫ7 Ϫ 8 Write as two equations.x ϭ x ϭ Solve each equation.Equation with Irrational RootsSolve x2Ϫ 10x ϩ 25 ϭ 12 by using the Square RootProperty.x2Ϫ 10x ϩ 25 ϭ 12 Original equation(x Ϫ 5)2ϭ 12 Factor the trinomial.x Ϫ 5 ϭ Square Root Propertyx ϭAdd 5 to each side.√12 ϭ 2√3x ϭ 5 ϩ or x ϭ 5 Ϫ Write as two equations.x Ϸ x Ϸ Use a calculator.Check Your Progress Solve by using the Square RootProperty.a. x2Ϫ 16x ϩ 64 ϭ 25 b. x2Ϫ 4x ϩ 4 ϭ 85–5• Solve quadraticequations by using theSquare Root Property.• Solve quadraticequations bycompleting the square.MAIN IDEASCompleting the SquareSquare Root PropertyFor any real number n, ifx2ϭ n, then x ϭ Ϯ√ n.KEY CONCEPTTEKS 2A.2The studentunderstands theimportance of the skillsrequired to manipulatesymbols in order to solveproblems and uses thenecessary algebraicskills required to simplifyalgebraic expressionsand solve equations andinequalities in problemsituations. (B) Usecomplex numbers todescribe the solutionsof quadratic equations.TEKS 2A.8 The studentformulates equations andinequalities based onquadratic functions, usesa variety of methods tosolve them, and analyzesthe solutions in terms ofthe situation. (D) Solvequadratic equations andinequalities using graphs,tables, and algebraicmethods. Also addressesTEKS 2A.6(A) and2A.8(A).
    • 5–5Glencoe Algebra 2 141Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc. Complete the SquareFind the value of c that makes x2ϩ 16x ϩ c a perfectsquare. Then write the trinomial as a perfect square.Step 1 Find one half of 16.Step 2 Square the result of Step 1.Step 3 Add the result of Step 2to x2ϩ 16x.The trinomial can be written as.Check Your Progress Find the value of c that makesx2ϩ 6x ϩ c a perfect square. Then write the trinomial as aperfect square.Solve an Equation by Completing the SquareSolve x2ϩ 4x Ϫ 12 ϭ 0 by completing the square.x2ϩ 4x Ϫ 12 ϭ 0 Notice that x2ϩ 4x Ϫ 12 is not aperfect square.x2ϩ 4x ϭ Rewrite so the left side is of theform x 2ϩ bx.x2ϩ 4x ϩ 4 ϭ 12 ϩ 4 Add to each side.ϭ 16 Write the left side as a perfectsquare by factoring.x ϩ 2 ϭ Square Root Propertyx ϭ Subtract 2 from each side.x ϭ or x ϭ Write as two equations.x ϭ x ϭ Solve each equation.Completing the SquareTo complete the squarefor any quadraticexpression of the formx2ϩ bx, follow the stepsbelow.Step 1 Find one half ofb, the coefficientof x.Step 2 Square the resultin Step 1.Step 3 Add the result ofStep 2 to x2ϩ bx.KEY CONCEPTREMEMBER ITBe sure to add thesame constant to bothsides of the equationwhen solving equationsby completing thesquare.
    • 5–5142 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Check Your Progress Solve x2ϩ 5x Ϫ 6 ϭ 0 bycompleting the square.Equation with a 1Solve 3x2Ϫ 2x Ϫ 1 ϭ 0 by completing the square.3x2Ϫ 2x Ϫ 1 ϭ 0 Notice that 3x2Ϫ 2x Ϫ 1 isnot a perfect square.x2Ϫ 2_3x Ϫ 1_3ϭ 0 Divide by the coefficient ofthe quadratic term, 3.x2Ϫ 2_3x ϭ 1_3Add 1_3to each side.x2Ϫ 2_3x ϩ ϭ 1_3ϩ Since (Ϫ2_3и 1_2)2ϭ ,add to each side.ϭWrite the left side as aperfect square by factoring.Simplify the right side.ϭ Ϯ Square Root Propertyx ϭ Ϯ Add to each side.x ϭ or x ϭ Write as two equations.x ϭ x ϭ Solve each equation.The solution set is .Check Your Progress Solve 2x2ϩ 11x ϩ 15 ϭ 0 bycompleting the square.On the page for Lesson5-5, list the steps youwould use to solvew2Ϫ 8w Ϫ 9 ϭ 0 bycompleting the square.ORGANIZE IT
    • Glencoe Algebra 2 143Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Page(s):Exercises:HOMEWORKASSIGNMENTEquation with Complex SolutionsSolve x2ϩ 2x ϩ 3 ϭ 0 by completing the square.x2ϩ 2x ϩ 3 ϭ 0 Notice that x2ϩ 2x ϩ 3 ϭ 0 is not aperfect square.ϭ Ϫ3 Rewrite so the left side is of the formx2ϩ bx.x2ϩ 2x ϩ 1 ϭ Ϫ3 ϩ 1 Since (2_2)2, add 1 to each side.2ϭ Ϫ2 Write the left side as a perfect squareby factoring.ϭ Ϯ√Ϫ2 Square Root Propertyϭ Ϯi√2 √Ϫ1 ϭ ix ϭ Subtract 1 from each side.The solution set is . Noticethat these are imaginary solutions.Check A graph of the related function shows that theequation has no real solutions since the graph had no.Check Your Progress Solve x2ϩ 4x ϩ 5 ϭ 0 by completingthe square.5–5
    • 144 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.5–6• Solve quadraticequations by using theQuadratic Formula.• Use the discriminant todetermine the numberand types of roots of aquadratic equation.MAIN IDEASThe Quadratic Formula and the DiscriminantTwo Rational RootsSolve x2Ϫ 8x ϭ 33 by using the Quadratic Formula.First, write the equation in the form ax2ϩ bx ϩ c ϭ 0 andidentify a, b, and c.ax2ϩ x ϩ x ϭ 0x2Ϫ 8x ϭ 33 Ϫ Ϫ ϭ 0Then, substitute these values into the Quadratic Formula.x ϭϪb Ϯ √b2Ϫ 4ac__2aQuadratic Formulax ϭϪ(Ϫ8) Ϯ √(Ϫ8)2Ϫ 4(1)(Ϫ33)___2(1)Replace a with 1, b withϪ8, and c with Ϫ33.x ϭ2ϩ8 ϮSimplify.x ϭ28 ϮSimplify.x ϭ28 Ϯϭx ϭ2or x ϭ2Write as two equations.ϭ ϭ Simplify.Check Your Progress Solve x2ϩ 13x ϭ 30 by using theQuadratic Formula.Quadratic FormulaThe solutions of aquadratic equationof the formax2ϩ bx ϩ c ϭ 0, wherea 0, are given by thefollowing formula.x ϭϪb Ϯ √b2Ϫ 4ac__2aKEY CONCEPTTEKS 2A.8The studentformulatesequations and inequalitiesbased on quadraticfunctions, uses a varietyof methods to solve them,and analyzes the solutionsin terms of the situation.(A) Analyze situationsinvolving quadraticfunctions and formulatequadratic equationsor inequalities to solveproblems. (B) Analyzeand interpret thesolutions of quadraticequations usingdiscriminants and solvequadratic equationsusing the quadraticformula. (D) Solvequadratic equations andinequalities using graphs,tables, and algebraicmethods. Also addressesTEKS 2A.1(A), 2A.2(B),2A.6(A), and 2A.6(B).
    • Glencoe Algebra 2 145Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.5–6One Rational RootSolve x2Ϫ 34x ϩ 289 ϭ 0 by using the Quadratic Formula.Identify a, b, and c. Then, substitute these values into theQuadratic Formula.x ϭϪb Ϯ √b2Ϫ 4ac__2aQuadratic Formulax ϭ2(1)Ϫ(Ϫ34) ϮReplace a with 1,b with Ϫ34, andc with 289.x ϭ34 Ϯ √0_2Simplify.x ϭ 34_2or √0 ϭ 0Check Your Progress Solve x2Ϫ 22x ϩ 121 ϭ 0 by usingthe Quadratic FormulaIrrational RootsSolve x2Ϫ 6x ϩ 2 ϭ 0 by using the Quadratic Formula.x ϭϪb Ϯ √b2Ϫ 4ac__2aQuadratic Formulax ϭϪ(Ϫ6) Ϯ √(Ϫ6)2Ϫ 4(1)(2)___2(1)Replace a with 1, b withϪ6, and c with 2.x ϭ26 Ϯ Simplify.x ϭ2or2√28 ϭ √4 и 7 or 2√7The exact solutions are and .The approximate solutions are and .On the page forLesson 5-5, explainwhy factoring cannotbe used to solve thequadratic equation inExample 3.ORGANIZE ITREMEMBER ITThe QuadraticFormula can be usedto solve any quadraticequation.
    • 5–6146 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Check Your Progress Solve x2Ϫ 5x ϩ 3 ϭ 0 by using theQuadratic Formula.Complex RootsSolve x2ϩ 13 ϭ 6x by using the Quadratic Formula.First, write the equation in the form ax2ϩ bx ϩ c ϭ 0 andidentify a, b, and c.x2ϩ 13 ϭ 6x 1x2Ϫ 6x ϩ 13 ϭ 0x ϭϪb Ϯ √b2Ϫ 4ac__2aQuadratic FormulaϭϪ(Ϫ6) Ϯ √(Ϫ6)2Ϫ 4(1)(13)___2(1)Replace a with 1, b withϪ6, and c with 13.ϭϮ √__ Simplify.ϭ √Ϫ16 ϭ √16(Ϫ1) or 4iϭ Simplify.The solutions are the complex numbers and.Check Your Progress Solve x2ϩ 5 ϭ 4x by using theQuadratic Formula.
    • 5–6Glencoe Algebra 2 147Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Page(s):Exercises:HOMEWORKASSIGNMENTDescribe RootsFind the value of the discriminant for each quadraticequation. Then describe the number and type of rootsfor the equation.a. x2ϩ 3x ϩ 5 ϭ 0a ϭ , b ϭ , c ϭb2Ϫ 4ac ϭϭϭThe discriminant is a number, so there.b. x2Ϫ 11x ϩ 10 ϭ 0a ϭ , b ϭ , c ϭb2Ϫ 4ac ϭϭϭThe discriminant is a number, so there.Check Your Progress Find the value of thediscriminant for each quadratic equation. Thendescribe the number and type of roots for the equation.a. x2ϩ 8x ϩ 16 ϭ 0 b. x2ϩ 2x ϩ 7 ϭ 0c. x2ϩ 3x ϩ 1 ϭ 0 d. x2ϩ 4x Ϫ 12 ϭ 0DiscriminantConsider ax2ϩ bx ϩ c ϭ 0.• If b2Ϫ 4ac Ͼ 0 andb2Ϫ 4ac is a perfectsquare, then there aretwo real, rational roots.• If b2Ϫ 4ac Ͼ 0 andb2Ϫ 4ac is not aperfect square, thenthere are two real,irrational roots.• If b2Ϫ 4ac ϭ 0, thenthere is one real,rational root.• If b2Ϫ 4ac Ͻ 0, thenthere are two complexroots.KEY CONCEPT
    • 148 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.A function written in the form, y ϭ (x Ϫ h)2ϩ k, where(h, k) is the of the parabola and x ϭ h is its, is referred to as the vertex form.5–7• Analyze quadraticfunctions of the formy ϭ a(x Ϫ h)2ϩ k.• Write a quadraticfunction in the formy ϭ a(x Ϫ h)2ϩ k.MAIN IDEASAnalyzing Graphs of Quadratic FunctionsBUILD YOUR VOCABULARY (pages 121–122)Graph a Quadratic Function in Vertex FormAnalyze y ϭ (x Ϫ 3)2ϩ 2. Then draw its graph.The vertex is at (h, k) or and the axis of symmetry isx ϭ . The graph has the same shape as the graph ofy ϭ x2, but is translated 3 units right and 2 units up.Now use this information to draw the graph.Step 1 Plot the vertex, .Step 2 Draw the axis of symmetry,.Step 3 Find and plot two pointson one side of the axis ofsymmetry, such as (2, 3)and (1, 6).Step 4 Use symmetry to complete the graph.Check Your Progress Analyze y ϭ (x ϩ 2)2Ϫ 4. Thendraw its graph.xf(x)O(1, 6)(1, 6)(5, 6)(5, 6)(2, 3)(2, 3) (4, 3)(4, 3)(3, 2)(3, 2)f(x) ϭ (x Ϫ 3)2 ϩ 2On the page forLesson 5-7, sketch agraph of a parabola.Then sketch the graphof the parabola after avertical translation anda horizontal translation.ORGANIZE ITTEKS 2A.4 Thestudent connectsalgebraic andgeometric representationsof functions. (A) Identifyand sketch graphsof parent functions,including linear (f(x) ϭ x),quadratic (f(x) ϭ x2),exponential (f(x) ϭ ax),and logarithmic (f(x)ϭ loga x) functions,absolute value of x (f(x) ϭ|x|), square root of x (f(x)ϭ √x), and reciprocal ofx (f(x) ϭ 1_x). (B) Extendparent functions withparameters such as ain f(x) ϭ a_xand describethe effects of theparameter changeson the graph of parentfunctions.
    • 5–7Glencoe Algebra 2 149Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Vertex Form ParametersTEST EXAMPLE Which function has the widest graph?A y ϭ Ϫ12x2C y ϭ 1.2x2B y ϭ 12x2D y ϭ 0.12x2Read the Test ItemYou are given four answer choices, each of which is in vertexform.Solve the Test ItemThe value of a determines the width of the graph. SinceϪ12 ϭ and 1.2 Ͼ 1, choices A, B, and Cproduce graphs that are than y ϭ x2. Since0.12 Ͻ 1, choice D produces a graph that is thany ϭ x2. The answer is .Check Your Progress Analyze y ϭ (x ϩ 2)2Ϫ 4. Whichgraph is the graph of y ϭ (x ϩ 2)2Ϫ 4?A CB DWrite Equations in Vertex FormWrite each equation in vertex form. Then analyze thefunction.a. y ϭ x2ϩ 2x ϩ 4y ϭ x2ϩ 2x ϩ 4 Notice that x2ϩ 2x ϩ 4 isnot a perfect square.y ϭ Complete the square. Add(2_2)2or 1. Balance thisaddition by subtracting 1.y ϭWrite x2ϩ 2x ϩ 1 as aperfect square.Consider the vertex formof a quadratic function,y ϭ a(x Ϫ h)2ϩ k.Describe what happensto the graph as | a |increases.WRITE ITTEKS 2A.7 Thestudent interpretsand describes theeffects of changes in theparameters of quadraticfunctions in applied andmathematical situations.(A) Use characteristicsof the quadratic parentfunction to sketch therelated graphs andconnect between they ϭ ax2 ϩ bx ϩ c andthe y ϭ a(x Ϫ h)2 ϩ ksymbolicrepresentations ofquadratic functions. (B)Use the parent functionto investigate, describe,and predict the effectsof changes in a, h, and kon the graphs ofy ϭ a(x Ϫ h)2 ϭ k formof a function in appliedand purely mathematicalsituations. Alsoaddresses TEKS 2A.6(C)and 2A.8(A).
    • 5–7150 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.This function can be rewritten as y ϭ ϩ 3.So, h ϭ and k ϭ . The vertex is atand the axis of symmetry is x ϭ . Since a ϭ 1, thegraph opens and has the same shape as y ϭ x2butis translated unit left and units up.b. y ϭ Ϫ2x2Ϫ 4x ϩ 2y ϭ Ϫ2x2Ϫ 4x ϩ 2 Original equationy ϭ Ϫ2 ϩ 2 Group ax2Ϫ bx andfactor, dividing by a.y ϭ Ϫ2(x ϩ 2x ϩ 1) ϩ 2 Ϫ (Ϫ2)(1) Complete the squareby adding 1 inside theparentheses. Notice thatthis is an overall additionof Ϫ2(1). Balance thisaddition by subtractingϪ2(1).y ϭ Write x2ϩ 2x ϩ 1 as aperfect square.The vertex is at (Ϫ1, 4), and the axis of symmetry isx ϭ . Since a ϭ Ϫ2, the graph opensand is narrower than the graph of y ϭ x2. It is alsotranslated unit left and units up.Check Your Progress Write each function in vertexform. Then analyze the function.a. y ϭ x2ϩ 6x ϩ 5b. y ϭ Ϫ3x2Ϫ 6x ϩ 4
    • 5–7Glencoe Algebra 2 151Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Page(s):Exercises:HOMEWORKASSIGNMENTWrite an Equation Given a GraphWrite an equation for the parabola shown in the graph.The vertex of the parabola is at (1, 2), so h ϭ andk ϭ . Since (3, 4) is a point on the graph of the parabola,let x ϭ and y ϭ . Substitute these values into thevertex form of the equation and solve for .y ϭ a(x Ϫ h)2ϩ k Vertex form4 ϭ a(3 Ϫ 1)2ϩ 2 Substitute for y, for x,for h, and for k.4 ϭ Simplify.2 ϭ 4a Subtract 2 from each side.ϭ a Divide each side by 4.The equation of the parabola in vertex form is.Check Your Progress Write an equation for the parabolawhose vertex is at (2, 3) and passes through (Ϫ2, 1).
    • 152 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.You can graph quadratic inequalities in two variablesusing the same techniques you used to graphinequalities in two variables.5–8• Graph quadraticinequalities in twovariables.• Solve quadraticinequalities in onevariable.MAIN IDEASGraphing and Solving Quadratic InequalitiesGraph a Quadratic InequalityGraph y Ͼ x2Ϫ 3x ϩ 2.Step 1 Graph the related quadraticfunction, y ϭ x2Ϫ 3x ϩ 2.Since the inequality symbolis Ͼ, the parabola shouldbe dashed.Step 2 Test a point inside theparabola, such as (1, 2).y Ͼ x2Ϫ 3x ϩ 22 Ͼ2 Ͼ2 Ͼ ✔So, (1, 2) is a solution of the inequality.Step 3 Shade the region inside the parabola.Check Your Progress Graph y Ͻ Ϫx2ϩ 4x ϩ 2.BUILD YOUR VOCABULARY (pages 121–122)xyOy ϭ x2 Ϫ 3x ϩ 2TEKS 2A.6The studentunderstandsthat quadratic functionscan be representedin different ways andtranslates among theirvarious representations.(A) Determine thereasonable domain andrange values of quadraticfunctions, as well asinterpret and determinethe reasonableness ofsolutions to quadraticequations andinequalities.TEKS 2A.8 The studentformulates equations andinequalities based onquadratic functions, usesa variety of methods tosolve them, and analyzesthe solutions in terms ofthe situation. (A) Analyzesituations involvingquadratic functions andformulate quadraticequations or inequalitiesto solve problems.(D) Solve quadraticequations and inequalitiesusing graphs, tables, andalgebraic methods. Alsoaddresses TEKS 2A.2(A)
    • 5–8Glencoe Algebra 2 153Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Solve ax2ϩ bx ϩ c Ͼ 0Solve x2Ϫ 4x ϩ 3 Ͼ 0 by graphing.The solution consists of the x values for which the graphof the related quadratic function lies above the x-axis.Begin by finding the roots of the related equation.x2Ϫ 4x ϩ 3 ϭ 0 Related equationϭ 0 Factor.ϭ 0 or ϭ 0 Zero Product Propertyx ϭ x ϭ Solve each equation.Sketch the graph of the parabolathat has x-intercepts at 3 and 1.The graph lies above the x-axis to theleft of x ϭ 1 and to the right of x ϭ 3.The solution set is .Check Your Progress Solve each inequality bygraphing.a. x2ϩ 5x ϩ 6 Ͼ 0b. x2ϩ 6x ϩ 2 Յ 0Explain how you cancheck the solution set toa quadratic inequality.WRITE ITxyOy ϭ x2 Ϫ 4x ϩ 3On the page forLesson 5-8, use yourown words to describethe three steps forgraphing quadraticinequalities.ORGANIZE IT
    • 5–8154 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Page(s):Exercises:HOMEWORKASSIGNMENTSolve a Quadratic InequalitySolve x2ϩ x Յ 2 algebraically.First, solve the related equation x2ϩ x ϭ 2.x2ϩ x ϭ 2 Related quadratic equationϭ 0 Subtract 2 from each side.ϭ 0 Factor.ϭ 0 or ϭ 0 Zero Product Propertyx ϭ x ϭ Solve each equation.Plot the values on a number line. Use closed circles since thesesolutions are included. Note that the number line is separatedinto 3 intervals.4–3 –2 –1 0 1 2 3 5–5 –4x ≤ Ϫ2 x ≥ 1Ϫ2 ≤ x ≤ 1Test a value in each interval to see if it satisfies the originalinequality.x Յ Ϫ2 Ϫ2 Յ x Յ 1 x Ն 1Test x ϭ Ϫ3. Test x ϭ 0. Test x ϭ 2.x2ϩ x Յ 2 x2ϩ x Յ 2 x2ϩ x Յ 2(Ϫ3)2Ϫ 3 Յ?2 02ϩ 0 Յ?2 22ϩ 2 Յ?26 Յ 2 ϫ 0 Յ 2 ✔ 6 Յ 2 ϫThe solution set is .692 91 90 - 0 1 2 898 96Check Your Progress Solve x2ϩ 5x Ͻ Ϫ6 algebraically.
    • Glencoe Algebra 2 155Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.BRINGING IT ALL TOGETHERBUILD YOURVOCABULARYUse your Chapter 5 Foldable tohelp you study for your chaptertest.To make a crossword puzzle,word search, or jumblepuzzle of the vocabulary wordsin Chapter 5, go to:glencoe.comYou can use your completedVocabulary Builder(pages 121–122) to help yousolve the puzzle.VOCABULARYPUZZLEMAKERC H A P T E R5STUDY GUIDERefer to the graph at the right as you compe the followingsentences.1. The curve is called a .2. The line x ϭ Ϫ2 called the .3. The point (Ϫ2, 4) is called the .Determine whether each function has a maximum orminimum value. Then find the maximum or minimum valueof each function.4. ƒ(x) ϭ Ϫx2ϩ 2x ϩ 5 5. ƒ(x) ϭ 3x2Ϫ 4x Ϫ 2Solve each equation. If exact roots cannot be found, state theconsecutive integers between which the roots are located.6. x2Ϫ 2x ϭ 8 7. x2ϩ 5x Ϫ 7 ϭ 0xf(x)O(0, –1)(–2, 4)5-1Graphing Quadratic Functions5-2Solving Quadratic Equations by Graphing
    • Chapter BRINGING IT ALL TOGETHER5156 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.8. The solution of a quadratic equation by factoring is shownbelow. Give the reason for each step of the solution.x2Ϫ 10x ϭ Ϫ21 Original equationx2Ϫ 10x ϩ 21 ϭ 0(x Ϫ 3)(x Ϫ 7) ϭ 0x Ϫ 3 ϭ 0 or x Ϫ 7 ϭ 0x ϭ 3 x ϭ 7The solution set is .Write a quadratic equation with the given roots. Write theequation in the form ax2ϩ bx ϩ c ϭ 0, where a, b, and c areintegers.9. Ϫ4, Ϫ2 10. 3, 6Simplify.11. √Ϫ2 и √Ϫ10 12. (3 Ϫ 8i) Ϫ (5 ϩ 2i)13. (4 Ϫ i)(5 ϩ 2i) 14. 3 Ϫ i_2 ϩ i15. Solve 5x2ϩ 60 ϭ 0.5-3Solving Quadratic Equations by Factoring5-4Complex Numbers
    • Chapter BRINGING IT ALL TOGETHER5Glencoe Algebra 2 157Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.16. Solve x2ϩ 6x ϩ 9 ϭ 49 by using the Square Root Property.17. Solve x2Ϫ 2x ϩ 10 ϭ 5 by completing the square.18. When the dimensions of a cube are reduced by 2 inches oneach side, the surface area of the new cube is 486 squareinches. What were the dimensions of the original cube?The value of the discriminant for a quadratic equation withinteger coefficients is shown. Give the number and the typeof roots for the equation.Value ofDiscriminantNumber of Roots Type of Roots64Ϫ801523. Solve x2Ϫ 8x ϭ 2 by using the Quadratic Formula. Find exactsolutions.5-5Completing the Square5-6The Quadratic Formula and the Discriminant19.20.21.22.
    • Chapter BRINGING IT ALL TOGETHER5158 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.24. Match each graph with the description of the constants in theequation in vertex form.a. a Ͼ 0, h Ͼ 0, k Ͻ 0b. a Ͻ 0, h Ͻ 0, k Ͻ 0c. a Ͻ 0, h Ͻ 0, k Ͼ 0d. a Ͼ 0, h ϭ 0, k Ͻ 025. Solve 0 Ͻ x2Ϫ 6x ϩ 8 by graphing.Solve each inequality algebraically.26. x2Ϫ x Ͼ 2027. x2Ϫ 10x Ͻ Ϫ165-7Analyzing Graphs of Quadratic Functions5-8Graphing and Solving Quadratic Inequalitiesi.xyOii.xyOiii.xyOiv.xyO
    • Glencoe Algebra 2 159Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Check the one that applies. Suggestions to help you study aregiven with each item.C H A P T E R5ChecklistI completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.• You may want to take the Chapter 5 Practice Test on page 307of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 5 Study Guide and Reviewon pages 302–306 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 5 Practice Test onpage 307 of your textbook.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 5 Foldable.• Then complete the Chapter 5 Study Guide and Review onpages 302–306 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 5 Practice Test onpage 307 of your textbook.Student Signature Parent/Guardian SignatureTeacher SignatureVisit glencoe.com to accessyour textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 5.ARE YOU READY FORTHE CHAPTER TEST?
    • 160 Glencoe Algebra 2Use the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin this Interactive StudyNotebook to help you in taking notes.C H A P T E R6Begin with five sheets of notebook paper.Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Polynomial FunctionsStack sheets of paperwith edges 3_4-inch apart.Fold up the bottomedges to create equaltabs.Staple along the fold.Label the tabs withlesson numbers.NOTE-TAKING TIP: When you take notes, thinkabout the order in which the concepts are beingpresented. Write why you think the concepts werepresented in this sequence.
    • Glencoe Algebra 2 161Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Vocabulary TermFoundon PageDefinitionDescription orExampledegree of a polynomialdepressed polynomialdimensional analysisend behaviorleading coefficientLocation Principlepolynomial functionpolynomial in onevariableThis is an alphabetical list of new vocabulary terms you will learn in Chapter 6.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.C H A P T E R6BUILD YOUR VOCABULARY(continued on the next page)Chapter6
    • 162 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Vocabulary TermFoundon PageDefinitionDescription orExamplequadratic formrelative maximumrelative minimumscientific notationsimplifystandard notationsynthetic divisionsynthetic substitutionChapter BUILDING YOUR VOCABULARY6
    • • Use properties ofexponents to multiplyand divide monomials.• Use expressions writtenin scientific notation.MAIN IDEASGlencoe Algebra 2 1636–1Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.A monomial is an expression that is a number, a variable, orthe product of a number and one or more variables.The numerical factor of a monomial is the coefficient ofthe variable(s). The degree of a monomial is the sum of theexponents of its variables. A power is an expression of theform xn.Simplify Expressions with MultiplicationSimplify each expression.a. (Ϫ2a3b)(Ϫ5ab4)(Ϫ2a3b)(Ϫ5ab4)ϭ (Ϫ2 и a и a и a и b) Definition ofexponentsϭ Ϫ2(Ϫ5) и a и a и a и a и b и b и b и b и b CPropertyϭ 10a4b5Definition ofexponentsb. (3a5)(cϪ2)(Ϫ2aϪ4b3)(3a5)(cϪ2)(Ϫ2aϪ4b3)ϭ (3a5)(1_c2)(Ϫ2_aϪ4)(b3) Definitionof negativeexponentsϭ (3 и a и a и a и a и a)( 1_c и c)( Ϫ2_aa и a и a и a)(b и b и b)Definition ofexponentsϭ (3 и a/ и a/ и a/ и a/ и a)( 1_c и c)( Ϫ2_a/ и a/ и a/ и a/)(b и b и b)Cancel outcommon factors.ϭ Definition ofexponents andfractionsBUILD YOUR VOCABULARY (pages 161–162)Properties of ExponentsTEKS 2A.2The studentunderstands theimportance of the skillsrequired to manipulatesymbols in order to solveproblems and uses thenecessary algebraicskills required to simplifyalgebraic expressionsand solve equations andinequalities in problemsituations. (A) Use toolsincluding factoring andproperties of exponentsto simplify expressionsand to transform andsolve equations.
    • 6–1164 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Simplify Expressions with DivisionSimplify s2_s10. Assume that s 0.s2_s10ϭ Subtract exponents.ϭ or A simplified expression cannotcontain negative exponents.Simplify Expressions with PowersSimplify each expression.a. (Ϫ3c2d5)3b. Ϫ2a_b25(Ϫ3c2d5)3ϭ Ϫ2a_b25ϭ Ϫ25a5_(b2)5ϭ ϭCheck Your Progress Simplify each expression.a. (Ϫ3x2y)(5x3y5)b. x3_x7c. (x3)5d. (Ϫ2x2y3)5e.(Ϫ3x2_y3 )3f. (y_2)Ϫ3Write how you read theexpression a3. Then dothe same for z4.WRITE ITNegative ExponentsFor any real numbera 0 and any integer n,aϪnϭ 1_an and 1_aϪn ϭ an.Product of PowersFor any real number aand integers m and n,amи anϭ am ϩ n.Quotient of PowersFor any real numbera 0, and integersm and am_an ϭ am Ϫ n.Properties of PowersSuppose a and b are realnumbers and m and nare integers. Then thefollowing propertieshold.Power of a Power:(am)nϭ amnPower of a Product:(ab)mϭ ambmPower of a Quotient:(a_b)nϭ an_bn , b 0, and(a_b)Ϫnϭ (b_a)norbn_an , a 0, b 0KEY CONCEPTS
    • 6–1Page(s):Exercises:HOMEWORKASSIGNMENTGlencoe Algebra 2 165Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Simplify Expressions Using Several PropertiesSimplify (Ϫ3a5y_a6yb4 )5.Simplify the fraction before raising to the fifth power.(Ϫ3a5y_a6yb4 )5ϭ (Ϫ3a5y Ϫ 6y_b4 )5ϭ (Ϫ3aϪy_b4 )5ϭ orCheck Your Progress Simplify (Ϫ2a3n_a2nb5 )3.Divide Numbers in Scientific NotationBIOLOGY There are about 5 ϫ 106red blood cells inone milliliter of blood. A certain blood sample contains8.32 ϫ 106red blood cells. About how many millilitersof blood are in the sample?Divide the number of red blood cells in the sample by thenumber of red blood cells in 1 milliliter of blood.number of red blood cells in sample5 ϫ 106number of red blood cells in 1 milliliterϭ millilitersCheck Your Progress A petri dish contains 3.6 ϫ 105germs. A half hour later, there are 7.2 ϫ 107. How many timesas great is the amount a half hour later?On the page forLesson 6-1, write eachnew Vocabulary Builderword and its definition.Give an example ofeach word.ORGANIZE IT
    • 6–2166 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.A polynomial is a monomial or a sum of monomials.The polynomial x2ϩ 3x ϩ 1 is a trinomial because it hasthree unlike terms.A polynomial such as xy ϩ z3is a binomial because it hastwo unlike terms.• Add and subtractpolynomials.• Multiply polynomials.MAIN IDEASDegree of a PolynomialDetermine whether each expression is a polynomial. Ifit is a polynomial, state the degree of the polynomial.a. c4Ϫ 4√c ϩ 18This expression a polynomial because isnot a monomial.b. Ϫ16p5ϩ 3_4p2q7The expression a polynomial because each termis a monomial. The degree of the first term is , andthe degree of the second term is 2 ϩ 7 or 9. The degree ofthe polynomial is .Subtract and SimplifySimplify each expression.a. (2a3ϩ 5a Ϫ 7) Ϫ (a3Ϫ 3a ϩ 2).(2a3ϩ 5a Ϫ 7) Ϫ (a3Ϫ 3a ϩ 2)ϭ 2a3ϩ 5a Ϫ 7 Ϫ a3ϩ 3a Ϫ 2ϭ (2a3Ϫ a3) ϩ ϩϭb. (4x2Ϫ 9x ϩ 3) ϩ (Ϫ2x2Ϫ 5x Ϫ 6)(4x2Ϫ 9x ϩ 3) ϩ (Ϫ2x2Ϫ 5x Ϫ 6)ϭ 4x2Ϫ 9x ϩ 3 Ϫ 2x2Ϫ 5x Ϫ 6ϭ (4x2Ϫ 2x2) ϩ ϩϭ(pages 161–162)BUILD YOUR VOCABULARYOperations with PolynomialsTEKS 2A.2The studentunderstands theimportance of the skillsrequired to manipulatesymbols in order to solveproblems and uses thenecessary algebraicskills required to simplifyalgebraic expressionsand solve equations andinequalities in problemsituations. (A) Use toolsincluding factoring andproperties of exponentsto simplify expressionsand to transform andsolve equations.
    • 6–2Page(s):Exercises:HOMEWORKASSIGNMENTGlencoe Algebra 2 167Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Check Your Progress Determine whether eachexpression is a polynomial. If it is a polynomial, statethe degree of the polynomial.a. 1_2a2b3ϩ 3c5b. √c ϩ 2c. Simplify (3x2ϩ 2x Ϫ 3) Ϫ (4x2ϩ x Ϫ 5).Multiply and SimplifyFind Ϫy(4y2ϩ 2y Ϫ 3).Ϫy(4y2ϩ 2y Ϫ 3)ϭ Ϫy(4y2) Ϫ y(2y) Ϫ y(Ϫ3) Distributive Propertyϭ Multiply the monomials.Multiply PolynomialsFind (a2ϩ 3a Ϫ 4)(a ϩ 2).(a2ϩ 3a Ϫ 4)(a ϩ 2)ϭ a2(a ϩ 2) ϩ 3a(a ϩ 2) Ϫ 4(a ϩ 2) Distributive Propertyϭ a2и a ϩ a2и 2 ϩ ϩϪ Ϫ Distributive Propertyϭ a3ϩ 2a2ϩ 3a2ϩ 6a Ϫ 4a Ϫ 8 Multiply monomials.ϭ Combine like terms.Check Your Progress Find each product.a. Ϫx(3x3Ϫ 2x ϩ 5)b. (3p ϩ 2)(5p ϩ 1)c. (x2ϩ 3x Ϫ 2)(x ϩ 4)F0IL Method forMultiplying BinomialsThe product of twobinomials is the sumof the products of F thefirst terms, 0 the outerterms, I the inner terms,and L the last terms.On the pagefor Lesson 6-2, writeexamples of multiplyingtwo binomials.KEY CONCEPTREMEMBER ITThe prefix bi- meanstwo, so you canremember that abinomial has twounlike terms.The prefix tri- meansthree, so you canremember that atrinomial has threeunlike terms.
    • 6–3• Divide polynomialsusing long division.• Divide polynomialsusing synthetic division.MAIN IDEAS168 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Dividing PolynomialsIn your notes, explainhow to write theanswer to a longdivision problem thathas a quotient and aremainder.ORGANIZE ITDivide a Polynomial by a MonomialSimplify 5a2b Ϫ 15ab3ϩ 10a3b4___5ab.5a2b Ϫ 15ab3ϩ 10a3b4__5abϭ 5a2b_5abϪ 15ab3_5abϩ 10a3b4_5abSum ofquotientsϭ 5_5и a2 Ϫ 1b1 Ϫ 1Ϫ 15_5и a1 Ϫ 1b3 Ϫ 1ϩ 10_5и a3 Ϫ 1b4 Ϫ 1Divide.ϭ x1 Ϫ 1ϭ x0or 1Check Your Progress Simplify3x2y ϩ 6x5y2Ϫ 9x7y3__3x2y.Division AlgorithmUse long division to find (x2Ϫ 2x Ϫ 15) Ϭ (x Ϫ 5).x ϩ 3x Ϫ 5 ͤxෆ2ෆϪෆ 2ෆxෆෆϪ 1ෆ5ෆ(Ϫ) x(x Ϫ 5) ϭ x2Ϫ 5xϪ2x Ϫ (Ϫ5x) ϭ 3x(Ϫ) 3(x Ϫ 5) ϭ 3x Ϫ 15The quotient is .Check Your Progress Use long division to find(x2ϩ 5x ϩ 6) Ϭ (x ϩ 3).TEKS 2A.2The studentunderstands theimportance of the skillsrequired to manipulatesymbols in order to solveproblems and uses thenecessary algebraicskills required to simplifyalgebraic expressionsand solve equations andinequalities in problemsituations. (A) Use toolsincluding factoring andproperties of exponentsto simplify expressionsand to transform andsolve equations.
    • 6–3Glencoe Algebra 2 169Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Quotient with RemainderTEST EXAMPLE Which expression is equal to(a2Ϫ 5a ϩ 3)(2 Ϫ a)Ϫ1?A a ϩ 3 C Ϫa – 3 ϩ 3_2 Ϫ aB Ϫa ϩ 3 ϩ 3_2 Ϫ aD Ϫa ϩ 3 Ϫ 3_2 Ϫ aRead the Test ItemSince the second factor has an exponent of , this is aproblem.(a2Ϫ 5a ϩ 3)(2 Ϫ a)Ϫ1ϭ a2Ϫ 5a ϩ 3__2 Ϫ aSolve the Test ItemϪa ϩ 3Ϫa ϩ 2 ͤaෆ2ෆϪෆ 5ෆaෆෆϩ 3ෆ(Ϫ)a2Ϫ 2aϪ3a ϩ 3(Ϫ)Ϫ3a ϩ 6Ϫ3The quotient is , and the remainder is .Therefore, (a2Ϫ 5a ϩ 3)(2 Ϫ a)Ϫ1ϭ Ϫa ϩ 3 Ϫ 3_2 Ϫ a. Theanswer is .Synthetic DivisionUse synthetic division to find (x3Ϫ 4x2ϩ 6x Ϫ 4) Ϭ (x Ϫ 2).Step 1 With the terms ofthe dividend in descendingorder by degree, write justthe coefficients.Steps 2 & 3 Write theconstant of the divisor x Ϫ r,2, to the left. Bring downthe first coefficient, 1.Multiply the first coefficientby r, 1 ؒ 2 ϭ 2. Write theproduct under the next coefficient and add.2 1 Ϫ4 6 Ϫ421x3Ϫ 4x2ϩ 6x Ϫ 41 Ϫ4 6 Ϫ4
    • 6–3Page(s):Exercises:HOMEWORKASSIGNMENT170 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Step 4 Multiply this sumby r, 2. Write the productunder the next coefficientand add.Step 5 Multiply this sum by r,2. Write the product under thenext coefficient and add.The numbers along the bottomare the coefficients of the quotient.So, the quotient is x2Ϫ 2x ϩ 2.Divisor with First Coefficient Other than 1Use synthetic division to find (4y4Ϫ 5y2ϩ 2y ϩ 4) Ϭ (2y Ϫ 1).Use division to rewrite the divisor so it has a first coefficient of 1.4y4Ϫ 5y2ϩ 2y ϩ 4ᎏᎏᎏ2y Ϫ 1ϭ(4y4Ϫ 5y2ϩ 2y ϩ 4) Ϭ 2ᎏᎏᎏ(2y Ϫ 1) Ϭ 2ϭy Ϫ 1_2The numerator does not have a y3term. So use a coefficient of 0.1_22 0 Ϫ5_21 2 y Ϫ r ϭ y Ϫ 1_2, so1 1_2Ϫ1 0 r ϭ 1_2.2 1 Ϫ2 0 2The result is 2y3ϩ y2Ϫ 2y ϩ 2_y Ϫ 1_2. Now simplify the fraction.2_y Ϫ 1_2ϭ 2 Ϭ (y Ϫ 1_2)ϭ 2 Ϭ ϭ 2 и orThe solution is .Check Your Progressa. Use long division to find (x2ϩ 5x ϩ 6) Ϭ (x ϩ 3).b. Use synthetic division to find (16y4Ϫ 4y2ϩ 2y ϩ 8) Ϭ (2y ϩ 1).2 1 Ϫ4 6 Ϫ4212 1 Ϫ4 6 Ϫ421
    • Glencoe Algebra 2 1716–4Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Find Degree and Leading CoefficientsState the degree and leading coefficient of eachpolynomial in one variable. If it is not a polynomial inone variable, explain why.a. 7z3Ϫ 4z2ϩ zThis is a polynomial in one variable. The degree is andthe leading coefficient is .b. 6a3Ϫ 4a2ϩ ab2This a polynomial in one variable. It containstwo variables, and .Check Your Progress State the degree and leadingcoefficient of each polynomial in one variable. If it isnot a polynomial in one variable, explain why.a. 3x3ϩ 2x2Ϫ 3b. 3x2ϩ 2xy Ϫ 5• Evaluate polynomialfunctions.• Identify general shapesof graphs of polynomialfunctions.MAIN IDEAS The leading coefficient is the coefficient of the term withthe degree.A common type of function is a power function, which hasan equation in the form , where a and b arereal numbers.BUILD YOUR VOCABULARYPolynomial FunctionsA Polynomial in OneVariable A polynomial ofdegree n in one variablex is an expression of theforma0xn ϩ a1xnϪ1 ϩ … ϩanϪ2x2 ϩ anϪ1x ϩ an,where the coefficientsa0, a1, a2, …, an,represent real numbers,a0 is not zero, and nrepresents anonnegative integer.KEY CONCEPT(pages 161–162)TEKS 2A.1 The student uses properties and attributes of functions and appliesfunctions to problem situations. (A) Identify the mathematical domains andranges of functions and determine reasonable domain and range values forcontinuous and discrete situations.TEKS 2A.6The studentunderstandsthat quadratic functionscan be representedin different ways andtranslates among theirvarious representations.(A) Determine thereasonable domainand range values ofquadratic functions,as well as interpretand determine thereasonableness ofsolutions to quadraticequations andinequalities.
    • 6–4172 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Evaluate a Polynomial FunctionNATURE Refer to the application at the beginningof Lesson 6-4 in your textbook. A sketch of thearrangement of hexagons shows a fourth ring of 18hexagons, a fifth ring of 24 hexagons, and a sixth ring of30 hexagons.a. Show that the polynomial function f(r) ϭ 3r2Ϫ 3r ϩ 1gives the total number of hexagons when r ϭ 4, 5,and 6.f(4) ϭ 3(4)2Ϫ 3(4) ϩ 1, or ;f(5) ϭ 3(5)2Ϫ 3(5) ϩ 1, or ;f(6) ϭ 3(6)2Ϫ 3(6) ϩ 1, or ;The total number of hexagons for four rings is 19 ϩ 18, or, five rings is , or 61, and six rings is61 ϩ 30, or . These the functional valuesfor r ϭ 4, 5, and 6, respectively.b. Find the total number of hexagons in a honeycombwith 20 rings.ƒ(r) ϭ 3r2Ϫ 3r ϩ 1 Original functionƒ(20) ϭ Replace r with 20.ϭ or Simplify.Check Your Progress Refer to Example 2.a. What are the total number of hexagons when r ϭ 7, 8,and 9?b. Find the total number of hexagons in a honeycomb with30 rings.Definition of aPolynomial FunctionA polynomial functionof degree n can bedescribed by an equationof the form P(x) ϭa0xn ϩ a1xnϪ1 ϩ … ϩanϪ2x2 ϩ anϪ1x ϩ an,where the coefficientsa0, a1, a2, …, an,represent real numbers,a0 is not zero, and nrepresents a nonnegativeinteger.KEY CONCEPT
    • 6–4Glencoe Algebra 2 173Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Function Values of VariablesFind b(2x Ϫ 1) Ϫ 3b(x) if b(m) ϭ 2m2ϩ m Ϫ 1.To evaluate b(2x Ϫ 1), replace m in b(m) with .b(m) ϭ 2m2ϩ m Ϫ 1 Original functionb(2x Ϫ 1) ϭ 2(2x Ϫ 1)2ϩ (2x Ϫ 1) Ϫ 1 Replace m with 2x Ϫ 1.ϭ ϩ 2x Ϫ 1 Ϫ 1Evaluate 2(2x Ϫ 1)2.ϭ Simplify.To evaluate 3b(x), replace m with x in b(m), then multiply theexpression by .b(m) ϭ 2m2ϩ m Ϫ 1 Original function3b(x) ϭ 3(2x2ϩ x Ϫ 1) Replace m with x.ϭ Distributive PropertyNow evaluate b(2x Ϫ 1) Ϫ 3b(x).b(2x Ϫ 1) Ϫ 3b(x) ϭ ϪReplace b(2x Ϫ 1) and 3b(x)with evaluated expressions.ϭ 8x2Ϫ 6x Ϫ 6x2Ϫ 3x ϩ 3ϭ Simplify.Check Your Progress Find g(2x ϩ 1) Ϫ 2g(x) ifg(b) ϭ b2ϩ 3.Graphs of Polynomial FunctionsFor each graph, describe the end behavior, determinewhether it represents an odd-degree or an even-degreefunction, and state the number of real zeros.a.xOf(x) • ƒ(x) → as x → ϩϱ.• ƒ(x) → as x → Ϫϱ.• It is anpolynomial function.• The graph does not intersect thex-axis, so the function hasreal zeros.On the tab forLesson 6-4, sketch thegraph of a function thathas no real zeros, onereal zero, and two realzeros.ORGANIZE IT
    • b.xOf(x)• ƒ(x) → as x → ϩϱ.• ƒ(x) → as x → Ϫϱ.• It is anpolynomial function.• The graph intersects the x-axis atone point, so the function hasreal zero.Check Your Progress For each graph, describe theend behavior, determine whether it represents an odd-degree or an even-degree function, and state the numberof real zeros.a. f(x)xOb. f(x)xO6–4Page(s):Exercises:HOMEWORKASSIGNMENT174 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Glencoe Algebra 2 1756–5Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.On the tab forLesson 6-5, explainhow knowing the endbehavior of a graph willassist you in completingthe sketch of the graph.ORGANIZE ITGraph a Polynomial FunctionGraph ƒ(x) ϭ Ϫx3Ϫ 4x2ϩ 5 bymaking a table of values.This is an odd degree polynomial witha negative leading coefficient, soƒ(x) → as x → Ϫϱ andƒ(x) → as x → ϩϱ.The graph intersects the x-axisat points indicating thatthere are real zeros.Check Your Progress Graph ƒ(x) ϭ x3ϩ 2x2ϩ 1 by makinga table of values.• Graph polynomialfunctions and locatetheir real zeros.• Find the maxima andminima of polynomialfunctions.MAIN IDEAS x ƒ(x)Ϫ4Ϫ3Ϫ2Ϫ1012xOf(x) ϭ Ϫx3 Ϫ 4x2 ϩ 5f(x)Analyzing Graphs of Polynomial FunctionsTEKS 2A.1 The student uses properties and attributes of functions and appliesfunctions to problem situations. (A) Identify the mathematical domains andranges of functions and determine reasonable domain and range values forcontinuous and discrete situations.
    • 6–5176 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Locate Zeros of a FunctionDetermine consecutive values of x between whicheach real zero of the function ƒ(x) ϭ x4Ϫ x3Ϫ 4x2ϩ 1is located. Then draw the graph.Make a table of values. Since ƒ(x) is a 4th degree polynomialfunction, it will have between 0 and 4 zeros, inclusive. Lookat the value of ƒ(x) to locate the zeros. Then use the points tosketch the graph of the function.There are zeros between x ϭ and , x ϭand , x ϭ and , and x ϭ and .Check Your Progress Determine consecutivevalues of x between which each real zero of the functionƒ(x) ϭ x3Ϫ 4x2ϩ 2 is located. Then draw the graph.x ƒ(x)Ϫ2 9Ϫ1 Ϫ10 11 Ϫ32 Ϫ73 19xOf(x) ϭ x4 Ϫ x3 Ϫ 4x2 ϩ 1f(x)sign changesign changesign changesign changeLocation PrincipleSuppose y ϭ ƒ(x)represents a polynomialfunction and a and b aretwo numbers such thatƒ(a) Ͻ 0 and ƒ(b) Ͼ 0.Then the function hasat least one real zerobetween a and b.KEY CONCEPTA point on a graph is a relative maximum of a functionif no other nearby points have ay-coordinate. Likewise, a point is a relative minimum if noother nearby points have a y-coordinate.BUILD YOUR VOCABULARY (page 162)
    • 6–5Glencoe Algebra 2 177Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Maximum and Minimum PointsGraph ƒ(x) ϭ x3Ϫ 4x2ϩ 5.Estimate the x-coordinates atwhich the relative maximumand relative minimum occur.Make a table of values and graphthe function.The value of ƒ(x) at x ϭ is greater than the surroundingpoints, so it is a relative . The value of ƒ(x) atabout x ϭ is less than the surrounding points, so it is arelative .Check Your ProgressGraph ƒ(x) ϭ x3ϩ 3x2ϩ 2.Estimate thex-coordinates at whichthe relative maximumand relative minimumoccur.x ƒ(x)Ϫ2 Ϫ19Ϫ1 00 51 22 Ϫ33 Ϫ44 55 30zero at x ϭ Ϫ1indicates a relative maximumzero between x ϭ 1 and x ϭ 2indicates a relative minimumand zero between x ϭ 3 andx ϭ 4xOf(x) ϭ x3 Ϫ 4x2 ϩ 5f(x)
    • 6–5Page(s):Exercises:HOMEWORKASSIGNMENT178 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Graph a Polynomial ModelHEALTH The weight w, in pounds, of a patient during a7-week illness is modeled by the cubic equationw(n) ϭ 0.1n3Ϫ 0.6n2ϩ 110, where n is the number ofweeks since the patient became ill.a. Graph the equation. Describe the turning points ofthe graph and its end behavior.Make a table of values for the weeks 1–7. Plot the pointsand connect with a smooth curve.Relative minimum point at week .End behavior: w(n) increases asincreasesb. What trends in the patient’s weight does the graphsuggest? Is it reasonable to assume the trend willcontinue indefinitely?The patient lost weight for weeks; then gained weight.The trend may continue for a few weeks, but it isthat the patient’s weight will gain so quickly indefinitely.Check Your Progress The rainfall r, in inches permonth, during a 7-month period is modeled by ther(m) ϭ 0.01m3Ϫ 0.18m2ϩ 0.67m ϩ 3.23, where m is thenumber of months after March 1.a. Graph the equation. Describe theturning points of the graph andits end behavior.b. What trends in the amount of rainfall received by the towndoes the graph suggest?w w(n)12 108.434 106.856 110
    • Glencoe Algebra 2 1796–6Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.• Factor polynomials.• Simplify polynomialquotients by factoring.MAIN IDEASOn the page forLesson 6-6, write anordered list describingwhat you look for asyou factor a polynomial.ORGANIZE ITGCFFactor 10a3b2ϩ 15a2b Ϫ 5ab3.10a3b2ϩ 15a2b Ϫ 5ab3ϭ (2 и 5 и a и a и a и b и b) ϩ (3 и 5 и a и a и b) Ϫ (5 и a и b и b и b)ϭ (5ab и )ϩ (5ab и )Ϫ (5ab и ) The GCFis 5ab.ϭ Distributive PropertyGroupingFactor x3ϩ 5x2Ϫ 2x Ϫ 10.x3ϩ 5x2Ϫ 2x Ϫ 10ϭ ϩ Group to findthe GCF.ϭ x2ϩ (Ϫ2) Factor the GCF ofeach binomial.ϭ Distributive PropertyCheck Your Progress Factor each polynomial.a. 6x4y2ϩ 9x2y2Ϫ 3xy2b. x3Ϫ 3x2ϩ 4x Ϫ 12Factor PolynomialsFactor each polynomial.a. 12y3Ϫ 8y2Ϫ 20yThis trinomial does not fit any of the factoring patterns.First, factor out the .12y3Ϫ 8y2Ϫ 20y ϭ (3y2Ϫ 2y Ϫ 5) Factor out the GCF.Solving Polynomial EquationsTEKS 2A.2The studentunderstands theimportance of the skillsrequired to manipulatesymbols in order to solveproblems and uses thenecessary algebraicskills required to simplifyalgebraic expressionsand solve equations andinequalities in problemsituations. (A) Use toolsincluding factoring andproperties of exponentsto simplify expressionsand to transform andsolve equations.
    • 6–6180 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.The coefficient of the y terms must be 3 and Ϫ5 since3(Ϫ5) ϭ and 3 ϩ (Ϫ5) ϭ . Rewrite theexpression using Ϫ5y and 3y in place of Ϫ2y and factor bygrouping.3y2Ϫ 2y Ϫ 5 ϭ 3y2Ϫ 5y ϩ 3y Ϫ 5 Replace Ϫ2yϭ (3y2Ϫ 5y) ϩ (3y Ϫ 5) Associative Propertyϭ (3y Ϫ 5) ϩ (3y Ϫ 5) Factorϭ (3y Ϫ 5)(y ϩ 1) Distributive Propertyϭb. 64x6Ϫ y6This polynomial could be considered the difference of twoor the difference of two . Thedifference of two squares should always be done before thedifference of two cubes. This will make the next step of thefactorization easier.64x6Ϫ y6ϭ (8x3ϩ y3)(8x3Ϫ y3) Difference of two squaresϭ (2x ϩ y)(4x2Ϫ 2xy ϩ y2)(2x Ϫ y)(4x2ϩ 2xy ϩ y2)Sum and difference of two cubesCheck Your Progress Factor each polynomial.a. 2x2ϩ x Ϫ 3 b. a3b3Ϫ 27Write an Expression in Quadratic FormWrite each expression in quadratic form, if possible.a. 2x6ϩ x3ϩ 9 ϩ (x3) ϩ 9 x6ϭb. x4ϩ 2x3ϩ 1 This cannot be written in quadratic formsince x4.Quadratic Form Anexpression that isquadratic in form can bewritten as au2ϩ bu ϩ cfor any numbers a, b,and c, a 0, where u issome expression in x.The expression au2ϩbu ϩ c is called thequadratic form of theoriginal expression.KEY CONCEPT
    • 6–6Glencoe Algebra 2 181Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Check Your Progress Write each expression inquadratic form, if possible.a. 2x4ϩ x2ϩ 3 b. x6ϩ x4ϩ 1c. x Ϫ 2x1_2 ϩ 3 d. x12ϩ 5Solve Polynomial Equationsa. Solve x4Ϫ 29x2ϩ 100 ϭ 0.x4Ϫ 29x2ϩ 100 ϭ 0 Original equationϪ 29(x2) ϩ100 ϭ 0 Write the expressionon the left in quadraticform.ϭ 0 Factor the trinomial.ϭ 0 Factor.Use the Zero Product Property.ϭ 0 or ϭ 0x ϭ x ϭϭ 0 or ϭ 0x ϭ x ϭ
    • 6–6Page(s):Exercises:HOMEWORKASSIGNMENT182 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.b. Solve x3ϩ 216 ϭ 0.x3ϩ 216 ϭ 0 Original equationx3ϩ ϭ 0 Sum of two cubes(x ϩ 6) ϭ 0 Factor.ϭ 0 or ϭ 0 Zero Product PropertyThe solution of the first equation is . The secondequation can be solved by using the Quadratic Formula.x ϭϪb Ϯ √b2Ϫ 4ac__2aQuadratic Formulax ϭϪ(Ϫ6) Ϯ √(Ϫ6)2Ϫ 4(1)(36)___2(1)Replace a with 1,b with Ϫ6, andc with 36.x ϭ6 Ϯ2Simplify.x ϭ6 Ϯ2or6 Ϯ2Multiply √108and √Ϫ1.x ϭ Simplify.Check Your Progress Solve each equation.a. x4Ϫ 10x2ϩ 9 ϭ 0b. x3ϩ 8 ϭ 0
    • Glencoe Algebra 2 1836–7Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.• Evaluate functions usingsynthetic substitution.• Determine whether abinomial is a factor ofa polynomial by usingsynthetic substitution.Synthetic SubstitutionIf ƒ(x) ϭ 3x4Ϫ 2x3ϩ x2Ϫ 2, find ƒ(4).METHOD 1 Synthetic SubstitutionBy the Remainder Theorem, ƒ(4) should be the remainderwhen you divide the polynomial by x Ϫ 4.Notice that thereis no x term. Azero is placed inthis position asa placeholder.The remainder is . Thus, by using syntheticsubstitution, ƒ(4) ϭ .METHOD 2 Direct SubstitutionReplace x with 4.ƒ(x) ϭ 3x4Ϫ 2x3ϩ x2Ϫ 2 Original functionƒ(4) ϭ Replace x with 4.ƒ(4) ϭ or Simplify.By using direct substitution, ƒ(4) ϭ .Check Your Progress If ƒ(x) ϭ 2x3Ϫ 3x2ϩ 7, find ƒ(3).4 3 Ϫ2 1 0 Ϫ2MAIN IDEASKEY CONCEPTSRemainder TheoremIf a polynomial ƒ(x) isdivided by x Ϫ a, theremainder is theconstant ƒ(a), andDividend equals quotientƒ(x) ϭ q(x)times divisor plus remainder.и (x Ϫ a) ϩ ƒ(a).where q(x) is apolynomial with degreeone less than the degreeof ƒ(x).Factor TheoremThe binomial x Ϫ a is afactor of the polynomialƒ(x) if and only ifƒ(a) ϭ 0.When synthetic division is used to evaluate a function, it iscalled synthetic substitution.BUILD YOUR VOCABULARY (page 162)The Remainder and Factor TheoremsTEKS 2A.2The studentunderstands theimportance of the skillsrequired to manipulatesymbols in order to solveproblems and uses thenecessary algebraicskills required to simplifyalgebraic expressionsand solve equations andinequalities in problemsituations. (A) Use toolsincluding factoring andproperties of exponentsto simplify expressionsand to transform andsolve equations.
    • 6–7184 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Use the Factor TheoremShow that x Ϫ 3 is a factor of x3ϩ 4x2Ϫ 15x Ϫ 18. Thenfind the remaining factors of the polynomial.The binomial x Ϫ 3 is a factor of the polynomial if 3 is a zeroof the related polynomial function. Use the factor theorem andsynthetic division.Since the remainder is 0, is a factor of thepolynomial. The polynomial x3 ϩ 4x2 Ϫ 15x Ϫ 18 can befactored as . The polynomialis the depressed polynomial. Check to seeif this polynomial can be factored.x2ϩ 7x ϩ 6 ϭ Factor the trinomial.So, x3ϩ 4x2Ϫ 15x Ϫ 18 ϭ .Check Your Progress Show that x ϩ 2 is a factor ofx3ϩ 8x2ϩ 17x ϩ 10. Then find the remaining factors of thepolynomial.On the tab forLesson 6-7, use theFactor Theorem to showthat x ϩ 3 is a factor of3x3 Ϫ 3x2 Ϫ 36x.ORGANIZE ITWhen you divide a polynomial by one of its binomialfactors, the quotient is called a depressed polynomial.BUILD YOUR VOCABULARY (page 162)3 1 4 Ϫ15 Ϫ18
    • 6–7Page(s):Exercises:HOMEWORKASSIGNMENTGlencoe Algebra 2 185Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Find All FactorsGEOMETRY The volume of the rectangular prism isgiven by V(x) ϭ x3ϩ 7x2ϩ 2x Ϫ 40. Find the missingmeasures.??x Ϫ 2The volume of a rectangular prism is ᐉ ϫ w ϫ h.You know that one measure is , so x Ϫ 2 is a factorof V(x).2 1 7 2 Ϫ40The quotient is x2ϩ 9x ϩ 20. Use this to factor V(x).V(x) ϭ x3ϩ 7x2ϩ 2x Ϫ 40 Volume functionϭ (x2ϩ 9x ϩ 20) Factor.ϭ Factor the trinomialx2ϩ 9x ϩ 20.So the missing measures of the prism are and.Check Your Progress The volume of a rectangularprism is given by V(x) ϭ x3ϩ 6x2Ϫ x Ϫ 30. Find the missingmeasures.
    • 6–8186 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Determine Number and Type of RootsSolve each equation. State the number and typeof roots.a. x2ϩ 2x Ϫ 48 ϭ 0x2ϩ 2x Ϫ 48 ϭ 0 Original equationϭ 0 Factor.Use the Zero Product Property.ϭ 0 or ϭ 0x ϭ x ϭ Solve eachequation.This equation has two real roots, and .b. y4Ϫ 16 ϭ 0y4Ϫ 16 ϭ 0(y2ϩ 4)(y2Ϫ 4) ϭ 0(y2ϩ 4)(y ϩ 2)(y Ϫ 2) ϭ 0y2ϩ ϭ 0 or y ϩ ϭ 0 or y Ϫ ϭ 0y2ϭ y ϭ y ϭy ϭ Ϯ√Ϫ4 or Ϯ2iThis equation has two real roots, , and twoimaginary roots, .Check Your Progress Solve each equation. State thenumber and type of roots.a. x2Ϫ x Ϫ 12 ϭ 0b. a4Ϫ 81 ϭ 0• Determine the numberand type of roots for apolynomial equation.• Find the zeros of apolynomial function.MAIN IDEASFundamental Theoremof Algebra Everypolynomial equationwith degree greaterthan zero has at leastone root in the set ofcomplex numbers.Corollary A polynomialequation of the formP(x) ϭ 0 of degree n withcomplex coefficients hasexactly n roots in the setof complex numbers.On the tab forLesson 6-8, write thesekey concepts. Be sure toinclude examples.KEY CONCEPTSRoots and ZerosTEKS 2A.2The studentunderstands theimportance of the skillsrequired to manipulatesymbols in order to solveproblems and uses thenecessary algebraicskills required to simplifyalgebraic expressionsand solve equations andinequalities in problemsituations. (A) Use toolsincluding factoring andproperties of exponentsto simplify expressionsand to transform andsolve equations.
    • 6–8Glencoe Algebra 2 187Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Since there are two sign changes, there are 2 or 0 negativereal zeros. Make a chart of possible combinations.Check Your Progress State the possible number ofpositive real zeros, negative real zeros, and imaginary zerosof p(x) ϭ x4Ϫ x3ϩ x2ϩ x ϩ 3.Positive Real Negative Real ImaginaryTotalZeros Zeros Zeros2 2 60 62 60 6Find Numbers of Positive and Negative ZerosState the possible number of positive real zeros,negative real zeros, and imaginary zeros ofp(x) ϭ Ϫx6ϩ 4x3Ϫ 2x2Ϫ x Ϫ 1.Since p(x) has degree 6, it has 6 zeros. However, some of themmay be imaginary. Use Descartes’ Rule of Signs to determinethe number and type of real zeros. Count the number ofchanges in sign for the coefficients of p(x).p(x) ϭ Ϫx6ϩ 4x3Ϫ 2x2Ϫ x Ϫ 1Since there are two sign changes, there are 2 or 0 positive realzeros. Find p(Ϫx) and count the number of sign changes for itscoefficients.p(Ϫx) ϭ Ϫ(Ϫx)6ϩ 4(Ϫx)3Ϫ 2(Ϫx)2Ϫ (Ϫx) Ϫ 1ϭ Ϫx6Ϫ 4x3Ϫ 2x2ϩ x Ϫ 1Descartes’ Rule of SignsIf P(x) is a polynomialwith real coefficientswhose terms arearranged in descendingpowers of the variable,• the number of positivereal zeros of y ϭ P(x)is the same as thenumber of changes insign of the coefficientsof the terms, or is lessthan this by an evennumber, and• the number ofnegative real zeros ofy ϭ P(x) is the sameas the number ofchanges in sign of thecoefficients of theterms of P(Ϫx), or isless than this numberby an even number.KEY CONCEPT
    • 6–8188 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Use Synthetic Substitution to Find ZerosFind all of the zeros of f(x) ϭ x3Ϫ x2ϩ 2x ϩ 4.Since f(x) has degree 3, the function has three zeros. Todetermine the possible number and type of real zeros,examine the number of sign changes for f(x) and f(Ϫx).f(x) ϭ x3Ϫ x2ϩ 2x ϩ 4f(Ϫx) ϭ Ϫx3Ϫ x2Ϫ 2x ϩ 4The function has or positive real zeros and exactly1 negative real zero. Thus the function has eitherpositive real zeros and negative real zero orimaginary zeros and negative real zero.To find the zeros, list some possibilities and eliminatethose that are not zeros. Use a shortened form of syntheticsubstitution to find f(a) for several values of a.x 1 Ϫ1 2 4Ϫ3 1 Ϫ4 14 Ϫ38Ϫ2 1 Ϫ3 8 Ϫ12Ϫ1 1 Ϫ2 4 0From the table, we can see that one zero occurs at x ϭ .Since the depressed polynomial of this zero, x2Ϫ 2x ϩ 4, isquadratic, use the Quadratic Formula to find the roots of therelated quadratic equation, x2Ϫ 2x ϩ 4 ϭ 0.x ϭϪb Ϯ √ b2Ϫ 4ac__2aQuadratic FormulaϭReplace a with1, b with Ϫ2,and c with 4.ϭ2 Ϯ √ Ϫ12_2Simplify.ϭ2 Ϯ 2i√3_2or Simplify.(Ϫ) Ϯ ΂ ΃2Ϫ 4΂ ΃΂ ΃2΂ ΃
    • 6–8Page(s):Exercises:HOMEWORKASSIGNMENTGlencoe Algebra 2 189Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Thus, the function has one real zero at x ϭ , and twoimaginary zeros at x ϭ and x ϭ .Check Your Progress Find all the zeros off(x) ϭ x3Ϫ 3x2Ϫ 2x ϩ 4.Use Zeros to Write a Polynomial FunctionWrite a polynomial function of least degree with integralcoefficients, the zeros of which include 4 and 4 Ϫ i.Explore If 4 Ϫ i is a zero, then is also a zero. So,x Ϫ 4, x Ϫ (4 Ϫ i), and are factors ofthe polynomial function.Plan Write the polynomial function as a product of itsfactors: f(x) ϭ (x Ϫ 4)[x Ϫ (4 Ϫ i)][x Ϫ (4 ϩ i)]Solve Multiply the factors to find the polynomial function.f(x) ϭ (x Ϫ 4)x Ϫ (4 Ϫ i)[x Ϫ (4 ϩ i)]ϭ (x Ϫ 4) [(x Ϫ 4) ϩ i]ϭ (x Ϫ 4)ϭ (x Ϫ 4)ϭ (x Ϫ 4)ϭϭf(x) ϭ is a polynomial function ofleast degree with integral coefficients and zeros of 4, 4 Ϫ i, and4 ϩ i.Check Your Progress Write a polynomial function of leastdegree with integral coefficients whose zeros include 2 and1 ϩ i.
    • 6–9190 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Identify Possible ZerosList all of the possible rational zeros of each function.a. ƒ(x) ϭ 3x4Ϫ x3ϩ 4Ifp_q is a rational zero, then p is a factor of 4 and q is afactor of 3. The possible factors of p are , ,and . The possible factors of q are and .So,p_q .b. ƒ(x) ϭ x4ϩ 7x3Ϫ 15Since the coefficient of x4is the possible rational zerosmust be the factors of the constant term Ϫ15. So, the possiblerational zeros .Check Your Progress List all of the possible rationalzeros of each function.a. ƒ(x) ϭ 2x3ϩ x ϩ 6b. ƒ(x) ϭ x3 ϩ 3x ϩ 24Find Rational ZerosGEOMETRY The volume of a rectangular solid is 1120cubic feet. The width is 2 feet less than the height,and the length is 4 feet more than the height. Find thedimensions of the solid.x Ϫ 2 ftx ftx ϩ 4 ft• Identify the possiblerational zeros of apolynomial function.• Find all the rationalzeros of a polynomialfunction.MAIN IDEASRational Zero TheoremLet ƒ(x) ϭa0xn ϩ a1xnϪ1 ϩ … ϩanϪ2x2 ϩ anϪ1x ϩ anrepresent a polynomialfunction with integralcoefficients. Ifp_q is arational number insimplest form and is azero of y ϭ ƒ(x), then p isa factor of an and q is afactor of a0.Corollary (IntegralZero Theorem) Ifthe coefficients of apolynomial function areintegers such that a0 ϭ 1and an 0, any rationalzeros of the functionmust be factors of an.KEY CONCEPTSRational Zero TheoremTEKS 2A.2The studentunderstands theimportance of the skillsrequired to manipulatesymbols in order to solveproblems and uses thenecessary algebraicskills required to simplifyalgebraic expressionsand solve equations andinequalities in problemsituations. (A) Use toolsincluding factoring andproperties of exponentsto simplify expressionsand to transform andsolve equations.
    • 6–9Glencoe Algebra 2 191Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Let x ϭ the height, ϭ the width, and ϭ thelength. Write an equation for the volume.ᐉwh ϭ V Formula for volume(x Ϫ 2)(x ϩ 4)x ϭ 1120 Substitute.ϭ 1120 Multiply.x3ϩ 2x2Ϫ 8x Ϫ 1120 ϭ 0 Subtract 1120.The leading coefficient is 1, so the possible integer zeros arefactors of 1120, Ϯ1, Ϯ2, Ϯ4, Ϯ5, Ϯ7, Ϯ8, Ϯ10, Ϯ14, Ϯ16, Ϯ20,Ϯ28, Ϯ32, Ϯ35, Ϯ40, Ϯ56, Ϯ70, Ϯ80, Ϯ112, Ϯ140, Ϯ160, Ϯ224,Ϯ280, Ϯ560, and Ϯ1120. Since length can only be positive,we only need to check positive zeros. From Descartes’ Rule ofSigns, we also know there is only one positive real zero. Makea table and test possible real zeros.p 1 2 Ϫ8 Ϫ11202 1 3 Ϫ5 Ϫ11264 1 6 16 Ϫ10567 1 9 55 Ϫ73510 1 12 112 0Our zero is . Since there is only one positive real zero, wedo not have to test the other numbers. The other dimensionsare 10 ϩ 4 or feet and 10 Ϫ 2 or feet.Check Your ProgressThe volume of a rectangularsolid is 100 cubic feet. Thewidth is 3 feet less than theheight, and the length is 5 feetmore than the height. Findthe dimensions of the solid.Find All ZerosFind all of the zeros of ƒ(x) ϭ x4ϩ x3Ϫ 19x2ϩ 11x ϩ 30.From the corollary to the Fundamental Theorem of Algebra,we know there are exactly 4 complex roots.According to Descartes’ Rule of Signs, there are 2 or 0 positivereal roots and 2 or 0 negative real roots.
    • The possible rational zeros are.Make a table and test some possible rational zeros.Since ƒ(2) ϭ 0, you know that x ϭ 2 is a zero. The depressedpolynomial is .Since x ϭ 2 is a positive real zero, and there can only be 2or 0 positive real zeros, there must be one more positivereal zero. Test the next possible rational zeros on thedepressed polynomial.There is another zero at x ϭ 3. The depressed polynomial is.Factor x2ϩ 6x ϩ 5.x2ϩ 6x ϩ 5 ϭ 0 Write the depressedpolynomial.ϭ 0 Factor.ϭ 0 or ϭ 0 Zero Product Propertyx ϭ x ϭThere are two more real zeros at x ϭ and x ϭ .Check Your Progress Find all of the zeros ofƒ(x) ϭ x4ϩ 4x3Ϫ 14x2Ϫ 36x ϩ 45.p_q 1 3 Ϫ13 Ϫ1536–9Page(s):Exercises:HOMEWORKASSIGNMENT192 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.On the tab forLesson 6-9, write howyou would find thepossible rational zeros ofƒ(x) ϭ x3ϩ 6x2ϩ x ϩ 6.ORGANIZE IT p_q 1 1 Ϫ19 11 30012
    • BRINGING IT ALL TOGETHERBUILD YOURVOCABULARYUse your Chapter 6 Foldable tohelp you study for your chaptertest.To make a crossword puzzle,word search, or jumblepuzzle of the vocabulary wordsin Chapter 6, go to:glencoe.comYou can use your completedVocabulary Builder(pages 161–162) to help yousolve the puzzle.VOCABULARYPUZZLEMAKERC H A P T E R6STUDY GUIDEGlencoe Algebra 2 193Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Simplify. Assume that no variable equals 0.1. (3n4y3)(Ϫ2nyϪ5) 2.12(x2y)3_4(xy0)2Determine whether each expression is a polynomial. If theexpression is a polynomial, classify it by the number ofterms and state the degree of the polynomial.3. √3x 4. 4r4Ϫ 2r ϩ 15. 2ab ϩ 4ab2Ϫ 6ab36. 5x ϩ 4Simplify.7. (3a Ϫ 6) Ϫ (2a Ϫ 1) 8. (2x Ϫ 5)(3x ϩ 5)6-1Properties of Exponents6-2Operations with Polynomials
    • Chapter BRINGING IT ALL TOGETHER6194 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Simplify.9. (c3ϩ c2Ϫ 14c Ϫ 24) Ϭ (c Ϫ 4) 10.n2ϩ 3n Ϫ 2__n ϩ 26-3Dividing Polynomials11. Give the degree and leading coefficient of each polynomial.degree leading coefficienta. 10x3ϩ 3x2Ϫ x ϩ 7b. 7y2Ϫ 2y5ϩ y Ϫ 4y3c. 10012. Graph ƒ(x) ϭ x3Ϫ 6x2ϩ 2x ϩ 8 by making a table of values. Letx ϭ {Ϫ2, Ϫ1, 0, 1, 2, 3}. Then determine consecutive valuesof x between which each real zero is located. Estimate the x-coordinates at which the relative maxima and relative minimaoccur.6-4Polynomial Functions6-5Analyzing Graphs of Polynomial Functions
    • Chapter BRINGING IT ALL TOGETHER6Glencoe Algebra 2 195Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Factor completely. If the polynomial is not factorable, write prime.13. 3w2Ϫ 48 14. a3ϩ 5a Ϫ 3a2Ϫ 1515. Simplifyx2ϩ 7x ϩ 10__x2Ϫ 4. Assume that the denominator is not equal 0.Find ƒ(Ϫ2) for each function.16. ƒ(x) ϭ x3ϩ 4x2Ϫ 8x Ϫ 617. ƒ(x) ϭ x3ϩ 4x2ϩ 4xLet ƒ(x) ϭ x6Ϫ 2x5ϩ 3x4Ϫ 4x3ϩ 5x2ϩ 6x Ϫ 7.18. Write ƒ(Ϫx) in simplified form (with no parentheses).19. What are the possible numbers of positive real zeros of ƒ?negative real zeros of ƒ?20. List all the possible values of p, all the possible values of q, andall the possible rational zerosp_q for ƒ(x) ϭ x3Ϫ 2x2Ϫ 11x ϩ 12.6-6Solving Polynomial Equations6-7The Remainder and Factor Theorems6-8Roots and Zeros6-9Rational Zero Theorem
    • Check the one that applies. Suggestions to help you study aregiven with each item.ARE YOU READY FORTHE CHAPTER TEST?I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.• You may want to take the Chapter 6 Practice Test onpage 379 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 6 Study Guide and Reviewon pages 374–378 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 6 Practice Test onpage 379 of your textbook.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 6 Foldable.• Then complete the Chapter 6 Study Guide and Review onpages 374–378 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 6 Practice Test onpage 379 of your textbook.Visit glencoe.com toaccess your textbook,more examples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 6.Student Signature Parent/Guardian SignatureTeacher SignatureC H A P T E R6Checklist196 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.C H A P T E RChapter7Chapter4Glencoe Algebra 2 197Radical Equations and InequalitiesUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.C H A P T E R7NOTE-TAKING TIP: When you take notes, previewthe lesson and make generalizations about whatyou think you will learn. Then compare that withwhat you actually learned after each lesson.Begin with four sheets of grid paper..Fold in half along the width. On thefirst two sheets, cut 5 centimetersalong the fold at the ends. On thesecond two sheets, cut in the center,stopping 5 centimeters from the ends.Insert the first sheets through the secondsheets and align the folds. Label the pageswith lesson numbers.
    • Vocabulary TermFoundon PageDefinitionDescription orExamplecomposition offunctionsconjugatesextraneous solutionidentity functioninverse functioninverse relationlike radicalexpressions198 Glencoe Algebra 2This is an alphabetical list of new vocabulary terms you will learn in Chapter 7.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.C H A P T E R7BUILD YOUR VOCABULARYCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Chapter BUILD YOUR VOCABULARY7Glencoe Algebra 2 199Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Vocabulary TermFoundon PageDefinitionDescription orExamplenth rootone-to-oneprincipal rootradical equationradical inequalityrationalizing thedenominatorsquare root functionsquare root inequality
    • 200 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.7–1 Operations on FunctionsAdd and Subtract FunctionsGiven ƒ(x) ϭ 3x2ϩ 7x and g(x) ϭ 2x2Ϫ x Ϫ 1, find eachfunction.a. (ƒ ϩ g)(x)(ƒ ϩ g)(x) ϭ ƒ(x) ϩ g(x)ϭ ϩϭb. (f Ϫ g)(x)(f Ϫ g)(x) ϭ f(x) Ϫ g(x)ϭ ϪϭCheck Your Progress Given ƒ(x) ϭ 2x2ϩ 5x ϩ 2 andg(x) ϭ 3x2ϩ 3x Ϫ 4, find each function.a. (ƒ ϩ g)(x) b. (ƒ Ϫ g)(x)Multiply and Divide FunctionsGiven ƒ(x) ϭ 3x2Ϫ 2x ϩ 1 and g(x) ϭ x Ϫ 4, find eachfunction.a. (ƒ ؒ g)(x)(ƒ ؒ g)(x) ϭ ƒ(x) ؒ g(x)ϭ ΂ ΃΂ ΃ϭ 3x2(x Ϫ 4) Ϫ 2x(x Ϫ 4) ϩ 1(x Ϫ 4) DistributivePropertyϭ DistributivePropertyϭ Simplify.• Find the sum,difference, product,and quotient offunctions.• Find the compositionof functions.MAIN IDEASOperations withFunctionsSum(ƒ ϩ g)(x) ϭ ƒ(x) ϩ g(x)Difference(ƒ Ϫ g)(x) ϭ ƒ(x) Ϫ g(x)Product(ƒ ؒ g)(x) ϭ ƒ(x) ؒ g(x)Quotient΂ᎏgƒᎏ΃(x) ϭ ᎏgƒ((xx))ᎏ, g(x) 0KEY CONCEPTTEKS 2A.9 Thestudent formulatesequations andinequalities based onsquare root functions,uses a variety of methodsto solve them, andanalyzes the solutions interms of the situation. (B)Relate representationsof square root functions,such as algebraic,tabular, graphical, andverbal descriptions.(C) Determine thereasonable domain andrange values of squareroot functions, as well asinterpret and determinethe reasonablenessof solutions to squareroot equations andinequalities.
    • b. ΂ƒ_g΃(x)΂f_g΃(x) ϭf(x)_g(f)ϭCheck Your Progress Given ƒ(x) ϭ 2x2ϩ 3x Ϫ 1 andg(x) ϭ x ϩ 2, find each function.a. (ƒ ؒ g)(x) b. ΂ƒ_g΃(x)Evaluate Compositon of RelationsIf ƒ(x) ϭ {(2, 6), (9, 4), (7, 7), (0, Ϫ1)} and g(x) ϭ {(7, 0),(Ϫ1, 7), (4, 9), (8, 2)}, find ƒ ‫ؠ‬ g and g ‫ؠ‬ ƒ.To find ƒ ‫ؠ‬ g, evaluate g(x) first. Then use the range of g as thedomain of ƒ and evaluate ƒ(x).ƒ [g(7)] ϭ ƒ(0) or ƒ [g(4)] ϭ f(9) orƒ [g(Ϫ1)] ϭ ƒ(7) or ƒ [g(8)] ϭ f(2) orƒ ‫ؠ‬ g ϭTo find g ‫ؠ‬ ƒ, evaluate ƒ(x) first. Then use the range of ƒ as thedomain of g and evaluate g(x).g[ƒ(2)] ϭ g(6), which is g[ƒ(7)] ϭ g(7) org[ƒ(9)] ϭ g(4) or g[ƒ(0)] ϭ g(Ϫ1) org ‫ؠ‬ ƒ ϭCheck Your Progress If ƒ(x) ϭ {(1, 2), (0, Ϫ3), (6, Ϫ5),(2, 1)} and g(x) ϭ {(2, 0), (Ϫ3, 6), (1, 0), (6, 7)}, find ƒ ‫ؠ‬ gand g ‫ؠ‬ ƒ.7–1Glencoe Algebra 2 201Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 7–1202 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Simplify Composition of Functionsa. Find [ƒ ‫ؠ‬ g](x) and [g ‫ؠ‬ ƒ](x) for ƒ(x) ϭ 3x2Ϫ x ϩ 4 andg(x) ϭ 2x Ϫ 1.[ƒ ‫ؠ‬ g](x) ϭ ƒ[g(x)]ϭ ƒ΂ ΃ϭ Ϫ ϩϭ 3(4x2Ϫ 4x ϩ 1) Ϫ 2x ϩ 1 ϩ 4ϭ[g ‫ؠ‬ ƒ](x) ϭ g(ƒ(x)]ϭ g΂ ΃ϭ (3x2Ϫ x ϩ 4) Ϫϭb. Evaluate [ƒ ‫ؠ‬ g](x) and [g ‫ؠ‬ ƒ](x) for x ϭ Ϫ2.[ƒ ‫ؠ‬ g](x) ϭ 12x2Ϫ 14x ϩ 8[ƒ ‫ؠ‬ g](Ϫ2) ϭϭ[g ‫ؠ‬ ƒ](x) ϭ 6x2Ϫ 2x ϩ 7[g ‫ؠ‬ ƒ](Ϫ2) ϭϭCheck Your Progressa. Find [ƒ ‫ؠ‬ g](x) and [g ‫ؠ‬ ƒ](x) for ƒ(x) ϭ x2ϩ 2x ϩ 3 andg(x) ϭ x ϩ 5.b. Evaluate [ƒ ‫ؠ‬ g](x) and [g ‫ؠ‬ ƒ](x) for x ϭ 1.On the pagefor Lesson 7-1, write howyou would read[f ‫ؠ‬ g](x). Then explainwhich function, ƒ or g,you would evaluate first.KEY CONCEPTComposition ofFunctions Suppose ƒand g are functions suchthat the range of g is asubset of the domain ofƒ. Then the compositefunction ƒ ‫ؠ‬ g can bedescribed by theequation[ƒ ‫ؠ‬ g](x) ϭ ƒ[g(x)].
    • 7–1Page(s):Exercises:HOMEWORKASSIGNMENTUse Composition of FunctionTAXES Tracey Long has $100 deducted from everypaycheck for retirement. She can have this deductiontaken before state taxes are applied, which reducesher taxable income. Her state income tax rate is 4%. IfTracey earns $1500 every pay period, find the differencein her net income if she has the retirement deductiontaken before or after state taxes.Explore Let x ϭ Tracey’s income per paycheck, r(x) ϭ herincome after the deduction for retirement, and t(x) ϭher income after the deduction for state income tax.Plan Write equations for r(x) and t(x).$100 is deducted from every paycheck for retirement:r(x) ϭ x ϪTracey’s tax rate is 4%: t(x) ϭSolve If Tracey has her retirement deducted before taxes, thenher net income is represented by [t ‫ؠ‬ r](1500).[t ‫ؠ‬ r](1500) ϭ t(1500 Ϫ 100)ϭ tϭ 1400 Ϫ 0.04(1400)ϭIf Tracey has her retirement deducted after taxes, thenher net income is represented by [r ‫ؠ‬ t](1500).[r ‫ؠ‬ t](1500) ϭ r[1500 Ϫ 0.04(1500)]ϭ rϭ Ϫ 100ϭHer net pay is more by having her retirement deductiontaken before state taxes.Check Your Progress Brandi Smith has deducted $200from every paycheck for retirement. She can have thisdeduction taken before state taxes are applied, which reducesher taxable income. Her state income tax is 10%. If Brandiearns $2200 every pay period, find the difference in her netincome if she has the retirement deduction taken before orafter state taxes.Glencoe Algebra 2 203Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 204 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Find an Inverse RelationGEOMETRY The ordered pairs of the relation {(1, 3),(6, 3), (6, 0), (1, 0)} are the coordinates of the verticesof a rectangle. Find the inverse of this relation anddetermine whether the resulting ordered pairs arealso the coordinates of the vertices of a rectangle.To find the inverse of this relation,reverse the coordinates of the orderedpairs. The inverse of the relation is.Plotting the points shows that theordered pairs also describe thevertices of a rectangle. Notice that thegraph of the relation and the inverseare reflections over the graph of y ϭ x.Find an Inverse Functiona. Find the inverse of ƒ(x) ϭ Ϫ1_2x ϩ 1.Step 1 Replace ƒ(x) with y in the original equation.ƒ(x) ϭ Ϫ1_2x ϩ 1Step 2 Interchange x and yStep 3 Solve for y.x ϭ Ϫ1_2y ϩ 1 Inverseϭ Multiply each side by Ϫ2.ϭ Add 2 to each side.Step 4 Replace y with ƒϪ1(x).y ϭ ƒϪ1(x)ϭInverse Functions and Relations7–2• Find the inverse of afunction or relation.• Determine whether twofunctions or relationsare inverses.MAIN IDEASInverse Relations Tworelations are inverserelations if and only ifwhenever one relationcontains the element(a, b), the other relationcontains the element(b, a).Property of InverseFunctions Supposeƒ and ƒϪ1are inversefunctions. Then, ƒ(a) ϭ bif and only if ƒϪ1(b) ϭ a.KEY CONCEPTSf(x)xOy ϭ xTEKS 2A.9 The student formulates equations and inequalities based on squareroot functions, uses a variety of methods to solve them, and analyzes the solutions interms of the situation. (B) Relate representations of square root functions, suchas algebraic, tabular, graphical, and verbal descriptions.
    • Verify Two Functions are InversesDetermine whether ƒ(x) ϭ 3_4x Ϫ 6 and g(x) 4_3x ϩ 8 areinverse functions.Check to see if the compositions of ƒ(x) and g(x) are identityfunctions.[ƒ ‫ؠ‬ g](x) [g ‫ؠ‬ ƒ](x)ϭ ƒ ΂ ΃ ϭ g ΂ ΃ϭ΂4_3x ϩ 8΃ Ϫ ϭ΂3_4x Ϫ 6΃ϩϭ ϭϭ ϭThe functions are inverses since both compositions equal .a. The ordered pairs of the relation {(Ϫ3, 4), (Ϫ1, 5), (2, 3), (1, 1),(Ϫ2, 1)} are the coordinates of the vertices of a pentagon.Find the inverse of this relation and determine whetherthe resulting ordered pairs are also the coordinates of thevertices of a pentagon.b. Find the inverse of ƒ(x) ϭ 1_3x ϩ 6.c. Determine whether ƒ(x) ϭ 3x ϩ 1 and g(x) ϭ x Ϫ 1_3areinverse functions.7–2Glencoe Algebra 2 205Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Page(s):Exercises:HOMEWORKASSIGNMENTOn the page forLesson 7-2, sketchyour own graph oftwo relations that areinverses.ORGANIZE ITCheck Your Progress
    • Graph a Square Root FunctionGraph y ϭ √ 3_2x Ϫ 1. State the domain, range, and x- andy-intercepts.Since the radicand cannot be negative, identify the domainՆ 0 Write the expression inside theradicand as Ն 0.x Ն Solve for x.The x-intercept is .Make a table of values to graph thefunction. The domain is x Ն 2_3and therange is y Ն 0.The x-intercept is 2_3. There is noy-interceptCheck Your Progress Graph y ϭ √2x Ϫ 2. State thedomain, range, and x- and y-intercepts.206 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Square Root Functions and Inequalities7–3• Graph and analyzesquare root functions.• Graph square rootinequalities.MAIN IDEASx y2_301 0.712 1.413 1.874 2.245 2.556 2.83TEKS 2A.9 Thestudent formulatesequations andinequalities based onsquare root functions,uses a variety of methodsto solve them, andanalyzes the solutions interms of the situation. (A)Use the parent functionto investigate, describe,and predict the effectsof parameter changeson the graphs of squareroot functions anddescribe limitations onthe domains and ranges.(F) Analyze situationsmodeled by square rootfunctions, formulateequations or inequalities,select a method, andsolve problems. (G)Connect inverses ofsquare root functionswith quadratic functions.Also addresses TEKS2A.9(B).
    • PHYSICS When an object is spinning in a circular pathof radius 2 meters with velocity v, in meters per second,the centripetal acceleration a, in meters per secondsquared, is directed toward the center of the circle. Thevelocity v and acceleration a of the object are related bythe function v ϭ √2a.a. Graph the function. State thedomain and range.The domain is , and therange is .b. What would be the centripetal acceleration of anobject spinning along the circular path with a velocityof 4 meters per second?v ϭ √2a Original equationϭ √2a Replace v with .16 ϭ 2a Square each side.ϭ a Divide each side by 2.The centripetal acceleration would have to be metersper second squared. Check the reasonableness of this resultby comparing it to the graph.Check Your Progress The volume V and surface areaA of a soap bubble are related by the functionV ϭ 0.094√A3.a. Graph the function.State the domain andrange.b. What would the surface area be if the volume were 94 cubicunits?7–3Glencoe Algebra 2 207Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 7–3208 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Graph a Square Root Inequalitya. Graph y Ͼ √3x ϩ 5.Graph the related equationy ϭ √3x ϩ 5. Since the boundaryis not included, the graph shouldbe dashed.The domain includes values forx Ն Ϫ5_3. So the graph is to theright of x ϭ Ϫ5_3.Select a point and test its ordered pair. Test (0, 0)0 >0 > falseShade the region that does not include (0, 0).b. Graph y ϭ √4 ϩ3_2x.Graph the related equation y ϭ √4 ϩ 3_2x.The domain includes values forx ϭ Ϫ8_3. Test (4, 1).1 Յ1 Յ trueShade the region that includes (4, 1).Check Your Progress Graph y Յ √3 ϩ 3x.On the page forLesson 7-3, write howyou know if a boundaryshould be included ornot. Then write how youknow which region toshade.ORGANIZE ITPage(s):Exercises:HOMEWORKASSIGNMENTxOyy ϭ ͱස4 ϩ3–2xxOyy ϭ ͙ෆෆෆ3x ϩ 5
    • 5Find RootsSimplifya. Ϯ√16x6b. Ϫ√(q3ϩ 5)4Ϯ√16x6Ϫ√(q3ϩ 5)4ϭ Ϯ2ϭ Ϫ΂ ΃2ϭ ϭc.5√243a10b15d. √Ϫ45√243a10b15ϭ ΂ ΃5√Ϫ4 ϭϭ Since n is even and b isnegative, √Ϫ4 is not areal number.Check Your Progress Simplify.a. Ϯ√9x8b. Ϫ√(a3ϩ 2)6c.5√32x5y10d. √Ϫ16The inverse of raising a number to the nth power isfinding the nth root of a number. When there is more thanone real root, the root is called theprincipal root.Glencoe Algebra 2 209Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.7–4 nth Roots• Simplify radicals.• Use a calculator toapproximate radicals.MAIN IDEASBUILD YOUR VOCABULARY (pages 198–199)Definition of Square RootFor any real numbersa and b, if a2ϭ b, thena is a square root of b.Definition of nth RootFor any real numbersa and b, and anypositive integer n, ifanϭ b, then a is an nthroot of b.KEY CONCEPTSTEKS 2A.9 Thestudent formulatesequations andinequalities based onsquare root functions,uses a variety of methodsto solve them, andanalyzes the solutionsin terms of the situation.(C) Determine thereasonable domain andrange values of squareroot functions, as well asinterpret and determinethe reasonablenessof solutions to squareroot equations andinequalities. (D)Determine solutions ofsquare root equationsusing graphs, tables, andalgebraic methods.
    • 5Simplify Using Absolute ValueSimplify.a.6√t6Note that t is a sixth root of t6. The index is even, sothe principal root is nonnegative. Since t could be negative,you must take the absolute value of t to identify theprincipal root.6√t6ϭb.5√243(x ϩ 2)155√243(x ϩ 2)15ϭ ΄ ΅5Since the index is odd, you do not need absolute value.5√243(x ϩ 2)15ϭCheck Your Progress Simplify.a.4√x4 b.3√27(x ϩ 2)9FISH The relationship between the length and mass ofPacific halibut can be approximated by the equationL ϭ 0.463√M, where L is the length of the fish in metersand M is the mass in kilograms. Use a calculator toapproximate the length of a 30-kilogram Pacific halibut.L ϭ 0.463√M Original equationL ϭ 0.46 3 or about M ϭThe length of a 30-kilogram Pacific halibut is about.Check Your Progress The time T in seconds that it takesa pendulum to make a complete swing back and forth is givenby the formula T ϭ 2␲ 3 L_g , where L is the length of thependulum in feet and g is the acceleration due to gravity,32 feet per second squared. Find the value of T for a 2-foot-long pendulum.7–4210 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Page(s):Exercises:HOMEWORKASSIGNMENTOn the page forLesson 7-4, explainhow to simplify usingabsolute value. Includean example.ORGANIZE IT
    • Glencoe Algebra 2 211Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.7–5 Radical Expressions• Simplify radicalexpressions.• Add, subtract, multiply,and divide radicalexpressions.MAIN IDEASProduct Property ofRadicals For any realnumbers a and b andany integer n Ͼ 1,1. if n is even and aand b are bothnonnegative, thenn√ab ϭ n√a ؒ n√b, and2. if n is odd, thenn√ab ϭ n√a ؒ n√b.Quotient Property ofRadicals For any realnumbers a and b 0,and any integer n Ͼ 1,n√a_bؒn√a_n√b, if all rootsare defined.KEY CONCEPTSSimplify ExpressionsSimplify.a. √25a4b9√25a4b9ϭ (b4)2b Factor intosquares.ϭ √(b4)2√b Product Propertyϭ b4√b Simplify.b. √y8_x7√y8_x7ϭ √y8_x7Quotient Propertyϭ Factor into squares.ϭ√(y4)2_√(x3)2ؒ √xProduct Propertyϭy4_x3√x√(y4)2ϭ y4ϭy4_x3√xؒ√x_√xRationalize the denominator.ϭ √x ؒ √x ϭ xc.3√ 2_9x3√2_9xϭ3√2_3√9xQuotient Property΂ ΃2ؒ x΂ ΃2Preparation for TEKS 2A.9 The student formulates equations and inequalities basedon square root functions, uses a variety of methods to solve them, and analyzes thesolutions in terms of the situation. (D) Determine solutions of square root equationsusing graphs, tables, and algebraic methods.
    • ϭ3√2_3√9xؒ Rationalize the denominator.ϭ3√2 ؒ 3x2_3√9x ؒ 3x2Product Propertyϭ Multiply.ϭ3√27x3ϭ 3xCheck Your Progress Simplify each expression.a. √16x4y11b.√x3_y7c.3√ 2_3aMultiply RadicalsSimplify 53√100a2ؒ3√10a.53√100a2ؒ3√10a ϭ 5 ؒ3√100a2ؒ 10a Product Property ofRadicalsϭ 5 ؒ Factor into cubeswhere possible.ϭ 5 ؒ ؒ Product Property ofRadicalsϭ 5 ؒ ؒ or Multiply.7–5212 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Two radical expressions are called like radical expressions ifboth the indices and radicands are alike.BUILD YOUR VOCABULARY (page 198)
    • 7–5Glencoe Algebra 2 213Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.On the page forLesson 7-5, write yourown example of twolike radical expressions.Then find their sum.ORGANIZE ITCheck Your Progress Simplify 33√16a2ؒ 23√4a.Add and Subtract RadicalsSimplify 3√45 Ϫ 5√80 ϩ 4√20.3√45 Ϫ 5√80 ϩ 4√20ϭ 3√32ؒ 5 Ϫ 5√42ؒ 5 ϩ 4√22ؒ 5 Factor.ϭ 3√32ؒ √5 Ϫ 5√42ؒ √5 ϩ 4√22ؒ √5 Product Propertyϭ 3 ؒ √5 Ϫ 5 ؒ 4√5 ϩ 4 ؒ √5ϭ Multiply.ϭ Combine likeradicals.Check Your Progress Simplify 3√75 Ϫ 2√48 ϩ √3.Multiply RadicalsSimplify.a. (2√3 ϩ 3√5)(3 Ϫ √3)(2√3 ϩ 3√5)(3 Ϫ √3)ϭ 2√3 ؒ 3 Ϫ ϩ Ϫ 3√5 ؒ √3ϭ 6√3 Ϫ ϩ Ϫ 3√15ϭ 6√3 Ϫ Ϫ 3√15F O I L
    • b. (4√2 ϩ 7)(4√2 Ϫ 7)(4√2 ϩ 7)(4√2 Ϫ 7)ϭ 4√2 ؒ 4√2 Ϫ ϩ Ϫ 7 ؒ 7ϭ Ϫ 7 ؒ 4√2 ϩ 7 ؒ 4√2 Ϫϭ 32 Ϫ 49ϭCheck Your Progress Simplify each expression.a. (2√5 ϩ 4√6)(5 Ϫ √7)b. (3√5 Ϫ 2)(3√5 ϩ 2)Use a Conjugate to Rationalize a DenominatorSimplify2 ϩ √3_4 Ϫ √3.2 ϩ √3_4 Ϫ √3ϭ(2 ϩ √3)(4 ϩ √3)__(4 Ϫ √3)(4 ϩ √3)4 ϩ √3 is theconjugate of4 Ϫ √3.ϭFOILDifference ofsquares.ϭ Multiply.ϭ Combine liketerms.Check Your Progress Simplify3 ϩ √5_2 Ϫ √5.7–5Page(s):Exercises:HOMEWORKASSIGNMENT214 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.42Ϫ (√3)22 ؒ 4 ϩ 2√3 ϩ ϩ16 Ϫ 3ϩ 2√3 ϩ 4√3 ϩ
    • Glencoe Algebra 2 215Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.7–6 Rational Exponents• Write expressions withrational exponentsin radical form, andvice versa.• Simplify expressionsin exponential orradical form.MAIN IDEASb1_nFor any real numberb and for any positiveinteger n, b1_nϭ n√b,except when b Ͻ 0 andn is even.Rational ExponentsFor any nonzero realnumber b, and anyintegers m and n, withn Ͼ1,bm_nϭn√bmϭ (n√b)m,except when b Ͻ 0 andn is even.KEY CONCEPTSRadical and Exponential Formsa. Write a1_6in radical form.a1_6 ϭb. Write √w in exponential form.√w ϭCheck Your Progressa. Write x1_5 in radical form.b. Write 3√y in exponential form.Evaluate Expressions with Rational ExponentsEvaluate each expression.a. 49Ϫ1_249Ϫ1_2 ϭ bϪn ϭ 1_bnϭ 491_2ϭϭ Simplify.11TEKS 2A.4 The student connects algebraic and geometric representations offunctions. (C) Describe and analyze the relationship between a function andits inverse.
    • 7–6216 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.b. 322_5322_5 ϭ (25)2_5 32 ϭ 25ϭ Power of a Powerϭ Multiply exponents.ϭ 22ϭ 4Check Your Progress Evaluate each expression.a. 25Ϫ1_2 b. 163_4Simplify Expressions with Rational ExsponentsSimplify each expression.a. y1_7и y4_7y1_7 и y4_7 ϭ Multiply powers.ϭ Add exponents.b. xϪ2_3xϪ2_3 ϭ 1_x2_3bϪnϭ 1_bnϭ 1_x2_3и Multiply by .ϭ or x2_3и x1_3ϭ x2_3ϩ 1_3REMEMBER ITSince bm_n is definedas (b1_n)mor (bm)1_n,either can be used toevaluate an expression.Choose depending onthe given expression.
    • Check Your Progress Simplify each expression.a. x1_5 и x2_5 b. yϪ3_4Simplify Radical ExpressionsSimplify each expression.a.6√16_3√26√16_3√2ϭ 161_6_21_3Rational exponentsϭ 16 ϭ 24ϭ Power of a Powerϭ Quotient of Powersϭ or Simplify.b.6√4x46√4x4ϭ (4x4)1_6 Rational exponentsϭ (22)1_6 22ϭ 4ϭ 22(1_6) и Power of a Powerϭ Multiply.ϭ Simplify.7–6Glencoe Algebra 2 217Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.21_321_3On the page forLesson 7-6, write fourconditions that mustbe met in order for anexpression with rationalexponents to be insimplest form.ORGANIZE IT
    • c.y1_2 ϩ 1_y1_2 Ϫ 1y1_2 ϩ 1_y1_2 Ϫ 1ϭy1_2 ϩ 1_y1_2 Ϫ 1иy1_2 ϩ 1_y1_2 ϩ 1y1_2ϩ 1 is theconjugate of y1_2Ϫ 1.ϭy ϩ 2y1_2 ϩ 1__y Ϫ 1Multiply.Check Your Progress Simplify each expression.a.4√4_√2b.3√16x2c. x1_2 Ϫ 1_x1_2 ϩ 1TEST EXAMPLE If x is a positive number, then x1_3 и x1_2_x1_6ϭ ?A2√x3B3√x2C6√x5D x3_2x1_3 и x1_2_x1_6ϭ x5_6_x1_6Add the exponents in the numerator.ϭ x5_6 и Quotient of Powersϭ x2_3 orThe answer is .Check Your Progress If y is a positive number,theny3_4 y1_2_y1_4ϭ ?A y2B y4_5 C4√y5D 4√y7–6Page(s):Exercises:HOMEWORKASSIGNMENT218 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Glencoe Algebra 2 219Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.7–7 Solving Radical Equations and Inequalities• Solve equationscontaining radicals.• Solve inequalitiescontaining radicals.MAIN IDEASSolve Radical Equationsa. Solve √y Ϫ 2 Ϫ 1 ϭ 5.√y Ϫ 2 Ϫ 1 ϭ 5 Original equationϭ Add to isolate the radical.ϭ Square each side.ϭ Find the squares.y ϭ Add 2 to each side.b. Solve √x Ϫ 12 ϭ 2 Ϫ √x.√x Ϫ 12 ϭ 2 Ϫ √x Original equation(√x Ϫ 12)2ϭ (2 Ϫ √x )2Square each side.x Ϫ 12 ϭ Find the squares.Ϫ16 ϭ Ϫ4√x Isolate the radical.ϭ Divide each side by Ϫ4.ϭ Square each side.16 ϭ x Evaluate the squares.Since √16 Ϫ 12 , the solution does not checkand there isEquations with radicals that have variables in the radicandsare called radical equations.When you solve a radical equation and obtain a numberthat does not satisfy the original equation, the number iscalled an extraneous solution.A radical inequality is an inequality that has a variable ina radicand.(pages 198–199)BUILD YOUR VOCABULARYTEKS 2A.9The studentformulatesequations and inequalitiesbased on square rootfunctions, uses a varietyof methods to solve them,and analyzes the solutionsin terms of the situation.(C) Determine thereasonable domain andrange values of squareroot functions, as well asinterpret and determinethe reasonablenessof solutions to squareroot equations andinequalities. (D)Determine solutions ofsquare root equationsusing graphs, tables, andalgebraic methods. (E)Determine solutions ofsquare root inequalitiesusing graphs and tables.(F) Analyze situationsmodeled by square rootfunctions, formulateequations or inequalities,select a method, andsolve problems. Alsoaddresses TEKS 2A.4(A)and 2A.9(B).
    • Page(s):Exercises:HOMEWORKASSIGNMENTCheck Your Progress Solve.a. √x Ϫ 3 Ϫ 2 ϭ 6 b. √x ϩ 5 ϭ Ϫ1 Ϫ √xSolve a Cube Root EquationSolve (3y ϩ 1)1_3ϩ 5 ϭ 0.In order to remove the 1_3power, you must first isolate it andthen raise each side of the equation to the third power.(3y ϩ 1)1_3 ϩ 5 ϭ 0 Original equation(3y ϩ 1)1_3 ϭ Subtract 5 from each side.[(3y ϩ 1)1_3 ]3ϭ Cube each side.3y ϩ 1 ϭ Evaluate the cubes.y ϭ Simplify.Solve a Radical InequalitySolve √3x Ϫ 6 ϩ 4 ∞ 7.Find values of x for which the left side is defined.3x Ϫ 6 Ն 0 Radicand must be positive or 0.3x Ն 6x Ն 2Now solve √3x Ϫ 6 ϩ 4 Յ 7.√3x Ϫ 6 ϩ 4 Յ Original inequality√3x Ϫ 6 Յ Isolate the radical.Յ Eliminate the radical.3x Յ Add 6 to each side.x Յ Divide each side by 3.The solution is .Check Your Progress Solve.a. (2y ϩ 1)1_3 Ϫ 3 ϭ 0 b. √2x ϩ 5 Ϫ 2 Յ 97–7220 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.On the page forLesson 7-7, write yourown problem similar toExample 1. Then explainhow you eliminate theradical in order to solvethe equation.ORGANIZE IT
    • Glencoe Algebra 2 221Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.1. Find (ƒ ϩ g)(x), (ƒ Ϫ g)(x), (ƒ и g)(x), and (f_g)(x) forƒ(x) ϭ x2Ϫ 5x ϩ 2 and g(x) ϭ 3x ϩ 6.2. Find [g ‫ؠ‬ h](x) and [h ‫ؠ‬ g](x) for g(x) ϭ x2ϩ 8x Ϫ 7 andh(x) ϭ x Ϫ 3.3. Find the inverse of the function ƒ(x) ϭ Ϫ5x ϩ 4.4. Determine whether g(x) ϭ 2x ϩ 4 and ƒ(x) x_2ϩ 2 are inversefunctions.7-1Operations on Functions7-2Inverse Functions and RelationsBRINGING IT ALL TOGETHERBUILD YOURVOCABULARYUse your Chapter 7 Foldable tohelp you study for your chaptertest.To make a crossword puzzle,word search, or jumblepuzzle of the vocabulary wordsin Chapter 7, go to:glencoe.comYou can use your completedVocabulary Builder(pages 198–199) to help yousolve the puzzle.VOCABULARYPUZZLEMAKERC H A P T E R7STUDY GUIDECopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 7-3Square Root Functions and Inequalities222 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.5. Graph y Ͻ 2 ϩ √3x ϩ 4. Then state the domain and range ofthe function.6. Use a calculator to approximate3√Ϫ280 to three decimal places.Simplify.7.3√Ϫ64 8.8√x169. √100a1210.3√8y311. √49x6y812.3√125c6d15Simplify.13. √5_6x14. 2√45 Ϫ 7√8 ϩ √807-4nth Roots7-5Operations with Radical ExpressionsChapter BRINGING IT ALL TOGETHER7
    • Chapter BRINGING IT ALL TOGETHER7Glencoe Algebra 2 223Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Evaluate.15. (Ϫ32)1_5 16. 25Ϫ3_2 17. ( 1_64)Ϫ1_3Simplify.18. x1_4 и x5_4 19. (yϪ5_6)Ϫ1_520.10√36a2b10Solve each equation or inequality.21.3√5u Ϫ 2 ϭ Ϫ322. √4z Ϫ 3 ϭ √9z ϩ 223. 3 ϩ √2x Ϫ 1 Յ 624. √5x ϩ 4 ϩ 9 Ͼ 137-6Rational Exponents7-7Solving Radical Equations and Inequalities
    • Check the one that applies. Suggestions to help you study aregiven with each item.224 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.I completed the review of all or most lessons without using mynotes or asking for help.• You are probably ready for the Chapter Test.• You may want to take the Chapter 7 Practice Test on page 435of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 7 Study Guide and Reviewon pages 430–434 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 7 Practice Test onpage 435 of your textbook.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 7 Foldable.• Then complete the Chapter 7 Study Guide and Review onpages 430–434 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 7 Practice Test onpage 435 of your textbook.Student Signature Parent/Guardian SignatureTeacher SignatureARE YOU READY FORTHE CHAPTER TEST?Visit glencoe.com toaccess your textbook,more examples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 7.C H A P T E R7ChecklistCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Begin with a sheet of plain 81_2” by 11” paper.Fold in half lengthwiseleaving a1_2inch marginat the top. Fold again inthirds.Open. Cut along thefolds on the short tabto make three tabs.Label as shown.NOTE-TAKING TIP: Remember to always takenotes on your own. Don’t use someone else’snotes as they may not make sense.Rational Expressions and EquationsUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin this Interactive StudyNotebook to help you in taking notes.C H A P T E R8Glencoe Algebra 2 225Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Chapter8
    • Vocabulary TermFoundon PageDefinitionDescription orExampleasymptoteA-suhm(p)-TOHTcomplex fractionconstant of variationcontinuityKAHN-tuhn-OO-uh-teedirect variationinverse variationIHN-VUHRS226 Glencoe Algebra 2This is an alphabetical list of new vocabulary terms you will learn in Chapter 8.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.C H A P T E R8BUILD YOUR VOCABULARYCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Chapter BRINGING IT ALL TOGETHER8Glencoe Algebra 2 227Vocabulary TermFoundon PageDefinitionDescription orExamplejoint variationpoint discontinuityrational equationrational expressionrational functionrational inequalityCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • A ratio of two expressions is called arational expression.Simplify a Rational Expressiona. Simplify3y(y ϩ 7)__(y ϩ 7)(y2Ϫ 9).3y(y ϩ 7)__(y ϩ 7)(y2Ϫ 9)ϭ3y_y2Ϫ 9иy ϩ 71_y ϩ 71Factor.ϭ Simplify.b. Under what conditions is this expression undefined?To find out when this expression is undefined, completelyfactor the denominator.3y(y ϩ 7)__(y ϩ 7)(y2Ϫ 9)ϭ( y ϩ 7)3y(y ϩ 7)The values that would make the denominator equal to 0are , , and .Check Your Progressa. Simplifyx(x ϩ 5)__(x ϩ 5)(x2Ϫ 16).b. Under what conditions is this expression undefined?BUILD YOUR VOCABULARY (page 226)8–1• Simplify rationalexpressions.• Simplify complexfractions.MAIN IDEAS228 Glencoe Algebra 2Multiplying and Dividing Rational ExpressionsCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.TEKS 2A.2The studentunderstands theimportance of the skillsrequired to manipulatesymbols in order to solveproblems and uses thenecessary algebraicskills required to simplifyalgebraic expressionsand solve equations andinequalities in problemsituations. (A) Use toolsincluding factoring andproperties of exponentsto simplify expressionsand to transform and solveequations.
    • 8–1Glencoe Algebra 2 229Use the Process of EliminationTEST EXAMPLE For what value(s) of p isp2ϩ 2p Ϫ 3__p2Ϫ 2p Ϫ 15undefined?A 5 B Ϫ3, 5 C 3, Ϫ5 D 5, 1, Ϫ3Read the Test ItemYou want to determine which values of p make thedenominator equal to 0.Solve the Test ItemLook at the possible answers. Notice that the p term and theconstant term are both , so there will be onepositive solution and one negative solution. Therefore, you caneliminate choices A and D. Factor the denominator.p2Ϫ 2p Ϫ 15 ϭ Factor the denominator.p Ϫ 5 ϭ 0 or p ϩ 3 ϭ 0 Zero Product Propertyp ϭ p ϭ Solve each equation.Since the denominator equals 0 when p ϭ , the answeris .Check Your Progress For what values of p isp2ϩ 5p ϩ 6__p2ϩ 8p ϩ 15undefined?A Ϫ5, 3, 2 B Ϫ5 C 5 D Ϫ5, Ϫ3Simplify by Factoring Out Ϫ1Simplify a4b Ϫ 2a4ᎏᎏ2a3Ϫ a3b.a4b Ϫ 2a4_2a3Ϫ a3bϭ Factor the numeratorand the denominator.ϭa4a(Ϫ1)(2 Ϫ b)1__a31(2 Ϫ b)1b Ϫ 2 ϭ Ϫ(Ϫb ϩ 2) orϪ1(2 Ϫ b)ϭ or Ϫa Simplify.Multiply Rational ExpressionsSimplify each expression.a. 8x_21y3и7y2_16x3Rational Expressions• To multiply tworational expressions,multiply thenumerators and thedenominators.• To divide two rationalexpressions, multiplyby the reciprocal of thedivisor.KEY CONCEPTCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 8x_21y3и7y2_16x3ϭ21и 21и 21и x1и 71и y1и y1___3 и 71и y1и y1и y и 21и 21и 21и 2 и x1и x и xFactor.ϭ or Simplify.b.10ps2_3c2dϬ5ps_6c2d210ps2_3c2dϬ5ps_6c2d2ϭ10ps2_3c2dи 6c2d2_5psMultiply by thereciprocal of thedivisor.ϭ Factor.ϭ Simplify.ϭ Simplify.Check Your Progress Simplify each expression.a.x4y Ϫ 3x4_3x3Ϫ x3yb.3x2y_20abϬ6xy_5a2b3Polynomials in the Numerator and DenominatorSimplify each expression.a. k Ϫ 3_k ϩ 1и 1 Ϫ k2__k2Ϫ 4k ϩ 3k Ϫ 3_k ϩ 1и 1 Ϫ k2__k2Ϫ 4k ϩ 3ϭ k Ϫ 3_k ϩ 1и Multiply by thereciprocal of the divisor.ϭ(k Ϫ 3)1(1 Ϫ k)Ϫ1(1 ϩ k)1__(k ϩ 1)1(k Ϫ 1)1(k Ϫ 3)11 ϩ k ϭ k ϩ 1,1 Ϫ k ϭ Ϫ1(k Ϫ 1)8–1230 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 8–1Page(s):Exercises:HOMEWORKASSIGNMENTGlencoe Algebra 2 231ϭ Simplify.b. 2d ϩ 6_d2ϩ d Ϫ 2Ϭ d ϩ 3__d2ϩ 3d ϩ 22d ϩ 6_d2ϩ d Ϫ 2Ϭ d ϩ 3__d2ϩ 4d ϩ 2ϭ 2d ϩ 6_d2ϩ d Ϫ 2и Multiply by thereciprocal of the divisor.ϭ2(d ϩ 3)1(d ϩ 2)1(d ϩ 1)__(d ϩ 2)1(d Ϫ 1)(d ϩ 3)1Factor.ϭ Simplify.Simplify a Complex FractionSimplifyx2_9x2Ϫ 4y2_x3_2y Ϫ 3x.x2_9x2Ϫ 4y2_x3_2y Ϫ 3xϭ x2_9x2Ϫ 4y2Ϭ x3_2y Ϫ 3xExpress as a divisionexpression.ϭ x2_9x2Ϫ 4y2и Multiply by thereciprocal of thedivisor.ϭx1и x1(2y Ϫ 3x)Ϫ1___(3x Ϫ 2y)1(3x ϩ 2y) x1и x1и xFactor.ϭ Simplify.Check Your Progress Simplify each expression.a. x Ϫ 3_x ϩ 2и x2ϩ 5x ϩ 6_x2Ϫ 9b.a2_a2Ϫ 9b2_a4_a ϩ 3bUnder the tab forRational Expressions,write how simplifyingrational expressions issimilar to simplifyingrational numbers.ORGANIZE ITCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 8–2232 Glencoe Algebra 2Adding and Subtracting Rational ExpressionsLCM of MonomialsFind the LCM of 15a2bc3, 16b5c2, and 20a3c6.15a2bc3ϭ Factor the firstmonomial.16b5c2ϭ Factor the secondmonomial.20a3c6ϭ Factor the thirdmonomial.LCM ϭ Use each factorthe greatest numberϭof times it appears asa factor and simplify.LCM of PolynomialsFind the LCM of x3Ϫ x2Ϫ 2x and x2Ϫ 4x ϩ 4.x3Ϫ x2Ϫ 2x ϭ Factor the firstpolynomial.x2Ϫ 4x ϩ 4 ϭ Factor the secondpolynomial.LCM ϭ Use each factor thegreatest number oftimes it appears asa factor.Check Your Progress Find the LCM of each set ofpolynomials.a. 6x2zy3, 9x3y2z2, 4x2zb. x3ϩ 2x2Ϫ 3x, x2ϩ 6x ϩ 9• Determine the LCM ofpolynomials.• Add and subtractrational expressions.MAIN IDEASCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.TEKS 2A.2The studentunderstands theimportance of the skillsrequired to manipulatesymbols in order to solveproblems and uses thenecessary algebraicskills required to simplifyalgebraic expressionsand solve equations andinequalities in problemsituations. (A) Use toolsincluding factoring andproperties of exponentsto simplify expressionsand to transform and solveequations.
    • 8–2Glencoe Algebra 2 233REMEMBER ITWhen adding andsubtracting rationalexpressions, find theLCD. Then, rewrite theexpressions using thecommon denominator.Finally, add or subtractthe fractions andsimplify the result.Monomial DenominatorsSimplify 5a2_6bϩ 9_14a2b2.5a2_6bϩ 9_14a2b2ϭ5a2ؒ6b ؒ 14a2b2ؒ9 ؒϩThe LCD is 42a2b 2.Find equivalentfractions that havethis denominator.ϭ42a2b242a2b2ϩSimplify eachnumerator anddenominator.ϭ 35a4b ϩ 27_42a2b2Add the numerators.Polynomial DenominatorsSimplify x ϩ 10_3x Ϫ 15Ϫ 3x ϩ 15_6x Ϫ 30.x ϩ 10_3x Ϫ 15Ϫ 3x ϩ 15_6x Ϫ 30ϭx ϩ 10 3x ϩ 15Ϫ Factor the denominators.ϭ2(x ϩ 10)_2 и 3(x Ϫ 5)Ϫ 3x ϩ 15_6(x Ϫ 5)The LCD is 6(x Ϫ 5).ϭ2(x ϩ 10) Ϫ (3x ϩ 15)__6(x Ϫ 5)Subtract the numerators.ϭ6(x Ϫ 5)Distributive Propertyϭ6(x Ϫ 5)Combine like terms.ϭϪ1(x Ϫ 5)1_6(x Ϫ 5)1or Simplify.Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 8–2234 Glencoe Algebra 2Check Your Progress Simplify each expression.a. 3x2_2yϩ 5_12xy2b. x ϩ 5_2x Ϫ 4Ϫ 3x ϩ 8_4x Ϫ 8Simplify Complex FractionsSimplify1_a ϩ 1_b_1_bϪ 1.1_a ϩ 1_b_1_bϪ 1 ϭ1_bϩϩ a_abThe LCD of the numerator is ab.The LCD of the denominator is b.ϭbabSimplify the numerator anddenominator.ϭ Ϭ Write as a divisionexpression.ϭ b ϩ a_ab1и b1_1 Ϫ bMultiply by the reciprocal of thedivisor.ϭ b ϩ a_a(1 Ϫ b)or Simplify.Under the tab forRational Expressions,write your own examplesimilar to Example 5.Then simplify yourrational expression. Givethe reason for each step.ORGANIZE ITCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 8–2Page(s):Exercises:HOMEWORKASSIGNMENTGlencoe Algebra 2 235Check Your Progress Simplify1_a Ϫ 1_b_1_bϩ 1_a.Use a Complex Fraction to Solve a ProblemCOORDINATE GEOMETRY Find the slope of the line thatpasses through P(3_k, 1_3)and Q(1_2, 2_k).m ϭy2 Ϫ y1_x2 Ϫ x1Definition of slopeϭ2_kϪ 1_3_1_2Ϫ 3_ky2 ϭ 2_k, y1 ϭ 1_3, x2 ϭ 1_2, x1 ϭ 3_kϭk Ϫ 66 Ϫ kThe LCD of the numerator is 3k.The LCD of the denominator is 2k.ϭ 6 Ϫ k_3kϬ Write as division expression.ϭ 6 Ϫ k_3kи 2k_k Ϫ 6Multiply by the reciprocal ofthe divisor.ϭϪ(k Ϫ 6)_3kи 2k_k Ϫ 66 Ϫ k ϭ Ϫ(k Ϫ 6)ϭ Ϫ1(k Ϫ 6)_3kи 2k_k Ϫ 61orThe slope is .Check Your Progress Find the slope of the line thatpasses through P(4_k, 1_4) and Q(1_5, 5_k).Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 8–3236 Glencoe Algebra 2A rational function is an equation of the formƒ(x) ϭp(x)_q(x), where p(x) and q(x) arefunctions and q(x) 0.The graphs of rational functions may have breaks incontinuity. This means that not all rational functions aretraceable. Breaks in can appear as avertical asymptote or as a point discontinuity.Graphing Rational FunctionsVertical Asymptotes and Point DiscontinuityDetermine the equations of any vertical asymptotesand the values of x for any holes in the graph ofƒ(x) ϭ x2Ϫ 4__x2ϩ 5x ϩ 6.Factor the numerator and denominator of the rationalexpression.x2Ϫ 4_x2ϩ 5x ϩ 6ϭThe function is undefined for x ϭ and .Since(x Ϫ 2)(x ϩ 2)1__(x ϩ 2)1(x ϩ 3)ϭ , x ϭ is a verticalasymptote and x ϭ is a hole in the graph.Check Your Progress Determine the equations of anyvertical asymptotes and the values of x for any holes in thegraph of f(x) ϭ x2Ϫ 9__x2ϩ 8x ϩ 15.• Determine the verticalasymptotes and thepoint discontinuity forthe graphs of rationalfunctions.• Graph rational functions.MAIN IDEASBUILD YOUR VOCABULARY (page 226)Vertical AsymptotesIf the rational expressionof a function is writtenin simplest form and thefunction is undefined forx ϭ a, then x ϭ a is avertical asymptote.Point DiscontinuityIf the original function isundefined for x ϭ a butthe rational expressionof the function insimplest form is definedfor x ϭ a, then there is ahole in the graph atx ϭ a.KEY CONCEPTSCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.TEKS 2A.10(A) Use quotientsof polynomialsto describe the graphsof rational functions,predict the effects ofparameter changes,describe limitationson the domains andranges, and examineasymptotic behavior.(C) Determine thereasonable domain andrange values of rationalfunctions, as well asinterpret and determinethe reasonablenessof solutions torational equationsand inequalities. Alsoaddresses TEKS 2A.1(A),2A.2(A), 2A.4(B),2A.10(B), and 2A.10(F).TEKS 2A.10 The student formulates equations and inequalities based on rationalfunctions, uses a variety of methods to solve them, and analyzes the solutions interms of the situation.
    • 8–3Glencoe Algebra 2 237Graph with a Vertical AsymptoteGraph ƒ(x) ϭ x_x ϩ 1.The function is undefined for x ϭ . Since x_x ϩ 1is in itssimplest form, x ϭ is a vertical asymptote. Make atable of values. Plot the points and draw the graph.x ƒ(x)Ϫ4 1.33Ϫ3 1.5Ϫ2 20 01 0.52 0.673 0.75Graph with Point DiscontinuityGraph ƒ(x) ϭ x2Ϫ 4_x Ϫ 2.Notice that x2Ϫ 4_x Ϫ 2ϭ(x ϩ 2)(x Ϫ 2)__x Ϫ 2xOf(x)or . Therefore, thegraph of f(x) ϭ x2Ϫ 4_x Ϫ 2is the graphƒ(x) ϭ with a hole at x ϭ .Check Your Progress Graph each rational function.a. f(x) ϭ x_x ϩ 3b. f(x) ϭ x2Ϫ 16_x ϩ 4Under the tab forRational Functions,write how you can tellfrom a rational functionwhere the breaks incontinuity will appearin the graph of thefunction.ORGANIZE ITCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 8–3Page(s):Exercises:HOMEWORKASSIGNMENT238 Glencoe Algebra 2Use Graphs of Rational FunctionsAVERAGE SPEED Use the situation and formula given inExample 4 of your textbook.a. Draw the graph of r2 ϭ 15 miles per hour.The function is R ϭ2r1(15)_r1 ϩ 15or R ϭ30r1_r1 ϩ 15. The verticalasymptote is r1 ϭ .Graph the verticalasymptote and thefunction. Notice that thehorizontal asymptote isR ϭ .b. What is the R-intercept of the graph?The R-intercept is .c. What domain and range values are meaningful in thecontext of the problem?In the problem context, the speeds are nonnegative values.Therefore, only values of r1 greater than or equal toand values of R between are meaningful.Check Your Progress A train travels at one velocityV1 for a given amount of time t1 and then anothervelocity V2 for a different amount of time t2. Theaverage velocity is given by V ϭV1t1 ϩ V2t2__t1 ϩ t2.a. Let t1 be the independentvariable and let V be thedependent variable. Drawthe graph if V1 ϭ 60 milesper hour, V2 ϭ 30 miles perhour, and t2 ϭ 1 hour.b. What is the V-intercept of thegraph?c. What values of t1 and V are meaningful in the context of theproblem?Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Glencoe Algebra 2 2398–4 Direct, Joint, and Inverse VariationDirect VariationIf y varies directly as x and y ϭ Ϫ15 when x ϭ 5,find y when x ϭ 3.Use a proportion that relates the values.y1_x1ϭy2_x2Direct proportion.ϭ y1 ϭ Ϫ15, x1 ϭ 5, and x2 ϭ 3ϭ Cross multiply.ϭ Simplify.ϭ y2Divide each side by 5.When x ϭ 3, the value of y ϭ .Joint VariationSuppose y varies jointly as x and z. Find y when x ϭ 10and z ϭ 5 if y ϭ 12 when x ϭ 3 and z ϭ 8.Use a proportion that relates the values.y1_x1z1ϭy2_x2z2Joint variation.ϭ y1 ϭ 12, x1 ϭ 3, z1 ϭ 8,x2 ϭ 10, and z2 ϭ 5ϭ Cross multiply.ϭ Simplify.ϭ y2Divide each side by 24.When x ϭ 10 and z ϭ , y ϭ .• Recognize and solvedirect and jointvariation problems.• Recognize and solveinverse variationproblems.MAIN IDEASDirect Variation y variesdirectly as x if there issome nonzero constantk such that y ϭ kx. k iscalled the constant ofvariation.Joint Variation y variesjointly as x and z if thereis some number k suchthat y ϭ kxz, wherek 0, x 0, and z 0.KEY CONCEPTSCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.TEKS 2A.10The studentformulatesequations and inequalitiesbased on rationalfunctions, uses a varietyof methods to solve them,and analyzes the solutionsin terms of the situation.(F) Analyze a situationmodeled by a rationalfunction, formulate anequation or inequalitycomposed of a linear orquadratic function, andsolve the problem. (G)Use functions to modeland make predictionsin problem situationsinvolving direct andinverse variation.Also addresses TEKS2A.1(A), 2A.2(A), 2A.4(B),2A.10(B), and 2A.10(F).
    • Page(s):Exercises:HOMEWORKASSIGNMENT8–4240 Glencoe Algebra 2Inverse VariationIf r varies inversely as t and r ϭ Ϫ6 when t ϭ 2,find r when t ϭ Ϫ7.Use a proportion that relates the values.r1_t2ϭr2_t1Inverse variationϭ r1 ϭ Ϫ6, t1 ϭ 2, and t2 ϭ Ϫ7ϭ Cross multiply.ϭ Simplify.ϭ Divide each side by Ϫ7.When t ϭ Ϫ7, r is or 15_7.Check Your Progressa. If y varies directly as x and y ϭ 12 when x ϭ Ϫ3, find ywhen x ϭ 7.b. Suppose y varies jointly as x and z. Find y when x ϭ 3 andz ϭ 2 if y ϭ 11 when x ϭ 5 and z ϭ 22.c. If a varies inversely as b and a ϭ 3 when b ϭ 8, find a whenb ϭ 6.Inverse Variation y variesinversely as x if there issome nonzero constant ksuch that xy ϭ k ory ϭ k_x.KEY CONCEPTCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 8–5Glencoe Algebra 2 241Classes of Function• Identify graphs asdifferent types offunctions.• Identify equationsas different types offunctions.MAIN IDEASIdentify a Function Given the GraphIdentify the type of function represented by each graph.a.xyOThe graph is a V shape, so it isfunctionb.xyOThe graph is a parabola, so it isfunction.Check Your Progress Identify the type of functionrepresented by each graph.a. yxOb. yxOMatch Equation with GraphSHIPPING CHARGES A chart gives the shipping ratesfor an Internet company. They charge $3.50 to ship lessthan 1 pound, $3.95 for 1 pound and over up to 2 pounds,and $5.20 for 2 pounds and over up to 3 pounds. Whichgraph depicts these rates?a.xyO543211 42 3b.xyO543211 42 3c.xyO543211 42 3The shipping rate is constant for x values from 0 to 1. Then itjumps at x ϭ and remains constant until x ϭ . TheCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.TEKS 2A.4 Thestudent connectsalgebraic andgeometric representationsof functions. (A) Identifyand sketch graphsof parent functions,including linear(f(x) ϭ x), quadratic(f(x) ϭ x2), exponential(f(x) ϭ ax), and logarithmic(f(x) ϭ loga x), functions,absolute value of x(f(x) ϭ x), square rootof x (f(x) ϭ √x), andreciprocal of x (f(x) ϭ 1_x).(B) Extend parentfunctions withparameters such as a inf(x) ϭ a_xand describethe effects of theparameter changes onthe graph of parentfunctions.
    • Page(s):Exercises:HOMEWORKASSIGNMENT8–5242 Glencoe Algebra 2REVIEW ITIn order for a squareroot to be a realnumber, what is trueabout the radicand ofthe square rootfunction? (Lesson 7-3)graph jumps again at x ϭ and remains constant untilx ϭ . The graph of this function looks like steps, so this is, the step or .Check Your Progress A ball is thrown into the air. Thepath of the ball is represented by the equation h(t) ϭ Ϫ16t2ϩ 5.Which graph represents this situation?a. b. c.Identify a Function Given its EquationIdentify the type of function represented by eachequation. Then graph the equation.a. y ϭ Ϫ3Since the equation has nox-intercept, it is the constantfunction. Determine somepoints on the graph andgraph it.b. y ϭ √9xSince the equation includesan expression with a squareroot, it is a square root function.Plot some points and use whatyou know about square rootgraphs to graph it.Check Your Progress Identify the type of functionrepresented by each equation. Then graph the equation.a. y ϭ x2Ϫ 9_x ϩ 3b. y ϭ Ϫ2xCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Glencoe Algebra 2 2438–6Any equation that contains one or more rationalexpressions is called a rational equation.Inequalities that contain one or more rational expressionsare called rational inequalities.Solving Rational Equations and InequalitiesSolve a Rational EquationSolve 5_24ϩ 2_3 Ϫ xϭ 1_4.The LCD for the three denominators is 24(3 Ϫ x).5_24ϩ 2_3 Ϫ xϭ 1_4Originalequation( 5_24ϩ 2_3 Ϫ x)ϭ 1_4Multiplyeach side by.241(3 Ϫ x)(5_241)ϩ 24(3 Ϫ x)1( 2_3 Ϫ x1)ϭ 1_41246(3 Ϫ x) Simplify.ϭ Simplify.ϭ Add.x ϭCheck Your Progress Solve 5_2ϩ 3_x Ϫ 1ϭ 1_2.BUILD YOUR VOCABULARY (page 227)• Solve rationalequations.• Solve rationalinequalities.MAIN IDEASREMEMBER ITRational equationsare often easier to solvewhen you eliminate thefractions first.Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.TEKS 2A.10The studentformulatesequations and inequalitiesbased on rationalfunctions, uses a varietyof methods to solve them,and analyzes the solutionsin terms of the situation.(C) Determine thereasonable domain andrange values of rationalfunctions, as well asinterpret and determinethe reasonablenessof solutions torational equations andinequalities. (F) Analyzea situation modeledby a rational function,formulate an equationor inequality composedof a linear or quadraticfunction, and solve theproblem. Also addressesTEKS 2A.2(A), 2A.10(B),and 2A.10(D).
    • 8–6244 Glencoe Algebra 2Elimination of a Possible SolutionSolvep2Ϫ p ϩ 1_p ϩ 1ϭp2Ϫ 7_p2Ϫ 1ϩ p.The LCD is p2Ϫ 1.p2Ϫ p ϩ 1_p ϩ 1ϭp2Ϫ 7_p2Ϫ 1ϩ p Originalequation(p2Ϫ 1)p Ϫ 1p2Ϫ p ϩ 1_p ϩ 11ϭp2Ϫ 7_p2Ϫ 11(p2Ϫ 1)1ϩ p(p2Ϫ 1)(p Ϫ 1)(p2Ϫ p ϩ 1) ϭ p2Ϫ 7 ϩ (p2Ϫ 1)p DistributivePropertyp3Ϫ p2ϩ p Ϫ p2ϩ p Ϫ 1 ϭ p2Ϫ 7 ϩ p3Ϫ p Simplify.Ϫ2p2ϩ 2p Ϫ 1 ϭ p2Ϫ p Ϫ 7 Simplify.ϭ Add(2p2 Ϫ 2p ϩ 1)to each side.0 ϭ Divide eachside by 3.0 ϭ Factor.ϭ 0 or ϭ 0 Zero ProductPropertyp ϭ p ϭ Solve eachequation.Since p ϭ Ϫ1 results in a zero in the denominator, eliminate Ϫ1.Check Your Progress Solve 1_x Ϫ 2ϭ 2x ϩ 1_x2ϩ 2x Ϫ 8ϩ 2_x ϩ 4.When solving a rationalequation, why must thesolutions be checked inthe original equationrather than in any ofthe derived equations?WRITE ITCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 8–6Glencoe Algebra 2 245MOWING LAWNS Tim and Ashley mow lawns together.Tim working alone could complete a particular job in 4.5hours, and Ashley could complete it alone in 3.7 hours.How long does it take to complete the job when theywork together?In 1 hour, Tim could complete of the job.In 1 hour, Ashley could complete of the job.In t hours, Tim could complete 1_4.5и t or of the job.In t hours Ashley could complete 1_3.7и t or of the job.Part completedby Tim pluspart completedby Ashley equals entire job.t_4.5ϩ t_3.7= 1t_4.5ϩ t_3.7ϭ 1 Original equation΂ ΃( t_4.5ϩ t_3.7) ϭ (1)΂ ΃ Multiply each side by16.65.8.2t ϭ 16.65 Simplify.t ≈ Divide each side by 8.2.It would take about hours working together to completethe job.Check Your Progress Libby and Nate clean together.Nate working alone could complete the job in 3 hours, andLibby could complete it alone in 5 hours. How long does it taketo complete the job when they work together?Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • SWIMMING Janine swims for 5 hours in a stream thathas a current of 1 mile per hour. She leaves her dock andswims upstream for 2 miles and then swims back to herdock. What is her swimming speed in still water?Words The formula that relates distance, time, and rateis d ϭ rt or d_r ϭ t.Variables Let r ϭ speed in still water. Then her speedwith the current is , and her speedagainst the current is .EquationTime going withthe current plusTime going againstthe current equalstotaltime.ϩ =Solve the equation.2_r Ϫ 1ϩ 2_r ϩ 1ϭ 5 Originalequation(r2Ϫ 1)( 2_r Ϫ 1ϩ 2_r ϩ 1)ϭ (r2Ϫ 1)5 Multiply eachside by r2Ϫ 1.(r2Ϫ 1)r ϩ 12_r Ϫ 11ϩ (r2Ϫ 1)r Ϫ 12_r ϩ 11ϭ 5(r2Ϫ 1) DistributivePropertyϭ 5r2Ϫ 5 Simplify.ϭ 5r2Ϫ 5 Simplify.0 ϭ Subtract 4rfrom each side.Use the Quadratic Formula to solve for r.x ϭϪb Ϯ √ b2Ϫ 4ac__2aQuadratic Formular ϭϪ(Ϫ4) Ϯ √(Ϫ4)2Ϫ 4(5)(Ϫ5)___2(5)x ϭ r, a ϭ 5, b ϭ Ϫ4, andc ϭ Ϫ5r ϭ104 ϮSimplify.r ϭ or Use a calculator.Since speed must be positive, it is miles per hour.8–6246 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 8–6Glencoe Algebra 2 247Check Your Progress Lynne swims for 1 hour in a streamthat has a current of 2 miles per hour. She leaves her dock andswims upstream for 3 miles and then back to her dock. Whatis her swimming speed in still water?Solve a Rational InequalitySolve 1_3sϩ 2_9sϽ 2_3.Step 1 Values that make the denominator equal to 0 areexcluded from the denominator. For this inequalitythe excluded value is 0.Step 2 Solve the related equation.1_3sϩ 2_9sϭ 2_3Related equation.( 1_3sϩ 2_9s) ϭ 2_3Multiply each sideby .ϭ Simplify.ϭ Add.ϭ Divide each sideby .Step 3 Draw vertical lines at the excluded value and at thesolution to separate the number line into regions.Now test a sample value in each region to determineif the values in the region satisfy the inequality.Ϫ2Ϫ3 Ϫ1 0 1 2 3excluded value solution of relatedequationUnder the tab forRational Expressions,write how simplifyingrational expressions issimilar to simplifyingrational numbers.ORGANIZE ITCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 8–6Page(s):Exercises:HOMEWORKASSIGNMENT248 Glencoe Algebra 2Test s ϭ Ϫ1.1_3(Ϫ1)ϩ 2_9(Ϫ1)2_3Ϫ 2_3Ͻ 2_3So, s Ͻ 0 is a solution.Test s ϭ 1_3.1_3(1_3)ϩ 2_9(1_3)2_3ϩ 2_3ᎏ ≮ 2_3So, 0 Ͻ s Ͻ 5_6is not a solution.Test s ϭ 1.1_3(1)ϩ 2_9(1)2_3ϩ 2_3Ͻ 2_3So s Ͼ 5_6is not a solution.Check Your Progress Solve 1_x ϩ 3_5xϽ 2_5.Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • BRINGING IT ALL TOGETHERBUILD YOURVOCABULARYUse your Chapter 8 Foldable tohelp you study for your chaptertest.To make a crossword puzzle,word search, or jumblepuzzle of the vocabulary wordsin Chapter 8, go to:glencoe.comYou can use your completedVocabulary Builder(pages 226–227) to help yousolve the puzzle.VOCABULARYPUZZLEMAKERC H A P T E R8STUDY GUIDEGlencoe Algebra 2 2491. Which expressions are complex fractions?i. 7_12ii.3_8_5_16iii. r ϩ 5_r Ϫ 5vi.z ϩ 1_z_z v.r2Ϫ 25_9_r ϩ 5_3Simplify each expression.2. 6r2s3_t3и 3s2t2_12rst3.3y2ϩ 3y Ϫ 6__4y Ϫ 8Ϭy2Ϫ 4__2y2Ϫ 6y ϩ 4Find the LCM of each set of polynomials.4. 4y, 9xy, 6y25. x2Ϫ 5x ϩ 6, x3Ϫ 4x2ϩ 4x8-1Multiplying and Dividing Rational Expressions8-2Adding and Subtracting Rational ExpressionsCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Chapter BRINGING IT ALL TOGETHER8250 Glencoe Algebra 2Simplify each expression.6. 6_4abϪ 3_ab27.5q_2pϩ 8For Exercises 8 and 9, refer to the two rational functions shown.I.xyOII.xyO8. Graph I has a at x ϭ .9. Graph II has a at x ϭ .10. Graph f(x) ϭ 3_x Ϫ 2.11. Suppose y varies inversely as x. Find y when x ϭ 4, y ϭ 8 whenx ϭ 3.8-4Direct, Joint, and Inverse Variation8-3Graphing Rational FunctionsCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Chapter BRINGING IT ALL TOGETHER8Glencoe Algebra 2 251Which type of variation, direct or inverse, is represented byeach graph?12.xyO13.xyOMatch each graph below with thetype of function it represents.Some types may be used morethan once and others not at all.14.xyO15.xyO16.xyO17.xyOSolve each equation or inequality.18. 1_2xϩ x ϩ 1_xϭ 4 19.y ϩ 1_y Ϫ 1Ϫ 2_y Ϫ 1ϭy_318. 3_z ϩ 2Ϫ 6_zϾ 08-5Classes of Functionsa. square root b. absolute valuec. rational d. greatest integere. constant f. identity8-6Solving Rational Equations and InequalitiesCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Check the one that applies. Suggestions to help you study aregiven with each item.C H A P T E R8Checklist252 Glencoe Algebra 2I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.• You may want take the Chapter 8 Practice Test on page 493of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 8 Study Guide and Reviewon pages 489–492 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 8 Practice Test onpage 493.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 8 Foldable.• Then complete the Chapter 8 Study Guide and Review onpages 489–492 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 8 Practice Test onpage 493.ARE YOU READY FORTHE CHAPTER TEST?Visit glencoe.com to accessyour textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 8.Student Signature Parent/Guardian SignatureTeacher SignatureCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Exponential and Logarithmic RelationsCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Use the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin this Interactive StudyNotebook to help you in taking notes.C H A P T E R9NOTE-TAKING TIP: When you take notes, listen orread for main ideas. Then record those ideas in asimplified form for future reference.Glencoe Algebra 2 253Chapter9Fold in half along the width. On thefirst sheet, cut 5 cm along the fold atthe ends. On the second sheet, cut in thecenter, stopping 5 cm from the ends.Insert the first sheet through the secondsheet and align the folds. Label the pageswith lesson numbers.Begin with five sheets of notebook paper.
    • 254 Glencoe Algebra 2This is an alphabetical list of new vocabulary terms you will learn in Chapter 9.As you complete the study notes for the chapter, complete you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.C H A P T E R9BUILD YOUR VOCABULARYCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Vocabulary TermFoundon PageDefinitionDescription orExampleChange of BaseFormulacommon logarithm[LAW-guh-rih-thuhm]exponential decay[EHK-spuh-NEHN-chuhl]exponential equationexponential functionexponential growthexponential inequalitylogarithm
    • Chapter BRINGING IT ALL TOGETHER9Glencoe Algebra 2 255Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Vocabulary TermFoundon PageDefinitionDescription orExamplelogarithmic equationLAW-guh-RIHTH-mihklogarithmic functionLAW-guh-RIHTH-mihklogarithmic inequalityLAW-guh-RIHTH-mihknatural base, enatural baseexponential functionnatural logarithmnatural logarithmicfunctionrate of decayrate of growth
    • 256 Glencoe Algebra 29–1 Exponential FunctionsGraph an Exponential FunctionSketch the graph of y ‫؍‬ 4x. Then state the function’sdomain and range.Make a table of values. Connect thepoints to sketch a smooth curve.The domain is , and the range is.Check Your Progress Sketch the graph of y ϭ 3x. Thenstate the function’s domain and range.• Graph exponentialfunctions.• Solve exponentialequations andinequalities.MAIN IDEASREMEMBER ITPolynomialfunctions like y ϭ x2have a variable for thebase and a constant forthe exponent.Exponential functionslike y ϭ 2xhave aconstant for the baseand a variable for theexponent.xOyIn general, an equation of the form , wherea 0, b Ͼ 0, and b 1, is called an exponential functionwith base b.x y ϭ 4xϪ2Ϫ1012BUILD YOUR VOCABULARY (page 254)TEKS 2A.11 Thestudent formulatesequations andinequalities basedon exponential andlogarithmic functions,uses a variety of methodsto solve them, andanalyzes the solutionsin terms of the situation.(B) Use the parentfunctions to investigate,describe, and predictthe effects of parameterchanges on the graphsof exponential andlogarithmic functions,describe limitations onthe domains and ranges,and examine asymptoticbehavior. (F) Analyze asituation modeled byan exponential function,formulate an equation orinequality, and solve theproblem. Also addressesTEKS 2A.4(A), 2A.11(C),2A.11(D), and 2A.11(E).Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 9–1Glencoe Algebra 2 257Identify Exponential Growth and DecayDetermine whether each function representsexponential growth or decay.a. y ϭ 10(4_3)xThe function represents exponential, since the base, .isb. y ϭ (0.7)xThe function represents exponential, since the base, ,is .Check Your Progress Determine whether eachfunction represents exponential growth or decay.a. y ϭ 10(2_5)xb. y ϭ (0.5)xWrite an Exponential FunctionPHONES In 1995, there were an estimated 28,154,000cellular telephone subscribers in the United States. By2005, this estimated number had risen to 194,633,000.a. Write an exponential function of the form y ϭ abxthat could be used to model the number of cellulartelephone subscribers in the U.S. Write the functionin terms of x, the number of years since 1995.For 1995, the time x equals , and the initial number ofcellular telephone subscribers y is 28,154,000. Thus they-intercept, and the value of a, is .Exponential Growthand Decay• If a Ͼ 0 and b Ͼ 1,the function y ϭ abxrepresents exponentialgrowth.• If a Ͼ 0 and 0 Ͻ b Ͻ 1,the function y ϭ abxrepresents exponentialdecay.KEY CONCEPTExponential equations are equations in whichoccur as .Exponential inequalities are inequalities involvingfunctions.BUILD YOUR VOCABULARY (page 254)Use the page forLesson 9-1. Write yourown example of anexponential growth andan exponential decayfunction. Then sketch agraph of each function.ORGANIZE ITCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 9–1258 Glencoe Algebra 2For 2005, the time x equals 2005 Ϫ 1995 or .Substitute these values and the value of a into anexponential function to approximate the value of b.y ϭ abxExponential function194,633,000 ϭ 28,154,000b10Replace x with 10, y with194,633,000, and a with28,154,000.Ϸ b10Divide each side by28,154,000.Ϸ b Take the 10th root of eachside.To find the 10th root of , use selection 5:x√1 underthe MATH menu on the TI-83 Plus.Keystrokes: 10 MATH 5 6.91 ENTER 1.21324302926Answer: An equation that models the number ofcellular telephone subscribers in the U.S. from1995 to 2005 is .b. Suppose the number of cellular subscribers continuesto increase at the same rate. Estimate the number ofU.S. subscribers in 2015.For 2015, the time x equals 2015 Ϫ 1995 or .y ϭ 28,154,000(1.213)xModeling equationy ϭ 28,154,000(1.213) Replace x with .y ϭ Use a calculator.The number of cellular phone subscribers will be aboutin 2015.Check Your Progress In 1991, 4.9% of Americans haddiabetes. By 2000, this percent had risen to 7.3%.a. Write an exponential function of the form y ϭ abxthatcould be used to model the percentage of Americans withdiabetes. Write the function in terms of x, the number ofyears since 1991.b. Suppose the percent of Americans with diabetes continuesto increase at the same rate. Estimate the percent ofAmericans with diabetes in 2010.Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 9–1Glencoe Algebra 2 259Page(s):Exercises:HOMEWORKASSIGNMENTSolve Exponential EquationsSolve each equation.a. 49n Ϫ 2‫؍‬ 256.49n Ϫ 2ϭ 256 Original equation49n Ϫ 2ϭ 44Rewrite 256 as 44so eachside has the same base.ϭ Property of Equality forExponential FunctionsAdd 2 to each side.n ϭ Divide each side by 9.b. 35xϭ 92x Ϫ 135xϭ 92x Ϫ 1Original equation35xϭ 32(2x Ϫ 1)Rewrite 9 as 32so each side hasthe same base.5x ϭ (2x Ϫ 1) Property of Equality for ExponentialFunctions5x ϭ Distributive Propertyx ϭ Subtract 4x from each side.Solve Exponential InequalitiesSolve 53 Ϫ 2kϾ 1_625.53 Ϫ 2kϾ 1_625Original inequality53 Ϫ 2kϾ 5Ϫ4Rewrite 1_625as 1_54or 5Ϫ4.Ͼ Property of Inequality forExponential FunctionsϾ Subtract 3 from each side.k Ͻ Divide each side by Ϫ2.Check Your Progress Solve.a. 23x ϩ 1ϭ 32 b. 52xϭ 252x Ϫ 1c. 33 Ϫ 2kϾ 1_27Property of Equality forExponential FunctionsIf b is a positive numberother than 1, thenbxϭ byif and onlyif x ϭ y.Property of Inequality forExponential FunctionsIf b Ͼ 1, then bxϾ byif and only if x Ͼ y, andbxϽ byif and only ifx Ͻ y.KEY CONCEPTSCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 260 Glencoe Algebra 2Logarithms and Logarithmic Functions9–2Logarithmic to Exponential FormWrite each equation in exponential form.a. log3 9 ϭ 2log3 9 ϭ 2b. log101_100ϭ Ϫ2log101_100ϭ Ϫ2Exponential to Logarithmic FormWrite each equation in logarithmic form.a. 53ϭ 12553ϭ 125b. 271_3 ϭ 3271_3 ϭ 3Check Your ProgressWrite each equation in exponential form.a. log2 8 ϭ 3 b. log31_9ϭ Ϫ2Write each equation in logarithmic form.c. 34ϭ 81 d. 811_2 ϭ 9• Evaluate logarithmicexpressions.• Solve logarithmicequations andinequalities.MAIN IDEASLogarithm with Base bLet b and x be positivenumbers, b 1. Thelogarithm of x with baseb is denoted logb x andis defined as theexponent y that makesthe equation byϭ x true.KEY CONCEPTTEKS 2A.11 Thestudent formulatesequations andinequalities basedon exponential andlogarithmic functions, usesa variety of methods tosolve them, and analyzesthe solutions in termsof the situation. (A)Develop the definition oflogarithms by exploringand describing therelationship betweenexponential functionsand their inverses. (D)Determine solutionsof exponential andlogarithmic equationsusing graphs, tables,and algebraic methods.Also addresses TEKS2A.4(A) and 2A.11(C).Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 9–2Glencoe Algebra 2 261Evaluate Logarithmic ExpressionsEvaluate log3 243.log3 243 ϭ y Let the logarithm equal y.ϭ Definition of logarithmϭ 243 ϭ 35ϭ y Property of Equality forExponential FunctionsCheck Your Progress Evaluate log10 100.Solve a Logarithmic EquationSolve log8 n ϭ4_3.log8 n ϭ 4_3Original equationn ϭ Definition of logarithmn ϭ 8 ϭ 23n ϭ Power of a Powern ϭ Simplify.Check Your Progress Solve log27 n ϭ 2_3.Solve a Logarithmic InequalitySolve log6 x Ͼ 3.log6 x Ͼ 3 Original inequalityLogarithmic to exponentialinequalitySimplify.Logarithmic toExponential InequalityIf b Ͼ 1, x Ͼ 0, andlogb x Ͼ y, then x Ͼ by.If b Ͼ 1, x Ͼ 0, andlogb x Ͻ y, then0 Ͻ x Ͻ by.Property of Equality forLogarithmic FunctionsIf b is a positive numberother than 1, thenlogb x ϭ logb y if andonly if x ϭ y.Property of Inequalityfor LogarithmicFunctions If b Ͼ 1, thenlogb x Ͼ logb y if andonly if x Ͼ y, and logbx Ͻ logb y if and onlyif x Ͻ y.KEY CONCEPTSCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 9–2262 Glencoe Algebra 2Page(s):Exercises:HOMEWORKASSIGNMENTCheck Your Progress Solve log3 x Ͻ 2.Solve Equations with Logarithms on Each SideSolve log4 x2ϭ log4 (4x Ϫ 3).log4 x2ϭ log4 (4x Ϫ 3) Original equationx2ϭ Property of Equalityfor LogarithmicFunctionsx2Ϫ 4x ϩ 3 ϭ 0 Subtract 4x and add3 to each side.ϭ 0 Factor.ϭ 0 or ϭ 0Zero ProductPropertyx ϭ x ϭSolve eachequation.Solve Inequalities with Logarithms on Each SideSolve log7 (2x ϩ 8) Ͼ log7 (x ϩ 5).log7 (2x ϩ 8) Ͼ log7 (x ϩ 5) Original inequalityϾ Property of Inequality forLogarithmic Functionsx Ͼ Addition and SubtractionProperties of InequalitiesWe must exclude all values of x such that 2x ϩ 8 Յ 0 orx ϩ 5 Յ 0. Thus the solution set is , ,and . This compound inequality simplifies to.Check Your Progress Solve.a. log5 x2ϭ log5 (x ϩ 6)b. log5 (2x ϩ 6) Ͼ log3 (x ϩ 2)Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Glencoe Algebra 2 2639–3 Properties of LogarithmsUse the Product PropertyUse log5 2 Ϸ 0.4307 to approximate the value of log5 250.log5 250 ϭ log5 (53ؒ 2) Replace 250 with53ؒ 2.ϭ ϩ Product Propertyϭ ϩ Inverse Propertyof Exponents andLogarithmsϷ Replace log5 2.Use the Quotient PropertyUse log6 8 Ϸ 1.1606 and log6 32 Ϸ 1.9343 to approximatethe value of log6 4.log6 4 ϭ log6 (32_8 ) Replace 4 with 32_8.ϭ Ϫ Quotient PropertyϷ ϪorCheck Your Progressa. Use log2 3 Ϸ 1.5850 to approximate the value of log2 96.b. Use log5 4 Ϸ 0.8614 and log5 32 Ϸ 2.1534 to approximatethe value of log5 8.• Simplify and evaluateexpressions usingthe properties oflogarithms.• Solve logarithmicequations usingthe properties oflogarithms.MAIN IDEASProduct Propertyof Logarithms Thelogarithm of a productis the sum of thelogarithms of its factors.Quotient Propertyof Logarithms Thelogarithm of a quotientis the difference ofthe logarithms of thenumerator and thedenominator.KEY CONCEPTSCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.TEKS 2A.11 The student formulates equations and inequalities based on exponentialand logarithmic functions, uses a variety of methods to solve them, and analyzesthe solutions in terms of the situation. (D) Determine solutions of exponential andlogarithmic equations using graphs, tables, and algebraic methods.
    • 9–3264 Glencoe Algebra 2Power Property of LogarithmsGiven that log5 6 Ϸ 1.1133, approximate the value oflog5 216.log5 216 ϭ Replace 216 with 63.ϭ Power PropertyϷ Replace log5 6 with1.1133.Check Your Progress Given that log4 6 Ϸ 1.2925,approximate the value of log4 1296.Solve Equations Using Properties of LogarithmsSolve each equation.a. 4 log2 x Ϫ log2 5 ϭ log2 1254 log2 x Ϫ log2 5 ϭ log2 125 Original equationϪ log2 5 ϭ log2 125 Power Propertyϭ log2 125 Quotient Propertyϭ 125 Property of Equality forLogarithmic Functionsϭ Multiply each side by 5.x ϭ Take the 4th root ofeach side.Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Power Property ofLogarithms Thelogarithm of a poweris the product ofthe logarithm andthe exponent.Use the pagefor Lesson 9-3. Writeyour own examplesthat show the Product,Quotient, and PowerProperties of Logarithms.KEY CONCEPT
    • 9–3Page(s):Exercises:HOMEWORKASSIGNMENTGlencoe Algebra 2 265b. log8 x ϩ log8 (x Ϫ 12) ϭ 2log8 x ϩ log8 (x Ϫ 12) ϭ 2 Original equationlog8 [x(x Ϫ 12)] ϭ 2 Product Propertyx(x Ϫ 12) ϭ Definition of logarithmϭ 0Subtract 64 from eachside.ϭ 0 Factor.ϭ 0 or ϭ 0Zero ProductPropertyx ϭ x ϭSolve eachequation.Check Substitute each value into the original equation.log8 (Ϫ4) ϩ log8 [(Ϫ4) Ϫ 12] 2 Replace x with Ϫ4.log8 (Ϫ4) ϩ log8 ( ) 2Since log8 (Ϫ4) and log8 (Ϫ16) are , Ϫ4 isan extraneous solution and must be eliminated.log8 16 ϩ log8 (16 Ϫ 12) 2 Replace x with 16.log8 16 ϩ log8 4 2 16 Ϫ 12 ϭ 4log8 (16 и 4) 2 Product Propertylog8 2 16 и 4 ϭ 642 ϭ 2 ✓ Definition of logarithmThe only solution is .Check Your Progress Solve each equation.a. 2 log3 x Ϫ 2 log3 6 ϭ log3 4 b. log2 x ϩ log2 (x Ϫ 6) ϭ 4Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 266 Glencoe Algebra 29–4 Common LogarithmsFind Common LogarithmsUse a calculator to evaluate each expression to fourdecimal places.a. log 6 Keystrokes: LOG 6 ENTER .7781512504b. log 0.35 Keystrokes: LOG 0.35 ENTERϪ.4559319556Check Your Progress Use a calculator to evaluateeach expression to four decimal places.a. log 5 b. log 0.62Solve Logarithmic EquationsEARTHQUAKES Refer to Example 2 in your textbook.The San Fernando Valley earthquake of 1994 measured6.6 on the Richter scale. How much energy did thisearthquake release?log E ϭ 11.8 ϩ 1.5M Write the formula.log E ϭ 11.8 ϩ 1.5( ) Replace M with .log E ϭ Simplify.10 log E ϭ 1021.7Write each side using 10 asa base.E ϭ Inverse Property ofExponents and LogarithmsE ഠ Use a calculator.The amount of energy released was aboutergs.• Solve exponentialequations andinequalities usingcommon logarithms.• Evaluate logarithmicexpressions usingthe Change of BaseFormula.MAIN IDEASTEKS 2A.11 Thestudent formulatesequations andinequalities basedon exponential andlogarithmic functions, usesa variety of methods tosolve them, and analyzesthe solutions in termsof the situation. (C)Determine the reasonabledomain and range valuesof exponential andlogarithmic functions,as well as interpretand determine thereasonablenessof solutions toexponential andlogarithmic equationsand inequalities. (D)Determine solutionsof exponential andlogarithmic equationsusing graphs, tables, andalgebraic methods.Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 9–4Glencoe Algebra 2 267Check Your Progress The amount of energy E, in ergs,that an earthquake releases is related to its Richter scalemagnitude M by the equation log E ϭ 11.8 ϩ 1.5 M. In 1999an earthquake in Turkey measured 7.4 on the Richter scale.How much energy did this earthquake release?Solve Exponential Equations Using LogarithmsSolve 5xϭ 62.5xϭ 62 Original equationϭ Property of Equality forLogarithmic Functionsϭ Power Property of Logarithmsx ϭ Divide each side by log 5.x Ϸ Use a calculator.Check Your Progress Solve 3xϭ 17.Solve Exponential Inequalities Using LogarithmsSolve 27xϾ 35x Ϫ 3.27xϾ 35x Ϫ 3log 27xϾ log 35x Ϫ 3Ͼ7x log 2 Ͼ Ϫ7x log 2 Ϫ ϾREMEMBER ITWhen solving anexponential equationusing logarithms, thefirst step is oftenreferred to as taking thelogarithm of each side.Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 9–4268 Glencoe Algebra 2Page(s):Exercises:HOMEWORKASSIGNMENTϾ Ϫ3 log 3x ϽϪ3 log 3__7 log 2 Ϫ 5 log 3x ϽϪ3(0.4771)ᎏᎏᎏ7(0.3010) Ϫ 5(0.4771)x ϽCheck Your Progress Solve 53xϽ 10x Ϫ 2.Change of Base FormulaExpress log3 18 in terms of common logarithms.Then approximate its value to four decimal places.log3 18 ϭ Change of Base FormulaϷ Use a calculator.Check Your Progress Express log5 16 in terms of commonlogarithms. Then approximate its value to four decimal places.Change of Base FormulaFor all positive numbersa, b, and n, where a 1and b 1,loga n ϭlogb n_logb a.KEY CONCEPTCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Glencoe Algebra 2 269The number 2.71828 ... is referred toas the natural base, e.An function with base is calleda natural base exponential function.The with base is called thenatural logarithm.The natural logarithmic function, y ϭ In x, is theof the natural base exponential function,y ϭ ex.9–5 Base e and Natural LogarithmsEvaluate Natural Base ExpressionsUse a calculator to evaluate each expression to fourdecimal places.a. e0.5Keystrokes: 2nd [ex] 0.5 ENTER1.648721271b. eϪ8Keystrokes: 2nd [ex] Ϫ8 ENTER0.0003354626Check Your Progress Use a calculator to evaluateeach expression to four decimal places.a. e0.3b. eϪ2• Evaluate expressionsinvolving the naturalbase and naturallogarithms.• Solve exponentialequations andinequalities usingnatural logarithms.MAIN IDEAS(page 255)BUILD YOUR VOCABULARYTEKS 2A.11The studentformulatesequations and inequalitiesbased on exponentialand logarithmic functions,uses a variety of methodsto solve them, andanalyzes the solutionsin terms of the situation.(D) Determine solutionsof exponential andlogarithmic equationsusing graphs, tables,and algebraic methods.(F) Analyze a situationmodeled by anexponential function,formulate an equation orinequality, and solve theproblem. Also addressesTEKS 2A.11(A).Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 9–5270 Glencoe Algebra 2Evaluate Natural Logarithmic ExpressionsUse a calculator to evaluate each expression tofour decimal places.a. ln 3Keystrokes: LN 3 ENTER1.098612289b. ln1_4Keystrokes: LN 1 Ϭ 4 ENTERϪ1.386294361Check Your Progress Use a calculator to evaluateeach expression to four decimal places.a. ln 2 b. ln 1_2Write Equivalent ExpressionsWrite an equivalent exponential or logarithmicequation.a. exϭ 23exϭ 23 ϭϭb. ln x Ϸ 1.2528ln x Ϸ 1.2528 ϷϷCheck Your Progress Write an equivalentexponential or logarithmic equation.a. exϭ 6 b. ln x ϭ 2.25Write and evaluatethree logarithms, onewith base 2, one withbase e, and one withbase 10.WRITE ITCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 9–5Glencoe Algebra 2 271Solve Base e EquationsSolve 3eϪ2xϩ 4 ϭ 10.3eϪ2xϩ 4 ϭ 10 Original equationϭ Subtract 4 from each side.ϭ Divide each side by 3.ln eϪ2xϭ ln 2 Property of Equality for Logarithmsϭ Inverse Property of Exponentsand Logarithmsx ϭ Divide each side by Ϫ2.x Ϸ Use a calculator.Check Your Progress Solve 2eϪ2xϩ 5 ϭ 15.Solve Base e InequalitiesSAVINGS Suppose you deposit $700 into an accountpaying 3% annual interest, compounded continuously.a. What is the balance after 8 years?A ϭ PertContinuous compounding formulaϭ e(0.03)(8)Replace P with , r with 0.03,and t with 8.ϭ 700e0.24Simplify.Ϸ Use a calculator.The balance after 8 years would be .b. How long will it take for the balance in your accountto reach at least $1200?A Ն 1200 The balance is at least $1200.700e(0.03)tՆ 1200 Replace A with 700e(0.03)t.e(0.03)tՆ Divide each side by 700.ln e(0.03)tՆ ln 12_7Property of Inequality for LogarithmsՆ ln 12_7Inverse Property of Exponents andLogarithmst Նln 12_7_0.03Divide each side by 0.03.t Ն Use a calculator.It will take at least years for the balance to reach$1200.Use the page for Lesson9-5. On the same grid,sketch the graph fory ϭ exand y ϭ ln x.Then write how youcan tell that the twofunctions are inverses.ORGANIZE ITCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 9–5Page(s):Exercises:HOMEWORKASSIGNMENT272 Glencoe Algebra 2Check Your Progress Suppose you deposit $700 intoan account paying 6% annual interest, compoundedcontinously.a. What is the balance after 7 years?b. How long will it take for the balance in you account to reachat least $2500?Solve Natural Log Equations and InequalitiesSolve each equation or inequality.a. ln 3x ϭ 0.5ln 3x ϭ 0.5 Original equationeln 3xϭ e0.5Write each side using exponentsand base e.ϭ Inverse Property of Exponentsand Logarithmsx ϭ e0.5_3Divide each side by 3.x Ϸ Use a calculator.b. ln (2x Ϫ 3) Ͻ 2.5ln (2x Ϫ 3) Ͻ 2.5 Original inequalityeln(2x Ϫ 3)Ͻ e2.5Write each side using exponentsand base e.Ͻ Inverse Property of Exponentsand LogarithmsϽ Add 3 to each side.x Ͻ Divide each side by 2.x Ͻ Use a calculator.Check Your Progress Solve each equation orinequality.a. ln 2x ϭ 0.7 b. ln (x Ϫ 3) Ͼ 3Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Glencoe Algebra 2 2739–6Exponential Decay of the Form y ‫؍‬ a(1 ؊ r)tCAFFEINE A cup of coffee contains 130 milligrams ofcaffeine. If caffeine is eliminated from the body at arate of 11% per hour, how long will it take for 90% ofthis caffeine to be eliminated from a person’s body?y ϭ a(1 Ϫ r)tExponential decay formula13 ϭ 130(1 Ϫ 0.11)tReplace y with 13, a with130, and r with 0.11.ϭ Divide each side by 130.log ϭ log Property of Equality forLogarithmslog 0.10 ϭ Power Property forLogarithmsϭ t Divide each side bylog 0.89.Ϸ t Use a calculator.It will take approximately hours for 90% of thecaffeine to be eliminated from a person’s body.Check Your Progress Refer to Example 1. How long willit take for 80% of this caffeine to be eliminated from a person’sbody?Exponential Growth and DecayThe percent of decrease r is also referred to as therate of decay in the equation for exponential decay of theform y ϭ a(1 Ϫ r)t.BUILD YOUR VOCABULARY (page 255)• Use logarithms to solveproblems involvingexponential decay.• Use logarithms to solveproblems involvingexponential growth.MAIN IDEASTEKS 2A.11 Thestudent formulatesequations andinequalities basedon exponential andlogarithmic functions, usesa variety of methods tosolve them, and analyzesthe solutions in termsof the situation. (D)Determine solutionsof exponential andlogarithmic equationsusing graphs, tables,and algebraic methods.(F) Analyze a situationmodeled by anexponential function,formulate an equation orinequality, and solve theproblem.Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 9–6274 Glencoe Algebra 2Exponential Decay of the Form y ‫؍‬ ae ؊ktGEOLOGY The half-life of Sodium-22 is 2.6 years.a. What is the value of k and the equation of decay forSodium-22?y ϭ aeϪktExponential decay formula0.5a ϭ aeϪk(2.6)Replace y and t.0.5 ϭ eϪ2.6kDivide each side by a.ln ϭ ln Property of Equality forLogarithmic Functionsln 0.5 ϭ Inverse Property ofExponents and Logarithmsϭ k Divide each side by Ϫ2.6.ϭ k Use a calculator.The constant k for Sodium-22 is . The equationfor the decay of Sodium-22 is , where t isyears.b. A geologist examining a meteorite estimates thatit contains only about 10% as much Sodium-22 asit would have contained when it reached Earth’ssurface. How long ago did the meteorite reach thesurface of Earth?y ϭ aeϪ0.2666tDecay formula0.1a ϭ aeϪ0.2666tReplace y with 0.1a.0.1 ϭ eϪ0.2666tDivide each side by a.ln ϭ ln Property of Equality forLogarithmsln ϭ Inverse Propertyϭ t Divide each sideby Ϫ0.2666.Ϸ t Use a calculator.It was formed about years ago.Use the page for Lesson9-6. If an exponentialdecay problem involvesfinding a period of time,which model do youuse? Why?ORGANIZE ITCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 9–6Glencoe Algebra 2 275How can you tellwhether an exponentialequation involvesgrowth or decay?WRITE ITThe percent of increase r is also referred to as therate of growth in the equation for exponential growth ofthe form y ϭ a(1 ϩ r)t.BUILD YOUR VOCABULARY (page 255)Check Your Progress The half-life of radioactiveiodine used in medical studies is 8 hours.a. What is the value of k for radioactive iodine?b. A doctor wants to know when the amount of radioactiveiodine in a patient’s body is 20% of the original amount.When will this occur?Exponential Growth of the Form y ‫؍‬ a(1 ؉ r)tTEST EXAMPLE The population of a city of one millionis increasing at a rate of 3% per year. If the populationcontinues to grow at this rate, in how many years willthe population have doubled?A 4 years B 5 years C 20 years D 23 yearsy ϭ a(1 ϩ r)tGrowth formula2,000,000 ϭ 1,000,000(1 ϩ 0.03)tReplace y, a, and r.ϭ Divide each side by1,000,000.ln ϭ ln Property of Equalityfor Logarithmsln 2 ϭ Power Property ofLogarithmsln 2_ln 1.03ϭ t Divide.t Ϸ 23.45 Use a calculator.The population will have doubled in years.Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 9–6276 Glencoe Algebra 2Check Your Progress The population of a city of 10,000is increasing at a rate of 5% per year. If the populationcontinues to grow at this rate, in about how many years willthe population have doubled?A 10 years B 12 years C 14 years D 18 yearsExponential Growth of the Form y ϭ aektPOPULATION As of 2005, Nigeria had an estimatedpopulation of 129 million people, and the United Stateshad an estimated population of 296 million people.Assume that the populations of Nigeria and the UnitedStates can be modeled by N(t) ϭ 129e0.024tandU(t) ϭ 296e0.009t, respectively. According to thesemodels, when will Nigeria’s population be more thanthe population of the United States?You want to find t such that N(t) Ͼ U(t).N(t) Ͼ!U(t)129e0.024tϾ!296e0.009tReplace N(t) with129e0.024tand U(t)with 296e0.009t.ln 129e0.024tϾ!ln 296e0.009tProperty ofInequality forLogarithmsln 129 ϩ ln e0.024tϾ! ϩ ProductProperty ofLogarithmsln 129 ϩ 0.024t Ͼ! ϩ InverseProperty ofExponentsandLogarithmsln 129 Ͼ! t Subtract 296 and0.024t from eachside.Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Ͻ!t Divide each side byϪ0.015.Ͻ!t Use a calculator.After years or in , Nigeria’s population will begreater than the population of the U.S.Check Your Progress As of 2000, Saudi Arabia had anestimated population of 20.7 million people and the UnitedStates had an estimated population of 278 million people. Thegrowth of the populations of Saudi Arabia and the UnitedStates can be modeled by S(t) ϭ 20.7e0.0327tandU(t) ϭ 278e0.009t, respectively. According to these models, whenwill Saudi Arabia’s population be more than the population ofthe United States?9–6Page(s):Exercises:HOMEWORKASSIGNMENTGlencoe Algebra 2 277Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 278 Glencoe Algebra 2BRINGING IT ALL TOGETHERBUILD YOURVOCABULARYUse your Chapter 9 Foldable tohelp you study for your chaptertest.To make a crossword puzzle,word search, or jumblepuzzle of the vocabulary wordsin Chapter 9, go to:glencoe.comYou can use your completedVocabulary Builder(pages 254–255) to help yousolve the puzzle.VOCABULARYPUZZLEMAKERC H A P T E R9STUDY GUIDEDetermine whether each function represents exponentialgrowth or decay.1. y ϭ 0.2(3)x2. y ϭ 3(2_5)x3. y ϭ 0.4(1.01)4. Simplify 4xؒ 42x. 5. Solve 25y ϩ 3ϭ (1_5)y.6. What is the inverse of the function y ϭ 5x?7. What is the inverse of the function y ϭ log10 x?8. Evaluate log27 9. 9. Solve log8 x ϭ Ϫ1_3.9-1Exponential Functions9-2Logarithms and Logarithmic FunctionsCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Chapter BRINGING IT ALL TOGETHER9Glencoe Algebra 2 279State whether each of the following equations is trueor false. If the statement is true, name the property oflogarithms that is illustrated.10. log3 10 ϭ log3 30 Ϫ log3 311. log4 12 ϭ log4 4 ϩ log4 812. log2 81 ϭ 2log2 9Solve each equation.13. log5 14 Ϫ log5 (2x) ϭ log5 21 14. log2 x ϩ log2 (x ϩ 2) ϭ 3Match each expression from the first column with anexpression from the second column that has the same value.15. log2 216. log 1217. log3 118. log51_519. log 100020. Solve 62xϪ1ϭ 25x. Round to four decimal places.21. Express log8 5 in terms of common logarithms. Thenapproximate its value to four decimal places.9-3Properties of Logarithms9-4Common Logarithmsa. log4 1b. log2 8c. log10 12d. log5 5e. log 0.1Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Chapter BRINGING IT ALL TOGETHER9280 Glencoe Algebra 2Match each expression from the first column with its valuein the second column. Some choices may be used more thanonce or not at all.22. eln 523. ln 124. eln e25. ln e526. ln e27. ln (1_e )28. Solve ln (x Ϫ 8) ϭ 5. Round tofour decimal places.29. Evaluate eln 5.2.State whether each equation represents exponentialgrowth or decay.30. y ϭ 5e0.15t31. y ϭ 1000(1 Ϫ 0.05)t32. y ϭ 0.3eϪ1200t33. y ϭ 2(1 ϩ 0.0001)t34. Leroy bought a lawn mower for $1,200. It is expected todepreciate at a rate of 20% per year. What will be the value ofthe lawn mower in 5 years?35. The population of a school has increased at a steady rate eachyear from 375 students to 580 students in 8 years. Find theannual rate of growth.9-5Base e and Natural Logarithms9-6Exponential Growth and DecayI. 1II. 10III. Ϫ1IV. 5V. 0VI. eCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Glencoe Algebra 2 281Check the one that applies. Suggestions to help you study aregiven with each item.ARE YOU READY FORTHE CHAPTER TEST?I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.• You may want to take the Chapter 9 Practice Test onpage 557 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 9 Study Guide and Reviewon pages 552–556 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 9 Practice Test onpage 557 of your textbook.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 9 Foldable.• Then complete the Chapter 9 Study Guide and Review onpages 552–556 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 9 Practice Test onpage 557 of your textbook.Visit glencoe.com to accessyour textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 9.Student Signature Parent/Guardian SignatureTeacher SignatureC H A P T E R9ChecklistCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.282 Glencoe Algebra 2Use the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin this Interactive StudyNotebook to help you in taking notes.C H A P T E R10NOTE-TAKING TIP: When you take notes, includepersonal experiences that relate to the lesson andways in which what you have learned will be usedin your daily life.Cut each sheet ofgrid paper in halflengthwise. Cut thesheet of constructionpaper in half lengthwiseto form a front andback cover for thebooklet of grid paper.Staple all the sheetstogether to form along, thin notepadof grid paper.Begin with five sheets of notebook paper.Conic Sections
    • Glencoe Algebra 2 283Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Vocabulary TermFoundon PageDefinitionDescription orExampleasymptoteA-suhm(p)-tohtcentercircleconic sectionconjugate axisKAHN-jih-guhtdirectrixduh-REHK-trihksellipseih-LIHPSfociThis is an alphabetical list of new vocabulary terms you will learn in Chapter 10.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.C H A P T E R10BUILD YOUR VOCABULARYChapter10(continued on the next page)
    • Vocabulary TermFoundon PageDefinitionDescription orExamplefocusFOH-kuhshyperbolahy-PUHR-buh-luhlatus rectumLA-tuhs REHK-tuhmmajor axisminor axisparabolapuh-RA-buh-luhtransverse axisvertex284 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Chapter BUILD YOUR VOCABULARY10
    • Glencoe Algebra 2 285Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.10–1 Midpoint and Distance Formulas• Find the midpoint ofa segment on thecoordinate plane.• Find the distancebetween two points onthe coordinate plane.MAIN IDEASMidpoint Formula If a linesegment has endpoints at(x1, y1) and (x2, y2), thenthe midpoint of thesegment has coordinates΂x1 + x2ᎏ2,y1 + y2ᎏ2 ΃.Distance Formula Thedistance between twopoints with coordinates(x1, y1) and (x2, y2) isgiven byd ϭ √(x2 Ϫ x1)2ϩ (y2 Ϫ y1)2KEY CONCEPTSFind a MidpointCOMPUTERS A graphing program draws a line segmenton a computer screen so that its endpoints are at (5, 2)and (7, 8). What are the coordinates of its midpoint?΂x1 ϩ x2_2,y1 ϩ y2_2 ΃ ϭ ΂ , ΃ϭ ΂ , ΃ orThe coordinates of the midpoint areFind the Distance Between Two PointsFind the distance between P(Ϫ1, 4) and Q(2, Ϫ3).d ϭ √(x2 Ϫ x1)2ϩ (y2 Ϫ y1)2Distance Formulad ϭ Let (x1, y1) ϭ (Ϫ1, 4)and (x2, y2) ϭ (2, Ϫ3).d ϭ Subtract.d ϭ or Simplify.The distance between the two points is units.Check Your Progressa. Find the midpoint of the segment with endpoints at (3, 6)and (Ϫ1, Ϫ8).b. What is the distance between P(2, 3) and Q(Ϫ3, 1)?2 22 2TEKS 2A.5 The student knows the relationship between the geometric and algebraicdescriptions of conic sections. (B) Sketch graphs of conic sections to relate simpleparameter changes in the equation to corresponding changes in the graph.
    • Page(s):Exercises:HOMEWORKASSIGNMENTTEST EXAMPLE A coordinate grid is placed over ascale drawing of Jenny’s patio. A grill is located at(2, Ϫ3). A flowerpot is located at (Ϫ6, Ϫ1). A picnic tableis at the midpoint between the grill and the flowerpot.In coordinate units, about how far is it from the grill tothe picnic table?A 0.2 B 2.24 C 4.1 D 5Read the Test ItemThe question asks us to find the distance between the grill and themidpoint. Find the midpoint and then use the Distance Formula.Solve the Test ItemUse the Midpoint Formula to find the coordinates of the picnictable.midpoint ϭ (2 ϩ (Ϫ6)_2,(Ϫ3) ϩ (Ϫ1)_2 ) Midpoint Formulaϭ Simplify.Use the Distance Formula to find the distance between the grilland the picnic table .distance ϭ √[(Ϫ2) Ϫ 2]2ϩ [Ϫ2 Ϫ (Ϫ3)]2Distance Formulaϭ √( )2ϩ ( )2Subtract.ϭ or about Simplify.The answer is .Check Your Progress A coordinate grid is placed over afloor plan of the school. The gymnasium is located at (4, 6).The science lab is located at (Ϫ2, Ϫ4). The office is located atthe midpoint between the gymnasium and the science lab.In coordinate units, about how far is the office from thescience lab?A 1.4 B 2.8 C 9.1 D 10.310–1286 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • The graph of an equation of the form, a 0,is a parabola.Any figure that can be obtained by slicing ais called a conic section.A parabola can also be defined as the set of all points in aplane that are the same from a given pointcalled the focus and a given line called the directrix.Glencoe Algebra 2 287Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.10–2 ParabolasAnalyze the Equation of a ParabolaWrite y ϭ Ϫx2Ϫ 2x ϩ 3 in standard form. Identify thevertex, axis of symmetry, and direction of opening ofthe parabola.y ϭ Ϫx2Ϫ 2x ϩ 3 Original equationy ϭ Ϫ1(x2ϩ 2x) ϩ 3 Factor.y ϭ Ϫ1(x2ϩ 2x ϩ ᮀ) ϩ 3 Ϫ (Ϫ1)(ᮀ) Complete thesquare.y ϭ Ϫ1΂x2ϩ 2x ϩ ΃ ϩ 3 ϩ 1΂ ΃ Multiplyby Ϫ1.y ϭy ϭ (h, k) ϭ (Ϫ1, 4)The vertex of this parabola is located at , and theequation of the axis of symmetry is x ϭ . The parabolaopens downward.• Write equations ofparabolas in standardform.• Graph parabolas.MAIN IDEASEquation of a ParabolaThe standard form of theequation of a parabolawith vertex (h, k) andaxis of symmetry x ϭ his y ϭ a(x Ϫ h)2ϩ k.• If a Ͼ 0, k is theminimum value of therelated function andthe parabola opensupward.• If a Ͻ 0, k is themaximum value of therelated function andthe parabola opensdownward.KEY CONCEPT(pages 283–284)BUILD YOUR VOCABULARYTEKS 2A.5The studentknows therelationship between thegeometric and algebraicdescriptions of conicsections. (A) Describea conic section asthe intersection of aplane and a cone. (B)Sketch graphs of conicsections to relate simpleparameter changesin the equation tocorresponding changesin the graph. (C) Identifysymmetries from graphsof conic sections. Alsoaddresses TEKS 2A.5(E),2A.7(A), and 2A.7(B).
    • 10–2288 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Check Your Progress Write y ϭ 2x2ϩ 4x ϩ 5 in standardform. Identify the vertex, axis of symmetry, and direction ofopening of the parabola.Graph Parabolasa. Graph y ϭ 2x2.For this equation, h ϭ 0 and k ϭ 0.The vertex is at the origin. Substitutepositive integers for x and find thecorresponding y-values.Since the graph is symmetric aboutthe y-axis, the points at ,, and are alsoon the parabola. Use all of thesepoints to draw the graph.b. Graph y ϭ 2(x Ϫ 1)2Ϫ 5.The equation is of the formy ϭ a(x Ϫ h)2ϩ k, where h ϭ 1and k ϭ Ϫ5. The graph of thisequation is the graph of y ϭ 2x2in part a translated unitright and units down.The vertex is now at .Check Your Progress Graph each equation.a. y ϭ Ϫ3x2b. y ϭ Ϫ3(x ϩ 1)2ϩ 3xyOxyOx y1 22 83 18
    • 10–2Glencoe Algebra 2289Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Graph an Equation Not in Standard FormGraph x ϩ y2ϭ 4y Ϫ 1.First write the equation in the form x ϭ a(y Ϫ k)2ϩ h.x ϩ y2ϭ 4y Ϫ 1x ϭ Ϫy2ϩ 4y Ϫ 1x ϭ Ϫ1΂ ΃ Ϫ 1x ϭ Ϫ1(y2Ϫ 4y ϩ ᮀ) Ϫ 1 Ϫ (Ϫ1)(ᮀ)x ϭ Ϫ1΂y2Ϫ 4y ϩ ΃ Ϫ 1 ϩ 1΂ ΃x ϭThen use the following information to draw the graph.vertex:axis of symmetry: y ϭfocus: (3 ϩ 1_4(Ϫ1), 2) or (11_4, 2)directrix: x ϭ or 31_4direction of opening: left, since a Ͻ 0length of the latus rectum: or unitCheck Your Progress Graph x Ϫ y2ϭ 6y ϩ 2.yxO(114, 2)(3, 2)x ϭ 3 14y ϭ 2Use the page forVocabulary. Makea sketch of eachvocabulary term inthis lesson.ORGANIZE ITThe line segment through the focus of a parabola andperpendicular to the axis of is called thelatus rectum.BUILD YOUR VOCABULARY (page 284)
    • Write and Graph an Equation for a ParabolaBRIDGES The 52-meter-long Hulme Arch Bridge inManchester, England, is supported by cables suspendedfrom a parabolic steel arch. The highest point of thearch is 25 meters above the bridge, and the focus of thearch is about 18 meters above the bridge.a. Let the bridge be the x-axis, and let the y-axis passthrough the arch’s vertex. Write an equation thatmodels the arch.The vertex is at (0, 25), so h ϭ and k ϭ .The focus is at (0, 12). Use the focus to find a.18 ϭ 25 ϩ 1_4ak ϭ 25; The focus y-coordinate is 18.ϭ 1_4aSubtract 25 from each side.ϭ 1 Multiply each side by 4a.a ϭ Divide each side by Ϫ28.An equation of the parabola is .b. Graph the equation.The length of the latus rectum is 1_a orunits, so the graph must passthrough (Ϫ14, 18) and (14, 18).According to the length of the bridge,the graph must pass through thepoints (Ϫ26, 0) and (26, 0). Use thesepoints and the information from parta to draw the graph.Check Your Progress The water stream follows aparabolic path. The highest point of the water streamis 61_4feet above the ground and the water hits theground 10 feet from the jet. The focus of the fountainis 51_2feet above the ground.a. Write an equation that modelsthe path of the water fountain.b. Graph the equation.10–2Page(s):Exercises:HOMEWORKASSIGNMENT290 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Glencoe Algebra 2 291Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.10–3 CirclesWrite an Equation Given the Center and RadiusLANDSCAPING The plan for a park puts the center ofa circular pond, of radius 0.6 miles, 2.5 miles east and3.8 miles south of the park headquarters. Write anequation to represent the border of the pond, usingthe headquarters as the origin.Since the headquarters is at , the center of the pondis at with radius 0.6 mile.(x Ϫ h)2ϩ (y Ϫ k)2ϭ r2Equation of a circle΂x Ϫ ΃2ϩ ΂y ϩ ΃2ϭ2(h, k) ϭ (2.5, Ϫ3.8),r ϭ 0.6(x Ϫ 2.5)2ϩ (y ϩ 3.8)2ϭ Simplify.The equation isCheck Your Progress The plan for a park puts the centerof a circular pond, of radius 0.5 mile, 3.5 miles west and 2.6miles north of the park headquarters. Write an equation torepresent the border of the pond, using the headquarters asthe origin.• Write equations ofcircles.• Graph circles.MAIN IDEAS A circle is the set of all points in a plane that arefrom a given in the plane,called the center.A line that the circle in exactlypoint is said to be tangent to the circle.Equation of a CircleThe equation of acircle with center (h, k)and radius r units is(x Ϫ h)2ϩ (y Ϫ k)2ϭ r2On the pagefor Circles, write thisequation. Be sure toexplain h, k, and r.KEY CONCEPTBUILD YOUR VOCABULARY (pages 283–284)TEKS 2A.5The studentknows therelationship betweenthe geometric andalgebraic descriptionsof conic sections. (B)Sketch graphs of conicsections to relate simpleparameter changesin the equation tocorresponding changesin the graph. (C) Identifysymmetries from graphsof conic sections.
    • 10–3292 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Write an Equation Given a DiameterWrite an equation for a circle if the endpoints of thediameter are at (2, 8) and (2, Ϫ2).First, find the center of the circle.(h, k) ϭ ΂x1 ϩ x2_2,y1 ϩ y2_2 ΃ Midpoint Formulaϭ ΂ , ΃ (x1, y1) ϭ (2, 8),(x2, y2) ϭ (2, Ϫ2)ϭ ΂ , ΃ Add.ϭ ΂ ΃ Simplify.Now find the radius.r ϭ √(x2 Ϫ x1)2ϩ (y2 Ϫ y1)2Distance Formular ϭ (x1, y1) ϭ (2, 8),(x2, y2) ϭ (2, 3)r ϭ Subtract.r ϭ or Simplify.The radius of the circle is units, so r2ϭ .Substitute h, k, and r2into the standard form ofthe equation of a circle. An equation of the circle isCheck Your Progress Write an equation for a circle if theendpoints of the diameter are at (3, 5) and (3, Ϫ7).2 2
    • 10–3Page(s):Exercises:HOMEWORKASSIGNMENTGlencoe Algebra 2 293Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Graph an Equation in Standard FormFind the center and radius of thecircle with equation x2ϩ y2ϭ 16.Then graph the circle.The center is at , and theradius is .The table lists some valuesfor x and y that satisfy theequation.Since the circle is centered atthe origin, it is symmetric aboutthe y-axis. Use these points andthe concept of symmetry tograph x2ϩ y2ϭ 16.Graph an Equation Not in Standard FormFind the center and radius of the circle with equationx2ϩ y2ϩ 6x Ϫ 7 ϭ 0. Then graph the circle.Complete the square.x2ϩ 6x ϩ ᮀ ϩ y2ϭ 7x2ϩ 6x ϩ ϩ y2ϭ 7 ϩϩ y2ϭThe center is at , and the radius is .Check Your Progress Graph each equation.a. x2ϩ y2ϭ 9 b. x2ϩ y2ϩ 8x Ϫ 4y ϩ 11 ϭ 0What must you do tothe equation of a circleif you want to graph thecircle on a calculator?WRITE ITxyOyxO(–3, 0)x y0 41 3.92 3.53 2.64 0
    • An ellipse is the set of all points in a plane such that thesum of the from two fixedis constant. The two fixed points are called the foci ofthe ellipse.The points at which the ellipse intersects its axes ofsymmetry determine two withendpoints on the ellipse called the major axis and theminor axis. The axes intersect at the center of the ellipse.10–4294 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.EllipsesWrite an Equation for a GraphWrite an equation for the ellipse.Find the values of a and b for theellipse. The length of the major axis ofany ellipse is 2a units. In this ellipse,the length of the major axis is thedistance between (0, 5) and (0, Ϫ5).10 units.2a ϭ Length of major axis ϭ 102a ϭ Divide each side by 2.The foci are located at (0, 4) and (0, Ϫ4), so c ϭ 4.Use the relationship between a, b, and c to find b.c2ϭ a2Ϫ b2Equation relating a, b, and cϭ c ϭ 4 and a ϭ 5b2ϭ Solve for b2.Since the major axis is vertical, substitute 25 for a2and 9 forb2in the formy2_a2ϩ x2_b2ϭ 1. An equation of the ellipse is.BUILD YOUR VOCABULARY (pages 283–284)xyO(0, 5)(0, 4)(0, –4)(0, –5)• Write equations ofellipses.• Graph ellipses.MAIN IDEASWrite the name for theendpoints of each axisof an ellipse.WRITE ITTEKS 2A.5 The student knows the relationship between the geometric and algebraicdescriptions of conic sections. (B) Sketch graphs of conic sections to relate simpleparameter changes in the equation to corresponding changes in the graph. (C)Identify symmetries from graphs of conic sections. Also addresses TEKS 2A.5(E).
    • 10–4Glencoe Algebra 2 295Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Write an Equation Given the Lengths of the AxesSOUND A listener is standing in an elliptical room150 feet wide and 320 feet long. When a speakerstands at one focus and whispers, the best placefor the listener to stand is at the other focus.a. Write an equation to model this ellipse, assumingthe major axis is horizontal and the center is at theorigin.The length of the major axis is 320 feet.2a ϭ 320 Length of major axis ϭ 3202a ϭ Divide each side by 2.The length of the minor axis is 150 feet.2b ϭ 150 Length of minor axis ϭ 1502b ϭ Divide each side by 2.Substitute a ϭ and b ϭ into the formx2_a2ϩy2_b2ϭ 1.An equation for the ellipses is .b. How far apart should the speaker and the listener bein this room?The two people should stand at the foci of the ellipse. Thedistance between the two foci is units.Check Your Progress Write an equation for the ellipseshown.yxO(0, –3)(0, 3)(0, 5)(0, –5)
    • c2ϭ a2Ϫ b2Equation relatinga, b, and c.c ϭ Take the square root ofeach side.c ϭ √a2Ϫ b2Multiply each side by 2.2c ϭ 2 Substitute a ϭ 160and b ϭ 75.2c ≈ Use a calculator.The points where two people should stand to hear eachother whisper are about feet apart.Check Your Progress A listener is standing in anelliptical room 60 feet wide and 120 feet long. When aspeaker stands at one focus and whispers, the best placefor the listener to stand is at the other focus.a. Write an equation to model this ellipse, assuming the majoraxis is horizontal and the center is at the origin.b. How far apart should the speaker and the listener be in thisroom?Graph an Equation in Standard FormFind the coordinates of the center and foci and thelengths of the major and minor axes of the ellipse withequation x2_36ϩy2_9ϭ 1. Graph the ellipse.The center of this ellipse is at (0, 0).Since a2ϭ 36, a ϭ and since b2ϭ 9, b ϭ .The length of the major axis is 2(6) or units, and thelength of the minor axis is 2(3) or . Since the x2term hasthe greatest denominator, the major axis is .c2ϭ a2Ϫ b2Equation relating a, b, and c.c2ϭ Ϫ a ϭ 6 and b ϭ 3c ϭ or Take a square root ofeach side.The foci are at and .10–4296 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.΂ ΃2؊ ΂ ΃2
    • 10–4Glencoe Algebra 2 297Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.You can use a calculator to find someapproximate nonnegative values for x and ythat satisfy the equation.Since the ellipse is centered at the origin, itis symmetric about the y-axis. So the pointsat and lie onthe graph.The ellipse is also symmetric about thex-axis, so the points at andalso lie on the graph.Graph the intercepts , , ,and , and draw the ellipse that passes throughthem and the other points. center: (0, 0); foci:(3√3, 0),(Ϫ3√3, 0); major axis: 12; minor axis: 6Check Your Progress Find the coordinates of the centerand foci and the lengths of the major and minor axes of theellipse with equation x2_25ϩy2_4ϭ 1. Then graph the equation.Use the page forEllipses. Make a sketchof each vocabulary termin this lesson.ORGANIZE ITx y0 31 2.962 2.833 2.604 2.245 1.666 0
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.10–4298 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Graph an Equation Not in Standard FormFind the coordinates of the center and foci and thelengths of the major and minor axes of the ellipse withequation x2ϩ 4y2Ϫ 6x Ϫ 16y Ϫ 11 ϭ 0. Graph the ellipse.Complete the square to write in standard form.x2ϩ 4y2Ϫ 6x Ϫ 16y Ϫ 11 ϭ 0x2Ϫ 6x ϩ ᮀ ϩ 4(y2Ϫ 4y ϩ ᮀ) ϭ 11 ϩ ᮀ ϩ 4(ᮀ)΂x2Ϫ 6x ϩ ΃ϩ 4΂y2Ϫ 4y ϩ ΃ϭ 11 ϩ 9 ϩ 4΂ ΃ϭ 1The center is , and the fociare located atand . Thelength of the major axis isunits, and the length of the minor axis is .Check Your Progress Find the coordinates of the centerand foci and the lengths of the major and minor axes of theellipse with equation 4x2ϩ 25y2ϩ 16x Ϫ 150y ϩ 141 ϭ 0. Graphthe ellipse.xyOPage(s):Exercises:HOMEWORKASSIGNMENT
    • Write an Equation for a GraphWrite an equation for the hyperbola shown.The center is .The value of a is the distance from thecenter to a vertex, or units. Thevalue of c is the distance from thecenter to a focus, or units.c2ϭ a2ϩ b2Equation relating a, b, and cϭ c ϭ 4, a ϭ 2ϭ Evaluate the squares.12 ϭ b2Solve for b2.Since the transverse axis is vertical, the equation is of theformy2_a2Ϫ x2_b2ϭ 1. Substitute the values for a2and b2.An equation of the hyperbola is .Check Your ProgressyxO(3, 0) (4, 0)(–3, 0)(–4, 0)Write an equation for the hyperbola.Graph an Equation in Standard FormFind the coordinates of the vertices and foci and theequations of the asymptotes for the hyperbola withequation x2Ϫ y2ϭ 1. Graph the hyperbola.The center of the hyperbola is at the origin. According to theequation, a2ϭ 1 and b2ϭ 1, a ϭ 1 and b ϭ 1. The coordinatesof the vertices are and .xyO(0, 4)(0, 2)(0, –4)(0, –2)• Write equations ofhyperbolas.• Graph hyperbolas.MAIN IDEASKEY CONCEPTEquations of Hyperbolaswith Centers at theOriginHorizontal TransverseAxis x2_a2Ϫy2_b2ϭ 1Vertical TransverseAxisy2_a2Ϫ x2_b2ϭ 1Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.10–5Glencoe Algebra 2 299Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.HyperbolasTEKS 2A.5 The student knows the relationship between the geometric and algebraicdescriptions of conic sections. (B) Sketch graphs of conic sections to relate simpleparameter changes in the equation to corresponding changes in the graph. (C)Identify symmetries from graphs of conic sections. Also addresses TEKS 2A.5(E).
    • c2ϭ a2ϩ b2Equation relating a, b, and c for a hyperbola.c2ϭ2ϩ2a ϭ 1, b ϭ 1c2ϭ 2 or c ϭ Simplify and take the square root.The equations of the asymptotes are y ϭ Ϯ b_ax or y ϭ Ϯx.Use a calculator to find some approximatenonnegative values for x and y that satisfythe equation. Since the hyperbola iscentered at the origin, it is symmetric aboutthe y-axis. Therefore, the points at (Ϫ5, 4.9),, (Ϫ3, 2.8), (Ϫ2, 1.7), andlie on the graph.The hyperbola is also symmetric about the x-axis, so the pointsat (Ϫ5, Ϫ4.9), (Ϫ4, 3.9), , (Ϫ2, Ϫ1.7), (3, Ϫ2.8),, and (5, Ϫ4.9) also lie on the graph.Draw a 2-unit by 2-unit square. Theasymptotes contain the diagonals of thesquare. Graph the vertices, which, inthis case, are the x-intercepts. Use theasymptotes as a guide to draw thehyperbola that passes through thevertices and the other points. Thegraph intersect theasymptotes.Check Your Progress Find the vertices and foci and theequations of the asymptotes for the hyperbola with equationx2_4 Ϫ y2ϭ 1. Then graph the hyperbola.10–5300 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.x y1 02 1.73 2.84 3.95 4.9
    • 10–5Page(s):Exercises:HOMEWORKASSIGNMENTGlencoe Algebra 2 301Under the page forHyperbolas, describetwo similarities betweenhyperbolas and ellipses.ORGANIZE ITCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Graph an Equation Not in Standard FormFind the coordinates of the vertices and foci and theequations of the asymptotes for the hyperbola withequation x2Ϫ y2ϩ 6x ϩ 10y Ϫ 17 ϭ 0. Then graph thehyperbola.Complete the square for each variable.x2Ϫ y2ϩ 6x ϩ 10y Ϫ 17 ϭ 0x2Ϫ 6x ϩ ᮀ Ϫ 1(y2Ϫ 10y ϩ ᮀ) ϭ 17 ϩ ᮀ Ϫ 1(ᮀ)x2ϩ 6x ϩ Ϫ 1΂y2Ϫ 10y ϩ ΃ϭ 17 ϩ Ϫ 1(25)Ϫ ϭThe vertices are and, and the foci are(√2 Ϫ 3, 5) and (Ϫ√2 Ϫ 3, 5).The equations of the asymptotes areor y ϭand y ϭ Ϫx ϩ 2.Check Your Progress Find the coordinates of the verticesand foci and the equations of the asymptotes for the hyperbolawith equation 9x2Ϫ 16y2Ϫ 72x Ϫ 64y ϩ 224 ϭ 0. Then graphthe hyperbola.xyO
    • 10–6302 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Conic SectionsRewrite an Equation of a Conic SectionWrite the equation y2ϭ 18 Ϫ 2x2in standard form. Statewhether the graph of the equation is a parabola, circle,ellipse, or hyperbola. Graph the equation.Write the equation in standard form.y2ϭ 18 Ϫ 2x2ϭ Isolate termsϭDivide eachside by .The graph is an ellipse with center at .Check Your Progress Write x2ϩ y2Ϫ 6x Ϫ 7 ϭ 0 instandard form. State whether the graph of the equation is aparabola, circle, ellipse, or hyperbola. Then graph the equation.Analyze an Equation of a Conic SectionWithout writing the equation in standard form, statewhether the graph of the equation is a parabola, circle,ellipse, or hyperbola.a. 3y2؊ x2؊ 9 ‫؍‬ 0A ϭ and C ϭSince ,the graph is a .• Write equations ofconic sections instandard form• Identify conic sectionsfrom their equations.MAIN IDEASxyOREMEMBER ITIf a ϭ b in theequation for an ellipse,the graph of theequation is a circle.TEKS 2A.5 The student knows the relationship between the geometric and algebraicdescriptions of conic sections. (D) Identify the conic section from a given equation.
    • 10–6Page(s):Exercises:HOMEWORKASSIGNMENTGlencoe Algebra 2 303Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.b. 2x2؉ 2y2؉ 16x ؊ 20y ‫؍‬ ؊32A ϭ and C ϭSince , the graph is a .c. y2؊ 2x ؊ 4y ؉ 10 ‫؍‬ 0A ϭ and C ϭSince , this graph is a .Check Your Progress Without writing the equation instandard form, state whether the graph of the equationis a parabola, circle, ellipse, or hyperbola.a. 2y2Ϫ x2ϩ 16 ϭ 0b. 3x2ϩ y2ϩ 15x Ϫ 21y ϭ Ϫ11c. y2Ϫ 3x ϩ 2y Ϫ 10 ϭ 0Under the page forConic Sections, sketchand label each of theconic sections. Thenwrite the standard formon the conic sectionbelow each label.ORGANIZE IT
    • 10–7304 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Solving Quadratic SystemsLinear-Quadratic SystemSolve the system of equations.4x2Ϫ 16y2ϭ 252y ϩ x ϭ 2You can use a graphing calculator to help visualize therelationships of the graphs of the equations and predict thenumber of solutions.Solve each equation for y to obtainy ϭ Ϯ√4x2Ϫ 25__4andy ϭ .Enter the functions on the Yϭ screen. The graph indicatesthat the hyperbola and the line intersect in one point. So, thesystem has one solution.Use substitution to solve the system.First, rewrite 2y ϩ x ϭ 2 as x ϭ 2 Ϫ 2y.4x2Ϫ 16y2ϭ 25 First equation in the system4΂ ΃2Ϫ 16y2ϭ 25 Substitute for x.ϩ 16 ϭ 25 Simplify.ϭ Subtract from each side.y ϭ Divide each side by .Now solve for x.x ϭ 2 Ϫ 2y Equation for x in terms of yx ϭ 2 Ϫ 2΂ ΃ Substitute the y-value.x ϭ Simplify.The solution is ΂ ΃.• Solve systems ofquadratic equationsalgebraically andgraphically.• Solve systems ofquadratic inequalitiesgraphically.MAIN IDEASBy the Square RootProperty, for any realnumber n, if x2= n,then x = ?(Lesson 5-4)REVIEW ITTEKS 2A.3 Thestudent formulatessystems ofequations and inequalitiesfrom problem situations,uses a variety of methodsto solve them, andanalyzes the solutions interms of the situations.(B) Use algebraicmethods, graphs, tables,or matrices, to solvesystems of equations orinequalities.2A.5 The studentknows the relationshipbetween the geometricand algebraic descriptionsof conic sections. (B)Sketch graphs of conicsections to relate simpleparameter changesin the equation tocorresponding changesin the graph.
    • 10–7Glencoe Algebra 2 305Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Check Your Progress Solve x2Ϫ y2ϭ 4 and 2y ϩ x ϭ 2.Quadratic-Quadratic SystemSolve the system of equations.x2ϩ y2ϭ 164x2ϩ y2ϭ 23A graphing calculator indicates that thecircle and ellipse intersect in four points.So, this system has four solutions.Use the elimination method to solve.Ϫx2Ϫ y2ϭ Ϫ16 Rewrite the first original equation.(ϩ) 4x2ϩ y2ϭ 23 Second original equationϭ 7 Add.ϭ Divide each side by 3.ϭ Ϯ Take the square root of eachside.Substitute 7_3and Ϫ 7_3in either of the original equationsand solve for y.x2ϩ y2ϭ 16 x2ϩ y2ϭ 16 Original equation(√7_3 )2ϩ y2ϭ 16 (Ϫ√7_3 )2ϩ y2ϭ 16 Substitute for x.y2ϭ y2ϭ Subtract fromeach sidey ϭ y ϭTake the squareroot of each side.The solutions are , ,, and .REMEMBER ITWhen graphingconic sections, pressZOOM 5. This windowgives the graphs a morerealistic look.
    • Page(s):Exercises:HOMEWORKASSIGNMENT10–7306 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Check Your Progress Solve x2ϩ y2ϭ 10 and3x2ϩ y2ϭ 28.System of Quadratic InequalitiesSolve the system of inequalities by graphing.y Ն x2ϩ 1x2ϩ y2Յ 9The graph of y Ն x2ϩ 1 is they ϭ x2ϩ 1 and theregion inside and above it. Shadethe region dark gray.The graph of x2ϩ y2Յ 9 is theinterior of x2ϩ y2ϭ 9.Shade the region medium gray.The intersection of these regions, shaded darker gray,represents the solution of the system of inequalities.Check Your Progress Solve y Ͻ Ϫx2ϩ 1 and x2ϩ y2Յ 4by graphing.xyOOn the page forQuadratic Systems,sketch five graphs ofsystems of quadraticequations. Write thenumber of solutionsbelow each graph.ORGANIZE IT
    • BRINGING IT ALL TOGETHERBUILD YOURVOCABULARYUse your Chapter 10 Foldable tohelp you study for your chaptertest.To make a crossword puzzle,word search, or jumblepuzzle of the vocabulary wordsin Chapter 10, go to:glencoe.comYou can use your completedVocabulary Builder(pages 283–284) to help yousolve the puzzle.VOCABULARYPUZZLEMAKERC H A P T E R10STUDY GUIDEGlencoe Algebra 2 307Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Consider the segment connecting the points (؊3, 5) and (9, 11).1. Find the midpoint of this segment.2. Find the length of the segment. Write your answer in simplifiedradical form.3. Circle P has a diameter CෆDෆ. If C is at (4, Ϫ3) and D is at (Ϫ3, 5),find the center of the circle and the length of the diameter.Write each equation in standard form.4. y ϭ 2x2Ϫ 8x ϩ 1 5. y ϭ Ϫ2x2ϩ 6x ϩ 16. y ϭ 1_2x2Ϫ 5x ϩ 1210-1Midpoint and Distance Formulas10-2Parabolas
    • Chapter BRINGING IT ALL TOGETHER10308 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.7. Write the equation of the circle with center (4, Ϫ3) and radius 5.8. The circle with equation (x ϩ 8)2ϩ y2ϭ 121 has centerand radius .9. a. In order to find the center and radius of the circle with equationx2ϩ y2ϩ 4x Ϫ 6y Ϫ 3 ϭ 0, it is necessary to .Fill in the missing parts of this process.x2ϩ y2ϩ 4x Ϫ 6y Ϫ 3 ϭ 0x2ϩ y2ϩ 4x Ϫ 6y ϭx2ϩ 4x ϩ ϩ y2Ϫ 6y ϩ ϭ ϩ ϩ΂x ϩ ΃2ϩ ΂y Ϫ ΃2ϭb. This circle has radius 4 and center at .10. Complete the table to describe characteristics of their graphs.Standard Formof Equationy2_25ϩ x2_16ϭ 1 x2_9ϩy2_4ϭ 1Direction ofMajor AxisDirection ofMinor AxisFociLength ofMajor AxisLength ofMinor Axis10-3Circles10-4Ellipses
    • Chapter BRINGING IT ALL TOGETHER10Glencoe Algebra 2 309Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Study the hyperbola graphed at the right.11. The center is .12. The value of a is .13. The value of c is .14. To find b2, solve ϭ ϩ .15. The equation in standard form for this hyperbola is .Name the conic section that is the graph of each equation.Give the coordinates of the vertex if the conic section is aparabola and of the center if it is a circle, an ellipse, or ahyperbola.16.(x Ϫ 3)2_36 ϩ(y ϩ 5)2_15ϭ 1 17. x ϭ Ϫ2(y ϩ 1)2ϩ 718. (x Ϫ 5)2Ϫ (y ϩ 5)2ϭ 1 19. (x ϩ 6)2ϩ (y Ϫ 2)2ϭ 1Draw a sketch to illustrate each of the following possibilities.20. a parabola and a 21. an ellipse and a 22. a hyperbola and aline that intersect circle that intersect line that intersectin 2 points in 4 points in 1 point10-5Hyperbolas10-6Conic Sections10-7Solving Quadratic SystemsxyO
    • C H A P T E R10Checklist310 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Check the one that applies. Suggestions to help you study aregiven with each item.ARE YOU READY FORTHE CHAPTER TEST?I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.• You may want to take the Chapter 10 Practice Test onpage 615 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 10 Study Guide and Reviewon pages 609–614 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 10 Practice Test onpage 615 of your textbook.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 10 Foldable.• Then complete the Chapter 10 Study Guide and Review onpages 609–614 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 10 Practice Test onpage 615 of your textbook.Visit glencoe.com toaccess your textbook,more examples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 10.Student Signature Parent/Guardian SignatureTeacher Signature
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Chapter11Chapter4C H A P T E R11Glencoe Algebra 2 311Sequences and SeriesUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.NOTE-TAKING TIP: When you take notes, writequestions you have about the lessons in the marginof your notes. Then include the answers to thesequestions as you work through the lesson.Begin with one sheet of 11” ϫ 17” paper andfour sheets of notebook paper.Fold the short sides of the11” ϫ 17” paper to meetin the middle.Fold the notebook paperin half lengthwise. Inserttwo sheets of notebookpaper in each tab andstaple the edges. Label withlesson numbers. Take notesunder the appropriate tabs.SequencesSeries
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Vocabulary TermFoundon PageDefinitionDescription orExamplearithmetic meansAR-ihth-MEH-tihkarithmetic sequencearithmetic seriescommon differencecommon ratiofactorialFibonacci sequencefih-buh-NAH-cheegeometric meansgeometric sequencegeometric series312 Glencoe Algebra 2This is an alphabetical list of new vocabulary terms you will learn in Chapter 11.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.C H A P T E R11BUILD YOUR VOCABULARY
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Chapter BUILD YOUR VOCABULARY11Glencoe Algebra 2 313Vocabulary TermFoundon PageDefinitionDescription orExampleindex of summationinduction hypothesisinfinite geometricseriesiterationIH-tuh-RAY-shuhnmathematicalinductionpartial sumPascal’s trianglepas-KALZrecursive formularih-KUHR-sihvsequenceseriessigma notationSIHG-muhterm
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.11–1314 Glencoe Algebra 2Arithmetic SequenceFind the Next TermsFind the next four terms of the arithmetic sequenceϪ8, Ϫ6, Ϫ4, . . . .Find the common difference d by subtracting 2 consecutiveterms.Ϫ6 Ϫ (Ϫ8) ϭ and Ϫ4 Ϫ (Ϫ6) ϭ So, d ϭ .Now add 2 to the third term of the sequence and then continueadding 2 until the next four terms are found.Ϫ4ϩ 2 ϩ 2 ϩ 2 ϩ 2Check Your Progress Find the next four terms of thearithmetic sequence 5, 3, 1, . . . .Find a Particular TermCONSTRUCTION The table belowshows typical costs for aconstruction company to rent acrane for one, two, three, or fourmonths. If the sequence continues,how much would it cost to rent thecrane for 24 months?An arithmetic sequence is a sequence in which eachafter the first is found by aconstant, called the common difference d, to theterm.BUILD YOUR VOCABULARY (pages 312–313)• Use arithmeticsequences.• Find arithmetic means.MAIN IDEASMonths Costs($)1 75,0002 90,0003 105,0004 120,000nth Term of an ArithmeticSequence The nth terman of an arithmeticsequence with firstterm a1 and commondifference d is givenby an ϭ a1 ϩ (n Ϫ 1)d,where n is any positiveinteger.On the tab forLesson 11-1, write yourown arithmetic sequence.KEY CONCEPTPreparation for TEKS P.4 The student uses sequences and series as well as toolsand technology to represent, analyze, and solve real-life problems. (A) Representpatterns using arithmetic and geometric sequences and series. (B) Use arithmetic,geometric, and other sequences and series to solve real-life problems.
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Explore Since the difference between any two successivecosts is $15,000, the costs form an arithmeticsequence with common difference .Plan You can use the formula for the nth term of anarithmetic sequence with a1 ϭ 75,000 andd ϭ 15,000 to find a24 the cost for 24 months.Solve an ϭ Formula for thenth terma24 ϭ 75,000 ϩ (24 Ϫ 1)(15,000) n ϭ 24,a1 ϭ 75,000,d ϭ 15,000a24 ϭ Simplify.It would cost to rent the crane for 24 months.Check Your Progress Refer to Example 2. How muchwould it cost to rent the crane for 8 months?Write an Equation for the nth TermWrite an equation for the nth term of the arithmeticsequence Ϫ8, Ϫ6, Ϫ4, . . . .In this sequence, a1 ϭ Ϫ8 and d ϭ 2. Use the nth formula towrite an equation.an ϭ a1 ϩ (n – 1)d Formula for the nth terman ϭ a1 ϭ Ϫ8, d ϭ 2an ϭ Distributive Propertyan ϭ Simplify.11–1Glencoe Algebra 2 315REVIEW ITWhen finding the valueof an expression, youmust follow the orderof operations. Briefly listthe order of operations.(Lesson 1-1)
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Check Your Progress Write an equation for the nth termof the arithmetic sequence 5, 3, 1, . . . .Find Arithmetic MeansFind the three arithmetic means between 21 and 45.You can use the nth term formula to find the commondifference. In the sequence 21, __, __, __, 45, . . . , a1 ϭand a5 ϭ .an ϭ a1 ϩ (n – 1)d Formula for the nth terma5 ϭ n ϭ 5, a1 ϭ 21ϭ a5 ϭ 45ϭ Subtract 21 from each side.ϭ d Divide each side by 4.Now use the value of d to find the three arithmetic means.21ϩ 6 ϩ 6 ϩ 6Check Your Progress Find the three arithmetic meansbetween 13 and 25.11–1Page(s):Exercises:HOMEWORKASSIGNMENT316 Glencoe Algebra 2The terms between any two nonsuccessive terms of ansequence are called arithmetic means.BUILD YOUR VOCABULARY (pages 312–313)
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.A series is an indicated sum of the of asequence. Since 18, 22, 26, 30 is an arithmetic sequence,18 ϩ 22 ϩ 26 ϩ 30 is an arithmetic series.Glencoe Algebra 2 31711–2 Arithmetic SeriesFind the Sum of an Arithmetic SeriesFind the sum of the first 20 even numbers, beginningwith 2.The series is 2 ϩ 4 ϩ 6 ϩ . . . ϩ 40. Since a1 ϭ ,a20 ϭ , and d ϭ , you can use either sum formulafor this series.Method 1Sn ϭ n_2(a1 ϩ an) Sum formulaS20 ϭ 20_2(2 ϩ 40) n ϭ 20, a1 ϭ 2, a20 ϭ 40S20 ϭ Simplify.S20 ϭ Multiply.Method 2Sn ϭ n_2[2a1 ϩ (n Ϫ 1)d] Sum formulaS2 ϭ 20_2[2(2) ϩ (20 Ϫ 1)2] n ϭ 20, a1 ϭ 2, d ϭ 2S2 ϭ Simplify.S2 ϭ Multiply.The sum of the first 20 even numbers is .Check Your Progress Find the sum of the first 15counting numbers, beginning with 1.Sum of an ArithmeticSeries The sum Sn ofthe first n terms of anarithmetic series isgiven bySn ϭ n_2[2a1 ϩ (n Ϫ 1)d] orSn ϭ n_2(a1 ϩ an).KEY CONCEPTBUILD YOUR VOCABULARY (page 313)• Find sums of arithmeticseries.• Use sigma notation.MAIN IDEASPreparation for TEKS P.4 The student uses sequences and series as well as toolsand technology to represent, analyze, and solve real-life problems. (A) Representpatterns using arithmetic and geometric sequences and series. (B) Use arithmetic,geometric, and other sequences and series to solve real-life problems.
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Find the First TermRADIO A radio station is giving away money every dayin the month of September for a total of $124,000. Theyplan to increase the amount of money given away by$100 each day. How much should they give away on thefirst day of September, rounded to the nearest cent?You know the values of n, Sn, and d. Use the sum formula thatcontains d.Sn ϭ n_2[2a1 ϩ (n Ϫ 1)d] Sum formulaS30 ϭ 30_2[2a1 ϩ (30 Ϫ 1)100] n ϭ 30, d ϭ 100ϭ 15(2a1 ϩ 2900) S30 ϭ 124,000124,000 ϭ ϩ Distributive Property80,500 ϭ 30a1 Subtract 43,500from each side.ϭ a1 Divide each side by30.They should give away the first day.Check Your Progress A television game showgives contestants a chance to win a total of $1,000,000 byanswering 16 consecutive questions correctly. If the value fromquestion to question increases by $5,000, how much is the firstquestion worth?Find the First Three TermsFind the first four terms of an arithmetic series inwhich a1 ϭ 14, an ϭ 29, and Sn ϭ 129.Step 1 Since you know a1, an, and Sn, useSn ϭ n_2(a1 ϩ an) to find n.Sn ϭ n_2(a1 ϩ an)ϭ n_2ϭϭ n11–2318 Glencoe Algebra 2Use the tab forLesson 11-2. Write anexample of an arithmeticsequence and anarithmetic series. Thenexplain the differencebetween the two.SequencesSeriesORGANIZE IT
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.11–2Glencoe Algebra 2 319Step 2 Find d.an ϭ a1 ϩ (n Ϫ 1)dϭ ϩ dϭϭ dStep 3 Use d to determine a2, a3, and a4.a2 ϭ ϩ ora3 ϭ ϩ ora4 ϭ ϩ orThe first four terms are , , , and .Check Your Progress Find the first three terms of anarithmetic series in which a1 ϭ 11, an ϭ 31 and Sn ϭ 105.A concise notation for writing out a series is called sigmanotation. The series 3 ϩ 6 ϩ 9 ϩ 12 ϩ . . . ϩ 30 can beexpressed as ∑n ϭ 1103n. The variable, in this case n, is calledthe index of summation.∑n ϭ 1103nBUILD YOUR VOCABULARY (page 313)}last value of nformula for the terms of the seriesfirst value of n
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.11–2Page(s):Exercises:HOMEWORKASSIGNMENT320 Glencoe Algebra 2Evaluate a Sum in Sigma NotationEvaluate ∑k ϭ 310(2k ϩ 1).Method 1 Find the terms by replacing k with 3, 4, …, 10.Then add.∑k ϭ 310(2k ϩ 1)ϭ [2(3) ϩ 1] ϩ [2(4) ϩ 1] ϩ [2(5) ϩ 1] ϩ [2(6) ϩ 1] ϩϩ ϩϩϭ ϩ ϩ ϩ ϩ ϩϩ ϩϭMethod 2 Since the sum is an arithmetic series, use theformula Sn ϭ n_2(a1 ϩ an). There are terms,a1 ϭ 2( )ϩ 1 or , and a8 ϭ 2( )ϩ 1or .Sn ϭ ϭ orCheck Your Progress Evaluate ∑i ϭ 510(2i ϩ 3).
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.A geometric sequence is a sequence in which each termafter the first is found by multiplying the previous termby a constant r called the common ratio.The missing term(s) between two nonsuccessive terms ofa geometric sequence are called geometric means.Glencoe Algebra 2 32111–3 Geometric SequencesFind the Next TermTEST EXAMPLE What is the missing term in thegeometric sequence 324, 108, 36, 12, ____?A 972 B 4 C 0 D Ϫ12Read the Test ItemSince 108_324ϭ 1_3, 36_108ϭ 1_3, and 12_36ϭ 1_3, the sequence has thecommon ratio of .Solve the Test ItemFind the missing term, multiply the last given term by 1_3:12(1_3) ϭ . The answer is .Check Your Progress Find the missing term in thegeometric sequence 100, 50, 25, ___.A 200 B 0 C 12.5 D Ϫ12.5Find a Term Given the First Term and the RatioFind the sixth term of a geometric sequence for whicha1 ϭ Ϫ3 and r ϭ Ϫ2.an ϭ a1. rn Ϫ 1 Formula for the nth terma6 ϭ и n ϭ 6, a1 ϭ Ϫ3, r ϭ Ϫ2a6 ϭ и (Ϫ25) ϭ Ϫ32a6 ϭ Multiply.BUILD YOUR VOCABULARY (pages 312–313)• Use geometricsequences.• Find geometric means.MAIN IDEASnth Term of a GeometricSequence The nth terman of a geometricsequence with first terma1 and common ratio r isgiven by an ϭ a1 и rn Ϫ 1,where n is any positiveinteger.KEY CONCEPTPreparation for TEKS P.4 The student uses sequences and series as well as toolsand technology to represent, analyze, and solve real-life problems. (A) Representpatterns using arithmetic and geometric sequences and series. (B) Use arithmetic,geometric, and other sequences and series to solve real-life problems.
    • 11–3322 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Check Your Progress Find the fifth term of a geometricsequence for which a1 ϭ 6 and r ϭ 2.Write an Equation for the nth TermWrite an equation for the nth term of the geometricsequence 5, 10, 20, 40, … .an ϭ a1rn Ϫ 1Formula for the nth terman ϭ ( )n Ϫ 1a1 ϭ , r ϭAn equation is .Check Your Progress Write an equation for the nth termof the geometric sequence 2, 6, 18, 54 . . . .Find a Term Given One Term and the RatioFind the seventh term of a geometric sequence forwhich a3 ϭ 96 and r ϭ 2.First find the value of a1.an ϭ a1rn Ϫ 1ϭϭ a1(2)2ϭ a1The seventh term is .Check Your Progress Find the sixth term of a geometricsequence for which a4 ϭ 27 and r ϭ 3.Use the tab forLesson 11-3. Write yourown example of anarithmetic sequence anda geometric sequence.Then explain thedifference betweenthe two.SequencesSeriesORGANIZE ITNow find a7.an ϭ a1rn Ϫ 1a7 ϭa7 ϭ
    • 11–3Page(s):Exercises:HOMEWORKASSIGNMENTFind Geometric MeansFind three geometric means between 3.12 and 49.92.In the sequence a1 is 3.12 and a5 is 49.92.an ϭ a1rn Ϫ 1 Formula for the nth terma5 ϭ n ϭ 5, a1 ϭ 3.12ϭ a5 ϭ 49.92ϭ r4Divide by 3.12.ϭ r Take the fourth root of each side.Use each value of r to find three geometric means.r ϭ 2 r ϭ Ϫ2a2 ϭ 3.12(2) or a2 ϭ 3.12(Ϫ2) ora3 ϭ 6.24(2) or a3 ϭ Ϫ6.24(Ϫ2) ora4 ϭ 12.48(2) or a4 ϭ 12.48(Ϫ2) orThe geometric means are , , andor , , and .Check Your Progress Find three geometric means between12 and 0.75.Glencoe Algebra 2 323Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 324 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.11–4 Geometric SeriesSum of a GeometricSeries The sum Sn ofthe first n terms of ageometric series is givenby Sn ϭa1 Ϫ a1rn_1 Ϫ rorSn ϭa1(1 Ϫ rn)_1 Ϫ r,where r 1.KEY CONCEPTFind the Sum of the First n TermsHEALTH Contagious disease can spread very quickly.Suppose five people are ill during the first week ofan epidemic, and each person who is ill spreads thedisease to four people by the end of the week. By theend of the sixth week of the epidemic, how many peoplehave been affected by the illness?This is a geometric series with a1 ϭ 5, r ϭ 4, and n ϭ 6.Sn ϭa1(1 Ϫ rn)_1 Ϫ rSum formulaϭ n ϭ 6, a1 ϭ 5, r ϭ 4ϭ Use a calculator.After 6 weeks, people have been affected.Evaluate a Sum Written in Sigma NotationEvaluate ∑n ϭ 1123 и 2n Ϫ 1.The sum is a geometric series.Sn ϭa1(1 Ϫ rn)_1 Ϫ rSum formulaS12 ϭ n ϭ 12, a1 ϭ 3, r ϭ 2S12 ϭ3(4095)_1212ϭ 4096S12 ϭ Simplify.Check Your Progressa. How many direct ancestors would a person have after7 generations?• Find sums of geometricseries.• Find specific terms ofgeometric series.MAIN IDEASREMEMBER ITThe sum inExample 2 can also befound by usingSn ϭa1 Ϫ a1rn_1 Ϫ r.Preparation for TEKS P.4 The student uses sequences and series as well as toolsand technology to represent, analyze, and solve real-life problems. (A) Representpatterns using arithmetic and geometric sequences and series. (B) Use arithmetic,geometric, and other sequences and series to solve real-life problems.
    • 11–4Glencoe Algebra 2 325Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Page(s):Exercises:HOMEWORKASSIGNMENTb. Evaluate ∑n ϭ 145 и 3n Ϫ 1.Use the Alternate Formula for a SumFind the sum of a geometric series for which a1 ϭ 7776,an ϭ 6, and r ϭ Ϫ1_6.Since you do not know the value of n, use Sn ϭa1 Ϫ anr_1 Ϫ r.Sn ϭa1 Ϫ anr_1 Ϫ rAlternate sum formulaϭ1 Ϫa1 ϭ 7776, an ϭ 6, and r ϭ Ϫ1_6ϭ 7777_7_6or Simplify.Check Your Progress Find the sum of a geometric seriesfor which a1 ϭ 64, an ϭ 729, and r ϭ Ϫ1.5.Find the First Term of a SeriesFind a1 in a geometric series for which S8 ϭ 765and r ϭ 2.Sn ϭa1(1 Ϫ rn)_1 Ϫ rSum formulaϭ S8 ϭ 765, r ϭ 2, and n ϭ 8765 ϭ 255a1 Simplify.ϭ a1 Divide each side by 255.Check Your Progress Find a1 in a geometric series forwhich S6 ϭ 364 and r ϭ 3.Use the tab forLesson 11-4. Writein words the sigmanotation that is usedin Example 2.SequencesSeriesORGANIZE IT
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.11–5326 Glencoe Algebra 2Infinite Geometric SeriesSum of an Infinite Geometric SeriesFind the sum of each infinite geometric series, if itexists.a. Ϫ4_3ϩ 4 Ϫ 12 ϩ 36 Ϫ 108 ϩ . . .Find the value of r to determine if the sum exists.a1 ϭ Ϫ4_3and a2 ϭ 4, so r ϭ or .Since Ϫ3 Ն 1, the sum exist.b. 3 Ϫ 3_2ϩ 3_4Ϫ 3_8ϩ . . .a1 ϭ and a2 ϭ , so r ϭ or Ϫ1_2.Since , the sum .Use the formula for the sum of an infinite geometric series.S ϭa1_1 Ϫ rSum formulaϭ 3_1 Ϫ (Ϫ1_2)a1 ϭ 3, r ϭ Ϫ1_2ϭ or Simplify.Check Your Progress Find the sum of each infinitegeometric series, if it exists.a. 2 ϩ 4 ϩ 8 ϩ 16 ϩ . . .If a geometric series has no last term, it is called an infinitegeometric series. For infinite series, Sn is called a partialsum of the series.BUILD YOUR VOCABULARY (page 312)• Find the sum of aninfinite geometric series.• Write repeatingdecimals as fractions.MAIN IDEASSum of an InfiniteGeometric Series Thesum S of an infinitegeometric series with–1 Ͻ r Ͻ 1 is given byS ϭa1_1 Ϫ r.KEY CONCEPTPreparation for TEKS P.4 The student uses sequences and series as well as toolsand technology to represent, analyze, and solve real-life problems. (C) Describelimits of sequences and apply their properties to investigate convergent anddivergent series.
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.11–5Page(s):Exercises:HOMEWORKASSIGNMENTGlencoe Algebra 2 327b. 1 ϩ 1_2ϩ 1_4ϩ . . .Infinite Series in Sigma NotationEvaluate ∑n ϭ 1∞5(1_2)n Ϫ 1.In this infinite geometric series, a1 ϭ 5 and r ϭ 1_2.S ϭa1_1 Ϫ rSum formulaϭ 5_1 Ϫ 1_2a1 ϭ 5, r ϭ 1_2ϭ or Simplify.Check Your Progress Evaluate ∑n ϭ 1ϱ2(1_3)n Ϫ 1.Write a Repeating Decimal a FractionWrite 0.2ෆ5ෆ as a fraction.S ϭ 0.2ෆ5ෆ Label the given decimal.S ϭ 0.25252525 . . . Repeating decimal100S ϭ Multiply each side by 100.99S ϭ Subtract the second equationfrom the third.S ϭ Divide each side by 99.Check Your Progress Write 0.3ෆ7ෆ a fraction.Under the tab forLesson 11-5, write 0.5ෆ4ෆas a fraction. Use eitherof the methods shownin Example 3 of yourtextbook.SequencesSeriesORGANIZE IT
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.11–6328 Glencoe Algebra 2Recursion and Special SequencesUse a Recursive FormulaFind the first five terms of the sequence in which a1 ϭ 5and an ϩ 1 ϭ 2an ϩ 7, n Ն 1.an ϩ 1 ϭ 2an ϩ 7 Recursive formulaa1 ϩ 1 ϭ 2a1 ϩ 7 n ϭ 1a2 ϭ 2΂ ΃ϩ 7 or a1 ϭ 5a2 ϩ 1 ϭ 2a2 ϩ 7 n ϭ 2a3 ϭ 2΂ ΃ϩ 7 or a2 ϭ 17a3 ϩ 1 ϭ 2a3 ϩ 7 n ϭ 3a4 ϭ 2΂ ΃ϩ 7 or a3 ϭ 41a4 ϩ 1 ϭ 2a4 ϩ 7 n ϭ 4a5 ϭ 2΂ ΃ϩ 7 or a4 ϭ 89Check Your Progress Find the first five terms of thesequence in which a1 ϭ 2 and an + 1 ϭ 3an ϩ 2, n Ն 1.The sequence 1, 1, 2, 3, 5, 8, 13, ... , where each term in thesequence after the second is the sum of the two previousterms, is called the Fibonacci sequence.The formula an ϭ an Ϫ 2 ϩ an Ϫ 1 is an example of a recursiveformula.• Recognize and usespecial sequences.• Iterate functions.MAIN IDEASBUILD YOUR VOCABULARY (pages 312–313)Use the tab forLesson 11-6. Describethe pattern in thesequence 1, 2, 6, 24,120, . . . . Then find thenext three terms of thesequence.SequencesSeriesORGANIZE ITPreparation for TEKS P.4 The student uses sequences and series as well as toolsand technology to represent, analyze, and solve real-life problems. (B) Use arithmetic,geometric, and other sequences and series to solve real-life problems.
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.11–6Page(s):Exercises:HOMEWORKASSIGNMENTGlencoe Algebra 2 329Iterate a FunctionFind the first three iterates x1, x2, and x3 of the functionƒ(x) ϭ 3x Ϫ 1 for an initial value of x0 ϭ 5.To find the first iterate x1, find the value of the functionfor x0 ϭ 5.x1 ϭ ƒ(x0) Iterate the function.ϭ ƒ x0 ϭ 5ϭ or Simplify.To find the second iterate x2, substitute x1 for x.x2 ϭ ƒ(x1) Iterate the function.ϭ ƒ x1 ϭ 14ϭ or Simplify.Substitute x2 for x to find the third iterate.x3 ϭ ƒ(x2) Iterate the function.ϭ ƒ x2 ϭ 41ϭ or Simplify.Check Your Progress Find the first three iterates x1,x2, and x3 of the function ƒ(x) ϭ 2x ϩ 1 for an initial valueof x0 ϭ 2.Iteration is the process of composing a withitself repeatedly.REVIEW ITIf ƒ(x) ϭ 4x ϩ 10 andg(x) ϭ 2x2 Ϫ 5, findƒ(g(x)). (Lesson 7-7)BUILD YOUR VOCABULARY (page 313)
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.11–7330 Glencoe Algebra 2The Binomial TheoremUse Pascal’s TriangleExpand (p ϩ q)5.Write the row of Pascal’s triangle corresponding to n ϭ 5.Use the patterns of a binomial expansion and the coefficientsto write the expansion of (p ϩ q)5.(p ϩ q)5ϭϭUse the Binomial TheoremExpand (t Ϫ s)8.The expression will have terms. Use the sequence1, 8_1, 8 и 7_1 и 2, 8 и 7 и 6_1 и 2 и 3, 8 и 7 и 6 и 5_1 и 2 и 3 и 4to find the coefficients for thefirst five terms. Use symmetry to find the remainingcoefficients.(t Ϫ s)8ϭ 1t8(Ϫs)0ϩ 8_1t7(Ϫs) ϩ 8 и 7_1 и 2t6(Ϫs)2ϩ 8 и 7 и 6_1 и 2 и 3t5(Ϫs)3ϩ8 и 7 и 6 и 5_1 и 2 и 3 и 4t4(Ϫs)4ϩ . . . ϩ 1t0(Ϫs)8ϭBinomial Theorem If n isa nonnegative integer,then(a ϩ b)nϭ 1anb0 ϩ n_1an Ϫ 1b1 ϩn(n Ϫ 1)_1 и 2an Ϫ 2b2 ϩn(n Ϫ 1)(n Ϫ 2)__1 и 2 и 3an Ϫ 3b3 ϩ. . . ϩ 1a0bn.KEY CONCEPTThe in powers of form apattern that is often displayed in a triangular formationknown as Pascal’s triangle.The factors in the coefficients of binomialinvolve special products called factorials.• Use Pascal’s triangleto expand powers ofbinomials.• Use the BinomialTheorem to expandpowers of binomials.MAIN IDEASBUILD YOUR VOCABULARY (pages 312–313)Use the tab forLesson 11-7. Refer toExample 1, and describewhat happens to theexponents for p as theexponents for q increase.SequencesSeriesORGANIZE ITPreparation for TEKS P.4 The student uses sequences and series as well astools and technology to represent, analyze, and solve real-life problems. (D) Applysequences and series to solve problems including sums and binomial expansion.
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.11–7Glencoe Algebra 2 331Check Your Progressa. Expand (x ϩ y)6.b. Expand (x Ϫ y)4.FactorialsEvaluate 6!_2!4!.6!_2!4!ϭ 6 и 5 и 4 и 3 и 2 и 1__2 и 1 и 4 и 3 и 2 и 1Note that 6! ϭ 6 и 5 и 4!, so6!_2!4!ϭ 6 и 5 и 4!_2!4!or 6 и 5_2 и 1.ϭ orCheck Your Progress Evaluate 7!_2!3!.Use a Factorial Form for the Binomial TheoremExpand (3x Ϫ y)4.(3x Ϫ y)4ϭϭ 4!_4!0!(3x)4(Ϫy)0ϩ 4!_3!1!(3x)3(Ϫy)1ϩ 4!_2!2!(3x)2(Ϫy)2ϩϩϭ 4 и 3 и 2 и 1__4 и 3 и 2 и 1 и 1(3x)4ϩ 4 и 3 и 2 и 1_3 и 2 и 1 и 1(3x)3(Ϫy) ϩ4 и 3 и 2 и 1_2 и 1 и 2 и 1(3x)2y2ϩ 4 и 3 и 2 и 1_1 и 3 и 2 и 1(3x)1(Ϫy3) ϩϭBinomial Theorem,Factored Form(a ϩ b)nϭ n!_n!0!anb0ϩn!_(n Ϫ 1)!1!an Ϫ 1b1ϩn!_(n Ϫ 2)!2! an Ϫ 2b2ϩ… ϩ n!_n!0!a0bnϩ∑k ϭ 0nn!_(n Ϫ k)!k!an Ϫ kbkKEY CONCEPT
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Check Your Progress Expand (2x ϩ y)4.Find a Particular TermFind the fourth term in the expansion of (a ϩ 3b)4.First, use the Binomial Theorem to write the expression insigma notation.(a ϩ 3b)4ϭ ∑k ϭ 044!_(4 Ϫ k)!k!a4 Ϫ k(3b)kIn the fourth term, k ϭ .4!_(4 Ϫ k)!k!a4 Ϫ k(3b)kϭ 4!_(4 Ϫ 3)!3!a4 Ϫ 3(3b)3k ϭ 3ϭ 4!_1!3!ϭϭ Simplify.Check Your Progress Find the fifth term in theexpansion of (x ϩ 2y)6.11–7Page(s):Exercises:HOMEWORKASSIGNMENT332 Glencoe Algebra 2
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Glencoe Algebra 2 33311–8 Proof and Mathematical InductionMathematical induction is used to statementsabout integers.• Prove statements usingmathematical induction.• Disprove statementsby finding acounterexample.MAIN IDEASSummation FormulaProve that 1 ϩ 3 ϩ 5 ϩ . . . ϩ (2n Ϫ 1) ϭ n2.Step 1 When n ϭ 1, the left side of the given equation isϪ or . The right side is or. Thus, the equation is true for n ϭ 1.Step 2 Assume 1 ϩ 3 ϩ 5 ϩ . . . ϩ (2k Ϫ 1) ϭ k2for a positiveinteger k.Step 3 Show that the given equation is true forn ϭ .1 ϩ 3 ϩ 5 ϩ . . . ϩ (2k Ϫ 1) ϩ (2(k ϩ 1) Ϫ 1)ϭAdd (2(k ϩ 1) Ϫ 1) toeach side.ϭ Add.ϭ Simplify.ϭ Factor.The last expression is the right side of the equation to beproved, where n has been replaced by k ϩ 1. Thus, theequation is true for n ϭ k ϩ 1.This proves that 1 ϩ 3 ϩ 5 ϩ . . . ϩ (2n Ϫ 1) ϭ n2is true for allpositive integers n.Mathematical InductionStep 1 Show that thestatement is true forsome integer n.Step 2 Assume that thestatement is true forsome positive integerk, where k Ն n. Thisassumption is called theinductive hypothesis.Step 3 Show that thestatement is true for thenext integer k ϩ 1.KEY CONCEPTBUILD YOUR VOCABULARY (page 313)Reinforcement of TEKS G.3 The student applies logical reasoning to justify andprove mathematical statements. (D) Use inductive reasoning to formulate a conjecture.
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.11–8334 Glencoe Algebra 2DivisibilityProve that 6n Ϫ 1 is divisible by 5 for all positiveintegers n.Step 1 When n ϭ 1, 6nϪ 1 ϭ or . Sinceis divisible by 5, the statement is true for n ϭ 1.Step 2 Assume that 6kϪ 1 is divisible by 5 for some positiveinteger k. This means that there is a whole number rsuch that .Step 3 Show that the statement is true for n ϭ k ϩ 1.ϭ Inductive hypothesis6kϭ Add 1 to each side.6 и 6kϭ 6 Multiply each side by 6.6k ϩ 1ϭ Simplify.6k ϩ 1Ϫ 1 ϭ Subtract 1 from each side.6k ϩ 1Ϫ 1 ϭ Factor.Since r is a whole number, 6r ϩ 1 is a whole number.Therefore, .Thus, the statement is true for . This proves that6nϪ 1 is divisible by 5.Check Your Progressa. Prove that 2 ϩ 4 ϩ 6 ϩ 8 ϩ . . . ϩ 2n ϭ n(n ϩ 1).b. Prove that 10nϪ 1 is divisible by 9 for all positive integers n.
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.11–8Page(s):Exercises:HOMEWORKASSIGNMENTCounterexampleFind a counterexample for the statement that n2ϩ n ϩ 5is always a prime number for any positive integer n.Check the first few positive integers.The value n ϭ is a counterexample for the formula.Check Your Progress Find a counterexample for theformula that 2n Ϫ 1 is always a prime number for any positiveinteger n.Glencoe Algebra 2 335n Formula Prime?1234On the tab forLesson 11-8, write a real-world statement. Thenfind a counterexampleto your statement.SequencesSeriesORGANIZE IT
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.BRINGING IT ALL TOGETHERBUILD YOURVOCABULARYUse your Chapter 11 Foldable tohelp you study for your chaptertest.To make a crossword puzzle,word search, or jumblepuzzle of the vocabulary wordsin Chapter 11, go to:glencoe.comYou can use your completedVocabulary Builder(pages 312–313) to help yousolve the puzzle.VOCABULARYPUZZLEMAKERC H A P T E R11STUDY GUIDE336 Glencoe Algebra 21. Find the next four terms of the arithmetic sequence 3, 6, 9, 12, . . . .2. Find the first five terms of the arithmetic sequence in which a1 ϭ 2and d ϭ 9.3. Write an equation for thenth term of the arithmeticsequence 10, 6, 2, Ϫ2, . . . .4. Find Sn for the arithmetic series in which a1 ϭ Ϫ6, n ϭ 18,and d ϭ 2.5. Find the sum of the arithmetic series 30 ϩ 25 ϩ . . . ϩ (Ϫ10).Find the sum of each arithmetic series.6. ∑j ϭ 39(6 Ϫ j) 7. ∑k ϭ 1025(2k ϩ 1)11-1Arithmetic Sequences11-2Arithmetic Series
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Chapter BRINGING IT ALL TOGETHER11Glencoe Algebra 2 3378. In the sequence 5, 8, 11, 14, 17, 20, the numbers 8, 11, 14, and17 are between 5 and 20.9. In the sequence 12, 4, 4_3, 4_9, 4_27, the numbers 4, 4_3, and 4_9arebetween 12 and 4_27.10. Find three geometric means between 4 and 324.11. Consider the formula Sn ϭa1(1 Ϫ rn)_1 Ϫ r. Suppose that youwant to use the formula to evaluate the sum ∑n ϭ 168(Ϫ2)n Ϫ 1.Indicate the values you would substitute into the formula inorder to find Sn.n ϭ a1 ϭ r ϭ rnϭ12. Find the sum of a geometric series for which a1 ϭ 5, n ϭ 9, andr ϭ 3.13. Consider the formula S ϭa1_1 Ϫ r. For what values of r does aninfinite geometric sequence have a sum?14. For the geometric series 2_3ϩ 2_9ϩ 2_27ϩ …, give the values of a1and r. Then state whether the sum of the series exists.11-3Geometric Sequences11-5Infinite Geometric Series11-4Geometric Series
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Chapter BRINGING IT ALL TOGETHER11338 Glencoe Algebra 215. Find the first five terms of the sequence in which a1 ϭ Ϫ3 andanϩ1 ϭ 2an ϩ 5.16. Find the first three iterates of ƒ(x) ϭ 3x Ϫ 2 for an initialvalue of x0 ϭ 4.Consider the expansion of (w ϩ z)5.17. How many terms does this expansion have?18. In the second term of the expansion, what is the exponentof w?19. In the fourth term of the expansion, what is the exponentof w?20. What is the last term of this expansion?Suppose that you wanted to prove that the followingstatement is true for all positive integers.3 ϩ 6 ϩ 9 ϩ . . . ϩ 3n ϭ3n(n ϩ 1)_221. Which statement shows that the statement is true for n ϭ 1?i. 3 ϭ 3 и 2 ϩ 1_2ii. 3 ϭ 3 и 1 и 2_2iii. 3 ϩ 1 ϩ 2_211-6Recursion and Special Sequences11-7The Binomial Theorem11-8Proof and Mathematical Induction
    • Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Check the one that applies. Suggestions to help you study aregiven with each item.C H A P T E R11ChecklistI completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.• You may want to take the Chapter 11 Practice Test onpage 679 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 11 Study Guide and Reviewon pages 674–678 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 11 Practice Test onpage 679 of your textbook.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 11 Foldable.• Then complete the Chapter 11 Study Guide and Review onpages 674–678 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 11 Practice Test onpage 679 of your textbook.Student Signature Parent/Guardian SignatureTeacher SignatureVisit glencoe.com toaccess your textbook,more examples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 11.Glencoe Algebra 2 339ARE YOU READY FORTHE CHAPTER TEST?
    • NOTE-TAKING TIP: When you take notes, look forwritten real-world examples in your everyday life.Comment on how writers use statistics to proveor disprove points of view and discuss the ethicalresponsibilities writers have when using statistics.Fold 2” tabs on eachof the short sides.Fold in half in bothdirections. Open andcut as shown.Refold alongthe width. Stapleeach pocket.Label pockets as TheCounting Principle,Permutations andCombinations,Probability, andStatistics.Begin with one sheet of 11” ϫ 17” paper.Probability and Statistics12Use the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.340 Glencoe Algebra 2C H A P T E R
    • Glencoe Algebra 2 341Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Vocabulary TermFoundon PageDefinitionDescription orExamplebinomial experimentcombinationcompound eventdependent andindependent eventseventinclusive eventsihn-KLOO-sihvmeasure of variationmutually exclusive eventsMYOO-chuh-leenormal distributionoutcomepermutationPUHR-myoo-TAY-shuhnThis is an alphabetical list of new vocabulary terms you will learn in Chapter12. As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.(continued on the next page)C H A P T E R12BUILD YOUR VOCABULARYChapter12
    • 342 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Vocabulary TermFoundon PageDefinitionDescription orExampleprobabilityprobability distributionrandomrandom variablerelative-frequencyhistogramsample spacesimple eventstandard deviationunbiased sampleuniform distributionunivariate datavarianceVEHR-ee-uhn(t)sChapter BUILD YOUR VOCABULARY12
    • Glencoe Algebra 2 343Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.12–1The set of all possible is called thesample space.An event consists of one or more outcomes of atrial. Events that do not affect each other are calledindependent events.Independent EventsFOOD A sandwich menu offers customers a choice ofwhite, wheat, or rye bread with one spread chosen frombutter, mustard, or mayonnaise. How many differentcombinations of bread and spread are possible?First note that the choice of the type of bread does not affectthe choice of the type of spread, so these events areindependent.Method 1 Tree DiagramW represents white, H, wheat, R, rye, B, butter,M, mustard, Y, mayonnaise.Bread HSpread M B Y MPossibleCombinations WB WY HB HY RB RYMethod 2 Make a TableMake a table in which each row represents a typeof bread and each column represents a type ofspread.Butter Mustard MayonnaiseWhite WMWheat HB HYRye RMThere are possible outcomes.• Solve problemsinvolving independentevents.• Solve problemsinvolving dependentevents.MAIN IDEASBUILD YOUR VOCABULARY (pages 341–342)The Counting PrincipleReinforcement of TEKS 7.10 The student recognizes that a physical ormathematical model can be used to describe the experimental and theoreticalprobability of real-life events. (A) Construct sample spaces for simple or compositeexperiments.
    • Check Your Progress A pizza place offers customers achoice of American, mozzarella, Swiss, feta, or provolone cheesewith one topping chosen from pepperoni, mushrooms, or sausage.How many different combinations of cheese and toppings arethere?Fundamental Counting PrincipleTEST EXAMPLE The Murray family is choosing from atrip to the beach or a trip to the mountains. The familycan select transportation from a car, plane, or train.How many different ways can they select a destinationfollowed by a means of transportation?A 2 B 5 C 6 D 9There are и or ways to choose a trip.The answer is .Check Your Progress For their vacation, the Esperfamily is going on a trip. They can select their transportationfrom a car, plane, or train. They can also select from 4 differenthotels. How many different ways can they select a means oftransportation followed by a hotel?A 8 B 12 C 16 D 7More than Two Independent EventsCOMMUNICATION How many answering machine codesare possible if the code is just 2-digits?The choice of any digit does not affect the other digit, so thechoices of digits are independent events.There are possible choices for the first digit andpossible choices for the second digit.So, there are и or possible different codes.12–1344 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Fundamental CountingPrinciple If event M canoccur in m ways andis followed by event Nthat can occur in n ways,then event M followedby event N can occur inm и n ways.KEY CONCEPT
    • 12–1Glencoe Algebra 2 345Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.With dependent events, the outcome of one eventaffect the outcome of another event.Page(s):Exercises:HOMEWORKASSIGNMENTBUILD YOUR VOCABULARY (page 341)Dependent EventsSCHOOL Refer to the table in Example 4 of yourtextbook. How many different schedules could a studentwho is planning to take only four different classes have?The choices of which class to schedule each period aredependent events.There are 4 classes that can be taken during the first period.That leaves 3 classes for the second period, 2 classes for thethird period, and so on.Period 1st 2nd 3rd 4thNumber of ChoicesThere are и и и or differentschedules for a student who is taking 4 classes.Check Your Progress How many different schedulescould a student have who is planning to take 5 differentclasses?Write and solve yourown examples ofindependent anddependent events.Place your work inThe Counting Principlepocket.ORGANIZE ITCheck Your Progress Many automated teller machines(ATM) require a 4-digit code to access an account. How manycodes are possible?
    • When a group of objects or people are arranged in acertain order, the arrangement is called a permutation.PermutationCOOKING Eight people enter the Best Pie contest. Howmany ways can blue, red, and yellow ribbons beawarded?Since each winner will receive a different ribbon, order isimportant. You must find the number of permutations of8 things taken 3 at a time.P(n, r) ϭn!_(n Ϫ r)! Permutation formulaϭ n ϭ 8, r ϭ 3ϭ Simplify.ϭ 8 и 7 и 6 и 51΋ и 41΋ и 31΋ и 21΋ и 11΋ᎏᎏᎏ51΋ и 41΋ и 31΋ и 21΋ и 11΋Divide.ϭPermutation with RepetitionHow many different ways can the letters of the wordBANANA be arranged?The second, fourth, and sixth letters are each A.The third and fifth letters are each N.Find the number of permutations of letters of whichof one letter and of another letter are the same.6!_3!2!ϭ or• Solve problemsinvolving permutations.• Solve problemsinvolving combinations.MAIN IDEASBUILD YOUR VOCABULARYPermutations Thenumber of permutationsof n distinct objectstaken r at a time is givenby P(n, r) ϭ n!_(n Ϫ r)!.Permutations withRepetitions The numberof permutations of nobjects of which p arealike and q are alike isn!_p!q!.KEY CONCEPTS(page 342)12–2346 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Permutations and CombinationsReinforcement of TEKS 7.10 The student recognizes that a physical ormathematical model can be used to describe the experimental and theoreticalprobability of real-life events. (A) Construct sample spaces for simple or compositeexperiments.
    • 12–2Glencoe Algebra 2 347Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.An arrangement or selection of objects in which order isnot important is called a combination.CombinationFive cousins at a family reunion decide that three ofthem will go to pick up a pizza. How many ways canthey choose the three people who will go?Since the order they choose the three people is not important,you must find the number of combinations of 5 cousins taken3 at a time.C(n, r) ϭ n!_(n Ϫ r)!r!Combination formulaC(5, 3) ϭ n ϭ 5 and r ϭ 3ϭ or Simplify.There are ways to choose the three cousins.BUILD YOUR VOCABULARY (page 341)Check Your Progressa. Ten people are competing in a swim race where 4 ribbonswill be given. How many ways can blue, red, green, andyellow ribbons be awarded?b. How many different ways can the letters of the wordALGEBRA be arranged?Combinations Thenumber of combinationsof n distinct objectstaken r at a time is givenby C(n, r) ϭ n!_(n Ϫ r)!r!.Write a real-world example of apermutation and acombination. Placeyour work in thePermutations andCombinations pocket.KEY CONCEPTS
    • 12–2Page(s):Exercises:HOMEWORKASSIGNMENT348 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Multiple EventsSix cards are drawn from a standard deck of cards. Howmany hands consist of two hearts and four spades?Multiply the number of ways to select two hearts and thenumber of ways to select four spades. Only the cards in thehand matter, not the order in which they were drawn, so use.C(13, 2) ؒ C(13, 4)ϭ 13!_(13 Ϫ 2)!2!и Combination formulaϭ и Subtract.ϭ и or Simplify.There are hands consisting of 2 hearts and4 spades.Check Your Progressa. Six friends at a party decide that three of them will go topick up a movie. How many ways can they choose threepeople to go?b. Thirteen cards are drawn from a standard deck of cards.How many hands consist of six hearts and seven diamonds?
    • Glencoe Algebra 2 349Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.12–3 ProbabilityProbabilityRoman has a collection of 26 books—16 are fiction and10 are nonfiction. He randomly chooses 8 books to takewith him on vacation. What is the probability that hehappens to choose 4 fiction and 4 nonfiction books?Step 1 Determine how many 8-book selections meet theconditions. Notice that order doesn’t matter.C(16, 4) Select 4 fiction books.C(10, 4) Select 4 nonfiction books.Step 2 Use the Fundamental Counting Principle to find s,the number of successes.C(16, 4) и C(10, 4) ϭ и orStep 3 Find the total number, s ϩ f, of possible 8-bookselections.C(26, 8) ϭ or s ϩ f ϭStep 4 Determine the probability.P(4 fiction and 4 nonfiction)ϭ s_s ϩ fProbability formulaϭ or aboutThe probability of selecting 4 fiction and 4 nonfiction is aboutor .Check Your Progress When threecoins are tossed, what is the probabilitythat exactly two are heads?The probability of an event is a that measuresthe chances of the event occurring.BUILD YOUR VOCABULARY (page 342)• Use combinations andpermutations to findprobability.• Create and usegraphs of probabilitydistributions.MAIN IDEASProbability of Successand Failure If an eventcan succeed in s waysand fail in f ways, thenthe probabilities ofsuccess, P(S), and offailure, P(F), are asfollows.• P(S) ϭ s_s ϩ f• P(F ) ϭ f_s ϩ fKEY CONCEPTReinforcement of TEKS 8.11 The student applies concepts of theoretical andexperimental probability to make predictions. (B) Use theoretical probabilities andexperimental results to make predictions and decisions.
    • OddsFor next semester, Alisa has signed up for English,precalculus, Spanish, geography, and chemistry classes.If class schedules are assigned randomly and each classis equally likely to be at any time of the day, what is theprobability that Alisa’s first two classes in the morningwill be precalculus and chemistry, in either order?Step 1 Determine how many schedule arrangements meetthe conditions. Notice that order matters.P(2, 2) Place the two earliest classes.P(3, 3) Place the other 3 classes.Step 2 Use the Fundamental Counting Principle to find thenumber of successes.P(2, 2) и P(3, 3) ϭ orStep 3 Find the total number, s ϩ f, of possible 5-classarrangements.P(5, 5) ϭ or s ϩ f ϭStep 4 Determine the probability.P(Precalculus, chemistry, followed by the other classes)ϭ s_s ϩ fProbability formulaϭ or about Substitute.The probability that Alisa’s first two classes are precalculusand chemistry is about or .Check Your Progress The chances of a male born in1980 to live to be at least 65 years of age are about 7 in 10.For females, the chances are about 21 in 25.a. What are the odds that a male born in 1980 will live toage 65?b. What are the odds that a female born in 1980 will live toage 65?Odds The odds that anevent will occur can beexpressed as the ratio ofthe number of ways itcan succeed to thenumber of ways it canfail. If an event cansucceed in s ways andfail in f ways, then theodds of success and offailure are as follows.• Odds of success ϭ s:f• Odds of failure ϭ f:sKEY CONCEPT12–3350 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Page(s):Exercises:HOMEWORKASSIGNMENT12–3Glencoe Algebra 2 351Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Probability With PermutationsUse the table and graph in Example 3 of your textbook.a. Use the graph to determine which outcomes are leastlikely. What are their probabilities?The least likely outcomes are and , with aprobability of for each.b. Use the table to find P(S ϭ 11). What other sum hasthe same probability?According to the table, the probability of a sum of 11 is. The other outcome with a probability of is.Check Your Progress Suppose two dice are rolled.The table and the relative frequency histogram showthe distribution of the sum of the numbers rolled.S ϭ Sum 2 3 4 5 6 7 8 9 10 11 12Probability1_361_181_121_95_361_65_361_91_121_181_36a. Use the graph Sum of Numbers Showing on the Dice54320Probability6 7 8 9 10 11 12Sum1653619112118136to determinewhichoutcomes arethe secondmost likely.What are theirprobabilities?b. Use the table to find P(S ϭ 4). Whatother sum has the same probability?c. What are the odds of rolling asum of 3?Write your ownprobability example.Then write your ownodds example. Explainthe difference betweenthe two. Place yourwork in the Probabilitypocket.ORGANIZE IT
    • 12–4352 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Multiplying ProbabilitiesTwo Independent EventsGerardo has 9 dimes and 7 pennies in his pocket. Herandomly selects one coin, looks at it, and replaces it.He then randomly selects another coin. What is theprobability that both of the coins he selects are dimes?P(both dimes)ϭ P(dime) и P(dime) Probability of independenteventsϭ и or Substitute and multiply.The probability is or about %.Three Independent EventsWhen three dice are rolled, what is the probabilitythat two dice show a 5 and the third die shows an evennumber?Let A be the event that the first dieshows a 5.P(A) ϭLet B be the event that the second dieshows a 5.P(B) ϭLet C be the event that the third dieshows an even number.P(C) ϭP(A, B, and C)ϭ P(A) и P(B) и P(C) Probability of Independenteventsϭ и и or Substitute and multiply.The probability that the first and second dice show a 5 and thethird die shows an even number is .Check Your Progressa. Gerardo has 9 dimes and 7 pennies in his pocket. Herandomly selects one coin, looks at it, and replaces it. Hethen randomly selects another coin. What is the probabilitythat both of the coins he selects are pennies?• Find the probabilityof two independentevents.• Find the probability oftwo dependent events.MAIN IDEASProbability of TwoIndependent Events Iftwo events, A and B,are independent, thenthe probability of bothevents occurring isP(A and B) ϭ P(A) и P(B).KEY CONCEPTReinforcement of TEKS 8.11 The student applies concepts of theoretical andexperimental probability to make predictions. (A) Find the probabilities of dependentand independent events. (B) Use theoretical probabilities and experimental results tomake predictions and decisions.
    • 12–4Glencoe Algebra 2 353Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Probability of TwoDependent Events Iftwo events, A and B,are dependent, thenthe probability of bothevents occurring isP(A and B) ϭP(A) и P(B following A).KEY CONCEPTb. When three dice are rolled, what is the probability that onedie is a multiple of 3, one die shows an even number, andone die shows a 5?Two Dependent EventsThe host of a game show draws chips from a bag todetermine the prizes for which contestants will play.Of the 20 chips, 11 show computer, 8 show trip, and1 shows truck. If the host draws the chips at randomand does not replace them, find the probability ofdrawing a computer, then a truck.P(C then T)ϭ P(C) и P(T following C) Dependent eventsϭ и or After the first chip is drawn,there are 19 left.The probability is or about .Check Your Progress Refer to Example 3. Find theprobability of drawing each group of prizes.a. a truck, then a trip b. two computersThree Dependent EventsThree cards are drawn from a standard deck of cardswithout replacement. Find the probability of drawing aheart, another heart, and a spade in that order.Since the cards are not replaced, the events are dependent.Let H represent a heart and S represent a spade.P(H, H, S) ϭ P(H) и P(H following H) и P(S following H and H)ϭ и и orThe probability is or about .Check Your Progress Refer to Example 4. Find theprobability of drawing a diamond, another diamond, andanother diamond in that order.Page(s):Exercises:HOMEWORKASSIGNMENT
    • An event that consists of two or more eventsis called a compound event.If two events cannot occur at the same , theyare called mutually exclusive events.Two Mutually Exclusive EventsSylvia has a stack of playing cards consisting of10 hearts, 8 spades, and 7 clubs. If she selects a card atrandom from this stack, what is the probability that itis a heart or club?These are mutually exclusive events since the card cannot beboth a heart and a club.P(heart or club)ϭ P(H) ϩ P(C) Mutually exclusive eventsϭ ϩ Substitute.ϭ Add.The probability that Sylvia selects a heart or a club is .Check Your Progress Sylvia has a stack of playing cardsconsisting of 10 hearts, 8 spades, and 7 clubs. If she selects acard at random from this stack, what is the probability that itis a spade or a club?• Find the probabilityof mutually exclusiveevents.• Find the probability ofinclusive events.MAIN IDEASProbability of MutuallyExclusive Events If twoevents, A and B, aremutually exclusive, thenthe probability that A orB occurs is the sum oftheir probabilities.KEY CONCEPTBUILD YOUR VOCABULARY (page 341)12–5354 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Adding Probabilities
    • 12–5Glencoe Algebra 2 355Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Three Mutually Exclusive EventsThe Film Club makes a list of 9 comedies and 5adventure movies they want to see. They plan to select4 titles at random to show this semester. What is theprobability that at least two of the films they select arecomedies?At least two comedies mean that the club may have ,, or comedies. The events are mutually exclusive.Add the probabilities of each type.P(at least 2 comdies)ϭ P(2 comedies) ϩ ϩϭC(9, 2) и C(5, 2)__C(14, 4)ϩ ϩϭ ϩ ϩϭThe probability of at least 2 comedies being shown thissemester is or about .Check Your Progress The Book Club makes a list of9 mysteries and 3 romance books they want to read. Theyplan to select 3 titles at random to read this semester. Whatis the probability that at least two of the books they select areromances?If two events are not , theyare called inclusive events.BUILD YOUR VOCABULARY (page 341)
    • Inclusive EventsThere are 2400 subscribers to an Internet serviceprovider. Of these, 1200 own Brand A computers,500 own Brand B, and 100 own both A and B. What isthe probability that a subscriber selected at randomowns either Brand A or Brand B?Since some subscribers own both A and B, the events areinclusive.P(A) ϭ P(B) ϭ P(both) ϭP(A or B) ϭ P(A) ϩ P(B) Ϫ P(A and B)ϭ ϩ Ϫϭ Substitute and simplify.The probability that a subscriber owns either A or Bis .Check Your Progress There are 200 students takingCalculus, 500 taking Spanish, and 100 taking both. There are1000 students in the school. What is the probability that astudent selected at random is taking Calculus or Spanish?Page(s):Exercises:HOMEWORKASSIGNMENT12–5356 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Probability of InclusiveEvents If two events,A and B, are inclusive,then the probability thatA or B occurs is the sumof their probabilitiesdecreased by theprobability of bothoccurring.KEY CONCEPT
    • Glencoe Algebra 2 357Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.12–6 Statistical MeasuresA number that describes a set of data is called a measureof central tendency because it represents theor middle of the data.Measures of variation or dispersion measure howor a set of data is.The standard deviation ␴ is the of thevariance.• Use measures of centraltendency to represent aset of data.• Find measures ofvariation for a set ofdata.MAIN IDEASBUILD YOUR VOCABULARYChoose a Measure of Central TendencySALARIES A new Internet company has 3 employeeswho are paid $300,000, ten who are paid $100,000, andsixty who are paid $50,000. Which measure of centraltendency best represents the pay at this company?Since most of the employees are paid $50,000, the highervalues are outliers.Thus, the or best represents thepay at this company.Check Your Progress In a cereal contest, there is 1Grand Prize of $1,000,000, 10 first prizes of $100, and 50second prizes of $10.a. Which measure of central tendency best represents theprizes?b. Which measure of central tendency would advertisers bemost likely to use?(pages 341–342)In your own words,what is an outlier?(Lesson 2-5)REVIEW ITReinforcement of TEKS 8.12 The student uses statistical procedures to describedata. (A) Select the appropriate measure of central tendency or range to describe aset of data and justify the choice for a particular situation.
    • Standard DeviationRIVERS This table shows the length in thousands ofmiles of some of the longest rivers in the world. Findthe standard deviation for these data.Find the mean. Add the data and divide by the number ofitems.xෆ ϭ4.16 ϩ 4.08 ϩ 2.35 ϩ 1.90 ϩ 1.78ᎏᎏᎏᎏ5ϭ thousand milesFind the variance.␴2ϭ(x1 Ϫ xෆ)2 ϩ (x2 Ϫ xෆ)2 ϩ ϩ (xn Ϫ xෆ)2ᎏᎏᎏᎏᎏnVarianceformula≈(4.16 Ϫ 2.85)2 ϩ (4.08 Ϫ 2.85)2 ϩ ϩ (1.78 Ϫ 2.85)2ᎏᎏᎏᎏᎏᎏᎏ5≈*** Simplify.≈ thousand milesFind the standard deviation.␴2 ϭ Take the square root of each side.␴ ≈ thousand milesCheck Your Progress A teacher has the following testscores: 100, 4, 76, 85, and 92. Find the standard deviation forthese data.RiverLength(thousands of miles)Nile 4.16Amazon 4.08Missouri 2.35Rio Grande 1.90Danube 1.78Standard Deviation If aset of data consists ofthe n values x1, x2, ..., xnand has mean x, thenthe standard deviation ␴is given by the followingformula.␴ ϭ(x1 Ϫ xෆ)2 ϩ (x2 Ϫ xෆ)2 ϩ ϩ (xn Ϫ xෆ)2ᎏᎏᎏᎏᎏnSuppose youhave a small standard ofdeviation for your testscores. Does this meanthat you have beenconsistent or inconsistent?Place your explanation inthe Statistics pocket.KEY CONCEPT12–6Page(s):Exercises:HOMEWORKASSIGNMENT358 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Glencoe Algebra 2 359Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.12–7 The Normal DistributionA curve that is , often called a bell curve,is called a normal distribution.A curve or histogram that is not symmetric represents askewed distribution.Classify a Data DistributionDetermine whether the data {31, 37, 35, 36, 34, 36, 32,36, 33, 32, 34, 34, 35, 34} appear to be positively skewed,negatively skewed, or normally distributed.Make a frequency table for the data. Then use the table tomake a histogram.Value 31 32 33 34 35 36 37FrequencySince the data are somewhatsymmetric, this is a.Check Your Progress Determine whether the data {7, 5,6, 7, 8, 4, 6, 8, 7, 6, 6, 4} shown in the histogram appear to bepositively skewed, negatively skewed, or normally distributed.• Determine whether aset of data appears tobe normally distributedor skewed.• Solve problemsinvolving normallydistributed data.MAIN IDEASREMEMBER ITTo help rememberhow skeweddistributions arelabeled, think about thelong tail being in thedirection of the skew.For example, a positivelyskewed distribution hasa long tail in thepositive direction.BUILD YOUR VOCABULARY (page 342)543210FrequencyValue4 5 6 7 8Reinforcement ofTEKS 8.12 Thestudent usesstatistical procedures todescribe data. (C) Selectand use an appropriaterepresentation forpresenting and displayingrelationships amongcollected data, includingline plots, line graphs,stem and leaf plots,circle graphs, bar graphs,box and whisker plots,histograms, and Venndiagrams, with and withoutthe use of technology.
    • Page(s):Exercises:HOMEWORKASSIGNMENTNormal DistributionStudents counted the number of candies in 100 smallpackages. They found that the number of candies perpackage was normally distributed with a mean of23 candies per package and a standard deviation of1 piece of candy.a. About how many packages had between 24 and 22candies?Draw a normal curve. Label20 21 22 23 24 25 2634%13.5%2%0.5% 0.5%2%13.5%34%the mean and positive andnegative multiples of thestandard deviation.The values of 22 and 24 are standard deviation belowand above the mean, respectively. Therefore, ofthe data are located here.100 и ϭ packages Multiply 100 by 0.68.About packages contained between 22 and 24 pieces.b. What is the probability that a package selected atrandom had more than 25 candies.The value 25 is standard deviations above themean. You know that about 100% Ϫ 95% or of thedata are more than one standard deviation away from themean. By the symmetry of the normal curve, half of 5% or, of the data are more than two standard deviationsabove the mean.The probability that a package selected at random has morethan 25 candies is about or .Check Your Progress Refer to Example 2.a. About how many packages had between25 and 21 candies?b. What is the probability that a packageselected at random had more than24 candies?Describe real-worldsituations where youwould expect the datato be positively skewed,negatively skewed, andnormally distributed.Place your explanationsin the Statistics pocket.ORGANIZE IT12–7360 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • Glencoe Algebra 2 361Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.12–8 Binomial ExperimentsBinomial TheoremIf a family has 4 children, what is the probability thatthey have 2 girls and 2 boys?There are two possible outcomes for the gender of each of theirchildren: boy or girl. The probability of a boy b is , andthe probability of a girl g is .(b ϩ g)4ϭ ϩ ϩ ϩ ϩThe term 6b2g2 represents 2 girls and 2 boys.P(2 girls and 2 boys)ϭ 6b2g26΂ ΃2΂ ΃2b , gSimplify.The probability of 2 boys and 2 girls is or .Check Your Progress If a family has 4 children, what isthe probability that they have 4 boys?• Use binomialexpansions to findprobabilities.• Find probabilities forbinomial experiments.MAIN IDEASUse the BinomialTheorem to expand(x Ϫ 3)5. (Lesson 11-7)REVIEW ITReinforcement of TEKS P.4 The student uses sequences and series as well astools and technology to represent, analyze, and solve real-life problems. (D) Applysequences and series to solve problems including sums and binomial expansion.
    • 12–9362 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Binomial ExperimentSALES A report said that approximately 1 out of 6 carssold in a certain year was green. Suppose a salespersonsells 7 cars per week.a. What is the probability that this salesperson will sellexactly 3 green cars in a week?The probability that a sold car is green is .The probability that a sold car is not green is .There are C(7, 3) ways to choose the three green carsthat sell.P(3 green cars)ϭ C(7, 3) ΂ ΃3΂ ΃4If he sells three greencars, he sells four thatare not green.ϭ ΂ ΃3΂ ΃4C(7, 3) ϭ 7!_4!3!ϭ Simplify.The probability that he will sell exactly 3 green cars isor about .b. What is the probability that this salesperson will sellat least 3 green cars in a week?Instead of adding the probabilities of selling exactly 3, 4, 5,6, and 7 green cars, it is easier to theprobabilities of selling exactly , , or greencars from .Binomial ExperimentsA binomial experimentexists if and only if all ofthese conditions occur.• There are exactly twopossible outcomes foreach trial.• There is a fixednumber of trials.• The trials areindependent.• The probabilities foreach trial are thesame.KEY CONCEPT
    • 12–8Page(s):Exercises:HOMEWORKASSIGNMENTP(at least 3 green cars)ϭ 1 Ϫ P(0 green cars) Ϫ P(1 green car) Ϫ P(2 green cars)ϭ 1 Ϫ C(7, 0)΂1_6΃0΂5_6΃7ϪϪϭ 1 Ϫ279,936Ϫ279,936Ϫ279,936ϭThe probability that this salesperson will sell at least threegreen cars in a week is or about .Check Your Progress A report said thatapproximately 1 out of 6 cars sold in a certain year wasgreen. Suppose a salesperson sells 7 cars per week.a. What is the probability that this salesperson will sellexactly 4 green cars in a week?b. What is the probability that this salesperson will sell atleast 2 green cars in a week?Glencoe Algebra 2 363Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
    • 12–9364 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Sampling and ErrorBiased and Unbiased SamplesState whether each method would produce a randomsample. Explain.a. surveying people going into an action movie to findout the most popular kind of movie; they will most likely think that action movies arethe most popular kind of movie.b. calling every 10th person on the list of subscribers toa newspaper to ask about the quality of the deliveryservice; no obvious bias exists in calling every 10thsubscriber.Check Your Progress State whether each methodwould produce a random sample. Explain.a. surveying people going into a football game to find out themost popular sportb. surveying every fifth person going into a mall to find outthe most popular kind of movieFind a Margin of ErrorIn a survey of 100 randomly selected adults, 37%answered “yes” to a particular question. What is themargin of error?ME ϭ 2p(1 Ϫ p)_n Formula for margin ofsampling errorϭ p ϭ 37% or 0.37,n ϭ 100≈ or 10% Use a calculator.• Determine whether asample is unbiased.• Find margins ofsampling error.MAIN IDEASMargin of Sampling ErrorIf the percent of peoplein a sample respondingin a certain way is p andthe size of the sample is n,then 95% of the time, thepercent of the populationresponding in that sameway will be betweenp Ϫ ME and p ϩ ME,whereME ϭ 2√p(1 Ϫ p)_n .KEY CONCEPTReinforcement of TEKS 8.13 The student evaluates predictions and conclusionsbased on statistical data. (A) Evaluate methods of sampling to determine validity of aninference made from a set of data.
    • This means that there is a 95% chance that the percent ofpeople in the whole population who would answer “yes”is between 37 Ϫ 10 or % and 37 ϩ 10 or %.Check Your Progress In a survey of 100 randomlyselected adults, 50% answered “no” to a particular question.What is the margin of error?Analyze a Margin of ErrorHEALTH In an earlier survey, 30% of the peoplesurveyed said they had smoked cigarettes in the pastweek. The margin of error was 2%. How many peoplewere surveyed?ME ϭ 2p(1 Ϫ p)_n Formula for marginof sampling error0.02 ϭ 20.3(1 Ϫ 0.3)__n ME ϭ 0.02, p ϭ 0.3ϭ Divide by 2.ϭ Square each side.n ϭ Multiply by n anddivide by 0.0001.n ϭ Use a calculator.Check Your Progress In an earlier survey, 25% of thepeople surveyed said they had exercised in the past week. Themargin of error was 2%. How many people were surveyed?12–9Page(s):Exercises:HOMEWORKASSIGNMENTGlencoe Algebra 2 365Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Write your own exampleof a biased and anunbiased survey. Placeyour work in theStatistics pocket.ORGANIZE IT
    • BRINGING IT ALL TOGETHERBUILD YOURVOCABULARYUse your Chapter 12 Foldable tohelp you study for your chaptertest.To make a crossword puzzle,word search, or jumblepuzzle of the vocabulary wordsin Chapter 12, go to:glencoe.comYou can use your completedVocabulary Builder(pages 335–336) to help yousolve the puzzle.VOCABULARYPUZZLEMAKERC H A P T E R12STUDY GUIDE366 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.A jar contains 6 red marbles, 4 blue marbles, and 3 yellowmarbles. Indicate whether the events described aredependent or independent.1. A marble is drawn out of the jar and is not replaced. A secondmarble is drawn.2. A marble is drawn out of the jar and is put back in. The jar isshaken. A second marble is drawn.3. A man owns two suits, ten ties, and eight shirts. How manydifferent outfits can he wear if each is made up of a suit, a tie,and a shirt?4. Indicate whether arranging five pictures in a row on a wallinvolves a permutation or a combination.Evaluate each expression.5. P(5, 3) 6. C(7, 2)12-1The Counting Principle12-2Permutations and Combinations
    • Chapter BRINGING IT ALL TOGETHER12Glencoe Algebra 2 367Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.A weather forecast says that the chance of rain tomorrowis 40%.7. Write the probability that it will raintomorrow as a fraction in lowest terms.8. What are the odds in favor of rain?9. Balls are numbered 1 through 15. Find the probability thata ball drawn at random will show a number less than 4. Thenfind the odds that a number less than 4 is drawn.A bag contains 4 yellow balls, 5 red balls, 1 white ball, and2 black balls. A ball is drawn from the bag and is notreplaced. A second ball is drawn.10. Tell which formula you would use to find the probability thatthe first ball is yellow and the second ball is black.a. P(Y and B) ϭP(Y)__P(Y) ϩ P(B)b. P(Y and B) ϭ P(Y) и P(B)c. P(Y and B) ϭ P(Y) и P(B following Y)11. Which equation shows the correct calculation of thisprobability?a. 1_3ϩ 2_11ϭ 17_33b. 1_3и 2_11ϭ 2_33c. 1_3ϩ 1_6ϭ 1_2d. 1_3и 1_6ϭ 1_1812. A pair of dice is thrown. What is the probability that both diceshow a number greater than 5?12-3Probability12-4Multiplying Probabilities
    • Chapter BRINGING IT ALL TOGETHER12368 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Marla took a quiz on this lesson that contained thefollowing problem. Her solution is shown.Each of the integers from 1 through 25 is written on a slip of paperand placed in an envelope. If one slip is drawn at random, what isthe probability that it is odd or a multiple of 5?P(odd) ϭ 13_25P(multiple of 5) ϭ 5_25or 1_5P(odd or multiple of 5) ϭ P(odd) ϩ P(multiple of 5)ϭ 13_25ϩ 5_25ϭ 18_2513. Why is Marla’s work incorrect?14. Show the corrected work.15. A card is drawn from a standard deck of 52 playing cards.What is the probability that an ace or a black card is drawn?Consider the data set {25, 31, 49, 52, 68, 79, 105}.16. Find the variance to the nearest tenth.17. Find the standard deviation to the nearest tenth.12-5Adding Probabilities12-6Statistical Measures
    • Chapter BRINGING IT ALL TOGETHER12Glencoe Algebra 2 369Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.12-7The Normal DistributionIndicate whether each of the following statements is trueor false.18. In a continuous probability distribution, there is a finitenumber of possible outcomes.19. Every normal distribution can be represented by a bellcurve.Indicate whether each of the following is a binomialexperiment or not a binomial experiment. If the experimentis not a binomial experiment, explain why.20. A fair coin is tossed 10 times and “heads” or “tails” is recordedeach time.21. A pair of dice is thrown 5 times and the sum of the numbersthat come up is recorded each time.Find each probability if a coin is tossed four times.22. P(exactly three heads) 23. P(exactly four heads)25. In a survey of 200 people, 36% voted in the last presidentialelection. Find the margin of sampling error.12-9Sampling and Error12-8Binomial Experiments
    • Check the one that applies. Suggestions to help you study aregiven with each item.C H A P T E R12Checklist370 Glencoe Algebra 2Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.• You may want to take the Chapter 12 Practice Test onpage 745 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 12 Study Guide and Reviewon pages 740–744 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 12 Practice Test onpage 745.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 12 Foldable.• Then complete the Chapter 12 Study Guide and Review onpages 740–744 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 12 Practice Test onpage 745.Visit glencoe.com to accessyour textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 12.Student Signature Parent/Guardian SignatureTeacher SignatureARE YOU READY FORTHE CHAPTER TEST?