1.
“Things to A detailed list of important theorems, Review these and you’ll know the 340–341
Know” formulas,and definitions most important material in the chapter!
from the chapter.
“You Should Be Contains a complete list of objectives by Do the recommended exercises and 341–342
Able to…” section,examples that illustrate the you’ll have mastery over the key
objective,and practice exercises that test material. If you get something wrong,
your understanding of the objective. review the suggested page numbers
and try again.
Review These provide comprehensive review and Practice makes perfect.These problems 342–345
Exercises practice of key skills,matched to the combine exercises from all sections,
Learning Objectives for each section. giving you a comprehensive review in
one place.
CHAPTER TEST About 15–20 problems that can be taken Be prepared. Take the sample practice 346
as a Chapter Test.Be sure to take the test under test conditions.This will
Chapter Test under test conditions—no get you ready for your instructor’s test.
notes! If you get a problem wrong,watch
the Chapter Test Prep video.
CUMULATIVE These problem sets appear at the end of These are really important. They will 346–347
REVIEW each chapter,beginning with Chapter 2. ensure that you are not forgetting
They combine problems from previous anything as you go. These will go a long
chapters,providing an ongoing way toward keeping you constantly
cumulative review. primed for the final exam.
CHAPTER The Chapter Project applies what you’ve The Project gives you an opportunity to 347–348
PROJECTS learned in the chapter.Additional projects apply what you’ve learned in the chapter
are available on the Instructor’s to solve a problem related to the
Resource Center (IRC). opening article. If your instructor allows,
these make excellent opportunities to
work in a group,which is often the
best way of learning math.
NEW!
Internetbased In selected chapters,a webbased The projects allow the opportunity for 347–348
Projects project is given. students to collaborate and use
mathematics to deal with issues that
come up in their lives.
Review “Study for Quizzes and Tests”
Feature Description Benefit Page
Chapter Reviews at the end of each chapter contain…
2.
As you begin, you may feel anxious about the number of theorems, definitions,
procedures, and equations. You may wonder if you can learn it all in time. Don’t
worry, your concerns are normal. This textbook was written with you in mind. If
you attend class, work hard, and read and study this book, you will build the
knowledge and skills you need to be successful. Here’s how you can use the book
to your benefit.
Read Carefully
When you get busy, it’s easy to skip reading and go right to the problems. Don’t. . .
the book has a large number of examples and clear explanations to help you break
down the mathematics into easytounderstand steps. Reading will provide you with
a clearer understanding, beyond simple memorization. Read before class (not after)
so you can ask questions about anything you didn’t understand.You’ll be amazed at
how much more you’ll get out of class if you do this.
Use the Features
I use many different methods in the classroom to communicate. Those methods,
when incorporated into the book, are called “features.” The features serve many
purposes, from providing timely review of material you learned before (just when
you need it), to providing organized review sessions to help you prepare for quizzes
and tests.Take advantage of the features and you will master the material.
To make this easier, I’ve provided a brief guide to getting the most from this
book. Refer to the “Prepare for Class,”“Practice,” and “Review” pages on the inside
front cover of this book. Spend fifteen minutes reviewing the guide and familiarizing
yourself with the features by flipping to the page numbers provided. Then, as you
read, use them.This is the best way to make the most of your textbook.
Please do not hesitate to contact me, through Pearson Education, with any
questions, suggestions, or comments that would improve this text. I look forward to
hearing from you, and good luck with all of your studies.
Best Wishes!
Michael Sullivan
To the Student
6.
Prentice Hall
PRECALCULUS
N I N T H E D I T I O N
Michael Sullivan
Chicago State University
Boston Columbus Indianapolis New York San Francisco Upper Saddle River
Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto
Delhi Mexico City Sao Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo
8.
To the Student ii
Preface to the Instructor xvi
Applications Index xxiii
Photo Credits xxx
1 Graphs 1
1.1 The Distance and Midpoint Formulas 2
1.2 Graphs of Equations in Two Variables; Intercepts;
Symmetry 9
1.3 Lines 19
1.4 Circles 34
Chapter Review 41
Chapter Test 43
Chapter Project 43
2 Functions and Their Graphs 45
2.1 Functions 46
2.2 The Graph of a Function 60
2.3 Properties of Functions 68
2.4 Library of Functions; Piecewisedefined Functions 80
2.5 Graphing Techniques:Transformations 90
2.6 Mathematical Models: Building Functions 103
Chapter Review 109
Chapter Test 114
Cumulative Review 115
Chapter Projects 115
3 Linear and Quadratic Functions 117
3.1 Linear Functions and Their Properties 118
3.2 Linear Models: Building Linear Functions from Data 128
3.3 Quadratic Functions and Their Properties 134
3.4 Build Quadratic Models from Verbal Descriptions
and from Data 146
3.5 Inequalities Involving Quadratic Functions 155
Chapter Review 159
Chapter Test 162
Cumulative Review 163
Chapter Projects 164
Table of Contents
ix
9.
4 Polynomial and Rational Functions 165
4.1 Polynomial Functions and Models 166
4.2 Properties of Rational Functions 188
4.3 The Graph of a Rational Function 199
4.4 Polynomial and Rational Inequalities 214
4.5 The Real Zeros of a Polynomial Function 220
4.6 Complex Zeros; Fundamental Theorem
of Algebra 233
Chapter Review 239
Chapter Test 243
Cumulative Review 243
Chapter Projects 244
5 Exponential and Logarithmic Functions 246
5.1 Composite Functions 247
5.2 OnetoOne Functions; Inverse Functions 254
5.3 Exponential Functions 267
5.4 Logarithmic Functions 283
5.5 Properties of Logarithms 296
5.6 Logarithmic and Exponential Equations 305
5.7 Financial Models 312
5.8 Exponential Growth and Decay Models; Newton’s Law;
Logistic Growth and Decay Models 322
5.9 Building Exponential, Logarithmic, and Logistic Models
from Data 332
Chapter Review 340
Chapter Test 346
Cumulative Review 346
Chapter Projects 347
6 Trigonometric Functions 349
6.1 Angles and Their Measure 350
6.2 Trigonometric Functions: Unit Circle Approach 363
6.3 Properties of the Trigonometric Functions 379
6.4 Graphs of the Sine and Cosine Functions 393
6.5 Graphs of the Tangent, Cotangent, Cosecant, and Secant
Functions 408
6.6 Phase Shift; Sinusoidal Curve Fitting 415
Chapter Review 426
Chapter Test 432
Cumulative Review 433
Chapter Projects 434
x Table of Contents
10.
7 Analytic Trigonometry 435
7.1 The Inverse Sine, Cosine, and Tangent Functions 436
7.2 The Inverse Trigonometric Functions (Continued) 448
7.3 Trigonometric Equations 454
7.4 Trigonometric Identities 464
7.5 Sum and Difference Formulas 472
7.6 Doubleangle and Halfangle Formulas 484
7.7 ProducttoSum and SumtoProduct Formulas 494
Chapter Review 498
Chapter Test 502
Cumulative Review 502
Chapter Projects 503
8 Applications of Trigonometric Functions 504
8.1 Right Triangle Trigonometry;Applications 505
8.2 The Law of Sines 517
8.3 The Law of Cosines 528
8.4 Area of a Triangle 535
8.5 Simple Harmonic Motion; Damped Motion;
Combining Waves 541
Chapter Review 550
Chapter Test 554
Cumulative Review 555
Chapter Projects 555
9 Polar Coordinates;Vectors 557
9.1 Polar Coordinates 558
9.2 Polar Equations and Graphs 567
9.3 The Complex Plane; De Moivre’s Theorem 582
9.4 Vectors 589
9.5 The Dot Product 603
9.6 Vectors in Space 610
9.7 The Cross Product 620
Chapter Review 625
Chapter Test 629
Cumulative Review 629
Chapter Projects 630
Table of Contents xi
11.
10 Analytic Geometry 631
10.1 Conics 632
10.2 The Parabola 633
10.3 The Ellipse 642
10.4 The Hyperbola 652
10.5 Rotation of Axes; General Form of a Conic 665
10.6 Polar Equations of Conics 673
10.7 Plane Curves and Parametric Equations 678
Chapter Review 691
Chapter Test 694
Cumulative Review 694
Chapter Projects 695
11 Systems of Equations and Inequalities 696
11.1 Systems of Linear Equations: Substitution
and Elimination 697
11.2 Systems of Linear Equations: Matrices 712
11.3 Systems of Linear Equations: Determinants 727
11.4 Matrix Algebra 736
11.5 Partial Fraction Decomposition 753
11.6 Systems of Nonlinear Equations 761
11.7 Systems of Inequalities 770
11.8 Linear Programming 777
Chapter Review 784
Chapter Test 789
Cumulative Review 790
Chapter Projects 790
12 Sequences; Induction; the Binomial Theorem 792
12.1 Sequences 793
12.2 Arithmetic Sequences 803
12.3 Geometric Sequences; Geometric Series 809
12.4 Mathematical Induction 820
12.5 The Binomial Theorem 824
Chapter Review 830
Chapter Test 833
Cumulative Review 833
Chapter Projects 834
xii Table of Contents
12.
Table of Contents xiii
13 Counting and Probability 835
13.1 Counting 836
13.2 Permutations and Combinations 841
13.3 Probability 850
Chapter Review 860
Chapter Test 862
Cumulative Review 863
Chapter Projects 863
14 A Preview of Calculus:The Limit,
Derivative,and Integral of a Function 865
14.1 Finding Limits Using Tables and Graphs 866
14.2 Algebra Techniques for Finding Limits 871
14.3 Onesided Limits; Continuous Functions 878
14.4 The Tangent Problem;The Derivative 885
14.5 The Area Problem;The Integral 892
Chapter Review 898
Chapter Test 902
Chapter Projects 903
Appendix A Review A1
A.1 Algebra Essentials A1
A.2 Geometry Essentials A14
A.3 Polynomials A22
A.4 Synthetic Division A32
A.5 Rational Expressions A36
A.6 Solving Equations A44
A.7 Complex Numbers; Quadratic Equations in the Complex
Number System A54
A.8 Problem Solving: Interest, Mixture, Uniform Motion,
Constant Rate Job Applications A62
A.9 Interval Notation; Solving Inequalities A72
A.10 nth Roots; Rational Exponents A81
13.
xiv Table of Contents
Appendix B Graphing Utilities B1
B.1 The Viewing Rectangle B1
B.2 Using a Graphing Utility to Graph Equations B3
B.3 Using a Graphing Utility to Locate Intercepts
and Check for Symmetry B5
B.4 Using a Graphing Utility to Solve Equations B6
B.5 Square Screens B8
B.6 Using a Graphing Utility to Graph Inequalities B9
B.7 Using a Graphing Utility to Solve Systems of Linear
Equations B9
B.8 Using a Graphing Utility to Graph a Polar Equation B11
B.9 Using a Graphing Utility to Graph Parametric
Equations B11
Answers AN1
Index I1
14.
Students have different goals, learning styles, and levels of preparation. Instructors
have different teaching philosophies, styles, and techniques. Rather than write one
series to fit all, the Sullivans have written three distinct series. All share the same
goal—to develop a high level of mathematical understanding and an appreciation
for the way mathematics can describe the world around us.The manner of reaching
that goal, however, differs from series to series.
Contemporary Series,Ninth Edition
The Contemporary Series is the most traditional in approach yet modern in its
treatment of precalculus mathematics. Graphing utility coverage is optional and can
be included or excluded at the discretion of the instructor: College Algebra,Algebra &
Trigonometry,Trigonometry, Precalculus.
Enhanced with Graphing Utilities Series,
Fifth Edition
This series provides a more thorough integration of graphing utilities into topics,
allowing students to explore mathematical concepts and foreshadow ideas usually
studied in later courses. Using technology, the approach to solving certain problems
differs from the Contemporary Series, while the emphasis on understanding concepts
and building strong skills does not: College Algebra, Algebra & Trigonometry,
Trigonometry, Precalculus.
Concepts through Functions Series,
Second Edition
This series differs from the others, utilizing a functions approach that serves as the
organizing principle tying concepts together. Functions are introduced early in
various formats. This approach supports the Rule of Four, which states that
functions are represented symbolically, numerically, graphically, and verbally. Each
chapter introduces a new type of function and then develops all concepts pertaining
to that particular function. The solutions of equations and inequalities, instead of
being developed as standalone topics, are developed in the context of the
underlying functions. Graphing utility coverage is optional and can be included or
excluded at the discretion of the instructor: College Algebra; Precalculus, with a Unit
Circle Approach to Trigonometry; Precalculus, with a Right Triangle Approach to
Trigonometry.
xv
Three Distinct Series
15.
xvi
A
s a professor of mathematics at an urban public
university for 35 years, I understand the varied needs
of students taking precalculus. Students range from
being underprepared, with little mathematical background
and a fear of mathematics, to being highly prepared and
motivated.For some,this is their final course in mathematics.
For others, it is preparation for future mathematics courses.
I have written this text with both groups in mind.
A tremendous benefit of authoring a successful series
is the broadbased feedback I receive from teachers and
students who have used previous editions. I am sincerely
grateful for their support. Virtually every change to this
edition is the result of their thoughtful comments and
suggestions. I hope that I have been able to take their ideas
and, building upon a successful foundation of the eighth
edition, make this series an even better learning and
teaching tool for students and teachers.
Features in the Ninth Edition
Rather than provide a list of features here, that
information can be found on the endpapers in the front of
this book.
This places the features in their proper context, as
building blocks of an overall learning system that has been
carefully crafted over the years to help students get the
most out of the time they put into studying. Please take the
time to review this and to discuss it with your students at
the beginning of your course. My experience has been
that when students utilize these features, they are more
successful in the course.
New to the Ninth Edition
• Chapter Projects, which apply the concepts of each
chapter to a realworld situation, have been enhanced
to give students an uptotheminute experience. Many
projects are new and Internetbased, requiring the
student to research information online in order to solve
problems.
• Author Solves It MathXL Video Clips—author Michael
Sullivan works by section through MathXL exercises
typically requested by students for more explanation
or tutoring. These videos are a result of Sullivan’s
experiences in teaching online.
• Showcase Examples are used to present examples in a
guided, stepbystep format. Students can immediately
see how each of the steps in a problem are employed.
The “How To” examples have a twocolumn format in
which the left column describes the step in solving the
problem and the right column displays the algebra
complete with annotations.
• Model It examples and exercises are clearly marked
with a icon.These examples and exercises are meant
to develop the student’s ability to build models from
both verbal descriptions and data. Many of the problems
involving data require the students to first determine
the appropriate model (linear, quadratic, and so on) to
fit to the data and justify their choice.
• Exercise Sets at the end of each section remain classified
according to purpose.The “Are You Prepared?” exercises
have been expanded to better serve the student who
needs a justintime review of concepts utilized in the
section. The Concepts and Vocabulary exercises have
been updated. These fillintheblank and True/False
problems have been written to serve as reading quizzes.
Mixed Practice exercises have been added where
appropriate. These problems offer a comprehensive
assessment of the skills learned in the section by asking
problems that relate to more than one objective. Some
times these require information from previous sections
so students must utilize skills learned throughout the
course. Applications and Extension problems have been
updated and many new problems involving sourced
information and data have been added to bring
relevance and timeliness to the exercises.The Explaining
Concepts: Discussion and Writing exercises have been
updated and reworded to stimulate discussion of
concepts in online discussion forums. These can also be
used to spark classroom discussion. In selected exercise
sets, Interactive Exercises are presented. These applets
provide a “handson” experience allowing students to
interact with Mathematics in an active learning
environment. Finally, in the Annotated Instructor’s
Edition, I have preselected problems that can serve as
sample homework assignments. These are indicated by a
blue underline, and they are assignable in MyMathLab®
if desired.
• The Chapter Review now identifies Examples to review
for each objective in the chapter.
Changes in the Ninth Edition
• CONTENT
❍ Chapter 2, Section 3 A new objective “Use a graph
to locate the absolute maximum and the absolute
minimum” has been added. The Extreme Value
Theorem is also cited here.
❍ Chapter 3, Section 3 A new objective “Find a
quadratic function given its vertex and one point”
has been added.
❍ Chapter 4, Section 1 A new objective “Build cubic
models from data” has been added.
Preface to the Instructor
16.
❍ Chapter 4, Section 5 Descartes’ Rule of Signs has
been removed as its value is redundant to the
information collected from other sources.
❍ Chapter 5, Section 3 The definition of an exponential
function has been broadened.
❍ Chapter 9, Section 5 More applications of decom
posing vectors have been added.
• ORGANIZATION
❍ Appendix A, Section A.3 The objective “Complete
the Square” has been relocated to here from
Section A.6.
❍ Chapter 7 The two sections on trigonometric equations,
Trigonometric Equations (I) and Trigonometric Equa
tions (II), have been consolidated into a new section in
Chapter 7, Section 3, entitled Trigonometric Equations.
In addition, trigonometric equations that utilize specific
identities have been woven into the appropriate sections
throughout the remainder of Chapter 7.
❍ Chapter 9 The material on applications of vectors
that was formerly in Section 5 on the Dot Product
has been moved to Section 4 to emphasize the
applications of the resultant vector.
Using the Ninth Edition Effectively
with Your Syllabus
To meet the varied needs of diverse syllabi, this book
contains more content than is likely to be covered in a
Precalculus course. As the chart illustrates, this book has
been organized with flexibility of use in mind. Within a
given chapter, certain sections are optional (see the detail
following the flow chart) and can be omitted without loss
of continuity.
Appendix A Review
This chapter consists of review material. It may be used as
the first part of the course or later as a justintime review
when the content is required. Specific references to this
chapter occur throughout the book to assist in the review
process.
Chapter 1 Graphs
This chapter lays the foundation for functions.
2
1Appendix A
Appendix B
10.1Ϫ10.4
10.5Ϫ10.7
9.1Ϫ9.3
3 4 5 6 12
14
1311
7
8
9.4Ϫ9.7
Chapter 2 Functions and Their Graphs
Perhaps the most important chapter. Section 2.6 is
optional.
Chapter 3 Linear and Quadratic Functions
Topic selection depends on your syllabus. Sections 3.2
and 3.4 may be omitted without a loss of continuity.
Chapter 4 Polynomial and Rational
Functions
Topic selection depends on your syllabus.
Chapter 5 Exponential and Logarithmic
Functions
Sections 5.1–5.6 follow in sequence. Sections 5.7, 5.8,
and 5.9 are optional.
Chapter 6 Trigonometric Functions
Section 6.6 may be omitted in a brief course.
Chapter 7 Analytic Trigonometry
Section 7.7 may be omitted in a brief course.
Chapter 8 Applications of Trigonometric
Functions
Sections 8.4 and 8.5 may be omitted in a brief course.
Chapter 9 Polar Coordinates;Vectors
Sections 9.1–9.3 and Sections 9.4–9.7 are independent and
may be covered separately.
Chapter 10 Analytic Geometry
Sections 10.1–10.4 follow in sequence. Sections 10.5, 10.6,
and 10.7 are independent of each other, but each requires
Sections 10.1–10.4.
Chapter 11 Systems of Equations and
Inequalities
Sections 11.2–11.7 may be covered in any order, but each
requires Section 11.1. Section 11.8 requires Section 11.7.
Chapter 12 Sequences; Induction;The Binomial
Theorem
There are three independent parts: Sections 12.1–12.3;
Section 12.4; and Section 12.5.
Chapter 13 Counting and Probability
The sections follow in sequence.
Chapter 14 A Preview of Calculus:The Limit,
Derivative,and Integral of a Function
If time permits, coverage of this chapter will give your
students a beneficial head start in calculus.
Preface to the Instructor xvii
17.
Acknowledgments
Textbooks are written by authors, but evolve from an idea
to final form through the efforts of many people. It was
Don Dellen who first suggested this book and series to me.
Don is remembered for his extensive contributions to
publishing and mathematics.
Thanks are due to the following people for their
assistance and encouragement to the preparation of this
edition:
• From Pearson Education:Anne Kelly for her substantial
contributions,ideas,and enthusiasm;Peggy Lucas,who is
a huge fan and supporter; Dawn Murrin, for her
unmatched talent at getting the details right; Sherry Berg
of Nesbitt Graphics, Inc., for her superb organizational
skills in directing production; Peggy McMahon for her
leadership in overseeing production; Chris Hoag for her
continued support and genuine interest; Greg Tobin for
his leadership and commitment to excellence; and the
Pearson Math and Science Sales team,for their continued
confidence and personal support of our books.
• As this book went to production,BobWalters,Production
Manager, passed away after a long and valiant battle
fighting lung disease. He was an old and dear friend—a
true professional in every sense of the word.
• Accuracy checkers: C. Brad Davis, who read the entire
manuscript and accuracy checked answers. His attention
to detail is amazing; Timothy Britt, for creating the
Solutions Manuals and accuracy checking answers.
• Reviewers: Larissa Williamson, University of Florida;
Richard Nadel, Florida International University;
Robin Steinberg, Puma CC; Mike Rosenthal, Florida
International University; Gerardo Aladro, Florida
International University; Tammy Muhs, Universty of
Central Florida; Val Mohanakumar, Hillsborough CC.
Finally, I offer my grateful thanks to the dedicated
users and reviewers of my books, whose collective insights
form the backbone of each textbook revision.
My list of indebtedness just grows and grows. And, if
I’ve forgotten anyone, please accept my apology. Thank
you all.
Roger Carlsen, Moraine Valley Community College
Elena Catoiu, Joliet Junior College
Mathews Chakkanakuzhi, Palomar College
Tim Chappell, Penn Valley Community College
John Collado, South Suburban College
Alicia Collins, Mesa Community College
Nelson Collins, Joliet Junior College
Jim Cooper, Joliet Junior College
Denise Corbett, East Carolina University
Carlos C. Corona, San Antonio College
Theodore C. Coskey, South Seattle Community College
Donna Costello, Plano Senior High School
Paul Crittenden, University of Nebraska at Lincoln
John Davenport, East Texas State University
Faye Dang, Joliet Junior College
Antonio David, Del Mar College
Stephanie Deacon, Liberty University
Duane E. Deal, Ball State University
Jerry DeGroot, Purdue North Central
Timothy Deis, University of WisconsinPlatteville
Joanna DelMonaco, Middlesex Community College
Vivian Dennis, Eastfield College
Deborah Dillon, R. L.Turner High School
Guesna Dohrman,Tallahassee Community College
Cheryl Doolittle, Iowa State University
Karen R. Dougan, University of Florida
Jerrett Dumouchel, Florida Community College
at Jacksonville
Louise Dyson, Clark College
Paul D. East, Lexington Community College
Don Edmondson, University of TexasAustin
Erica Egizio, Joliet Junior College
Jason Eltrevoog, Joliet Junior College
Christopher Ennis, University of Minnesota
Kathy Eppler, Salt Lake Community College
James Africh, College of DuPage
Steve Agronsky, Cal Poly State University
Grant Alexander, Joliet Junior College
Dave Anderson, South Suburban College
Richard Andrews, Florida A&M University
Joby Milo Anthony, University of Central Florida
James E.Arnold, University of WisconsinMilwaukee
Adel Arshaghi, Center for Educational Merit
Carolyn Autray, University of West Georgia
Agnes Azzolino, Middlesex County College
Wilson P Banks, Illinois State University
Sudeshna Basu, Howard University
Dale R. Bedgood, East Texas State University
Beth Beno, South Suburban College
Carolyn Bernath,Tallahassee Community College
Rebecca Berthiaume, Edison State College
William H. Beyer, University of Akron
Annette Blackwelder, Florida State University
Richelle Blair, Lakeland Community College
Kevin Bodden, Lewis and Clark College
Barry Booten, Florida Atlantic University
Larry Bouldin, Roane State Community College
Bob Bradshaw, Ohlone College
Trudy Bratten, Grossmont College
Tim Bremer, Broome Community College
Tim Britt, Jackson State Community College
Michael Brook, University of Delaware
Joanne Brunner, Joliet Junior College
Warren Burch, Brevard Community College
Mary Butler, Lincoln Public Schools
Melanie Butler,West Virginia University
Jim Butterbach, Joliet Junior College
William J. Cable, University of WisconsinStevens Point
Lois Calamia, Brookdale Community College
Jim Campbell, Lincoln Public Schools
xviii Preface to the Instructor
18.
Preface to the Instructor xix
Arthur Kaufman, College of Staten Island
Thomas Kearns, North Kentucky University
Jack Keating, Massasoit Community College
Shelia Kellenbarger, Lincoln Public Schools
Rachael Kenney, North Carolina State University
Debra Kopcso, Louisiana State University
Lynne Kowski, Raritan Valley Community College
Yelena Kravchuk, University of Alabama at
Birmingham
Keith Kuchar, Manatee Community College
Tor Kwembe, Chicago State University
Linda J. Kyle,Tarrant Country Jr. College
H.E. Lacey,Texas A & M University
Harriet Lamm, Coastal Bend College
James Lapp, Fort Lewis College
Matt Larson, Lincoln Public Schools
Christopher Lattin, Oakton Community College
Julia Ledet, Lousiana State University
Adele LeGere, Oakton Community College
Kevin Leith, University of Houston
JoAnn Lewin, Edison College
Jeff Lewis, Johnson County Community College
Janice C. Lyon,Tallahassee Community College
Jean McArthur, Joliet Junior College
Virginia McCarthy, Iowa State University
Karla McCavit,Albion College
Michael McClendon, University of Central Oklahoma
Tom McCollow, DeVry Institute of Technology
Marilyn McCollum, North Carolina State University
Jill McGowan, Howard University
Will McGowant, Howard University
Angela McNulty, Joliet Junior College
Laurence Maher, North Texas State University
Jay A. Malmstrom, Oklahoma City
Community College
Rebecca Mann,Apollo High School
Lynn Marecek, Santa Ana College
Sherry Martina, Naperville North High School
Alec Matheson, Lamar University
Nancy Matthews, University of Oklahoma
James Maxwell, Oklahoma State UniversityStillwater
Marsha May, Midwestern State University
James McLaughlin,West Chester University
Judy Meckley, Joliet Junior College
David Meel, Bowling Green State University
Carolyn Meitler, Concordia University
Samia Metwali, Erie Community College
Rich Meyers, Joliet Junior College
Eldon Miller, University of Mississippi
James Miller,West Virginia University
Michael Miller, Iowa State University
Kathleen Miranda, SUNY at Old Westbury
Chris Mirbaha,The Community College
of Baltimore County
Val Mohanakumar, Hillsborough Community College
Thomas Monaghan, Naperville North High School
Miguel Montanez, Miami Dade College,
Wolfson Campus
Ralph Esparza, Jr., Richland College
Garret J. Etgen, University of Houston
Scott Fallstrom, Shoreline Community College
Pete Falzone, Pensacola Junior College
W.A. Ferguson, University of IllinoisUrbana/Champaign
Iris B. Fetta, Clemson University
Mason Flake, student at Edison Community College
Timothy W. Flood, Pittsburgh State University
Robert Frank,Westmoreland County
Community College
Merle Friel, Humboldt State University
Richard A. Fritz, Moraine Valley
Community College
Dewey Furness, Ricke College
Randy Gallaher, Lewis and Clark College
Tina Garn, University of Arizona
Dawit Getachew, Chicago State University
Wayne Gibson, Rancho Santiago College
Robert Gill, University of Minnesota Duluth
Nina Girard, University of Pittsburgh at Johnstown
Sudhir Kumar Goel,Valdosta State University
Adrienne Goldstein, Miami Dade College,
Kendall Campus
Joan Goliday, Sante Fe Community College
Lourdes Gonzalez, Miami Dade College, Kendall Campus
Frederic Gooding, Goucher College
Donald Goral, Northern Virginia Community College
Sue Graupner, Lincoln Public Schools
Mary Beth Grayson, Liberty University
Jennifer L. Grimsley, University of Charleston
Ken Gurganus, University of North Carolina
James E. Hall, University of WisconsinMadison
Judy Hall,West Virginia University
Edward R. Hancock, DeVry Institute of Technology
Julia Hassett, DeVry InstituteDupage
Christopher HayJahans, University of South Dakota
Michah Heibel, Lincoln Public Schools
LaRae Helliwell, San Jose City College
Celeste Hernandez, Richland College
Gloria P. Hernandez, Louisiana State University
at Eunice
Brother Herron, Brother Rice High School
Robert Hoburg,Western Connecticut State University
Lynda Hollingsworth, Northwest Missouri State
University
Charla Holzbog, Denison High School
Lee Hruby, Naperville North High School
Miles Hubbard, St. Cloud State University
Kim Hughes, California State CollegeSan Bernardino
Ron Jamison, Brigham Young University
Richard A. Jensen, Manatee Community College
Glenn Johnson, Middlesex Community College
Sandra G. Johnson, St. Cloud State University
Tuesday Johnson, New Mexico State University
Susitha Karunaratne, Purdue University
North Central
Moana H. Karsteter,Tallahassee Community College
Donna Katula, Joliet Junior College
19.
xx Preface to the Instructor
Maria Montoya, Our Lady of the Lake University
Susan Moosai, Florida Atlantic University
Craig Morse, Naperville North High School
Samad Mortabit, Metropolitan State University
Pat Mower,Washburn University
A. Muhundan, Manatee Community College
Jane Murphy, Middlesex Community College
Richard Nadel, Florida International University
Gabriel Nagy, Kansas State University
Bill Naegele, South Suburban College
Karla Neal, Lousiana State University
Lawrence E. Newman, Holyoke Community College
Dwight Newsome, PascoHernando Community College
Denise Nunley, Maricopa Community Colleges
James Nymann, University of TexasEl Paso
Mark Omodt,AnokaRamsey Community College
Seth F. Oppenheimer, Mississippi State University
Leticia Oropesa, University of Miami
Linda Padilla, Joliet Junior College
E. James Peake, Iowa State University
Kelly Pearson, Murray State University
Dashamir Petrela, Florida Atlantic University
Philip Pina, Florida Atlantic University
Michael Prophet, University of Northern Iowa
Laura Pyzdrowski,West Virginia University
Neal C. Raber, University of Akron
Thomas Radin, San Joaquin Delta College
Aibeng Serene Radulovic, Florida Atlantic University
Ken A. Rager, Metropolitan State College
Kenneth D. Reeves, San Antonio College
Elsi Reinhardt,Truckee Meadows Community College
Jose Remesar, Miami Dade College,Wolfson Campus
Jane Ringwald, Iowa State University
Stephen Rodi,Austin Community College
William Rogge, Lincoln Northeast High School
Howard L. Rolf, Baylor University
Mike Rosenthal, Florida International University
Phoebe Rouse, Lousiana State University
Edward Rozema, University of Tennessee at Chattanooga
Dennis C. Runde, Manatee Community College
Alan Saleski, Loyola University of Chicago
Susan Sandmeyer, Jamestown Community College
Brenda Santistevan, Salt Lake Community College
Linda Schmidt, Greenville Technical College
Ingrid Scott, Montgomery College
A.K. Shamma, University of West Florida
Martin Sherry, Lower Columbia College
Carmen Shershin, Florida International University
Tatrana Shubin, San Jose State University
Anita Sikes, Delgado Community College
Timothy Sipka,Alma College
Charlotte Smedberg, University of Tampa
Lori Smellegar, Manatee Community College
Gayle Smith, Loyola Blakefield
Leslie Soltis, Mercyhurst College
John Spellman, Southwest Texas State University
Karen Spike, University of North Carolina
Rajalakshmi Sriram, OkaloosaWalton Community College
Katrina Staley, North Carolina Agricultural and Technical
State University
Becky Stamper,Western Kentucky University
Judy Staver, Florida Community CollegeSouth
Neil Stephens, Hinsdale South High School
Sonya Stephens, Florida A&M University
Patrick Stevens, Joliet Junior College
John Sumner, University of Tampa
Matthew TenHuisen, University of North
Carolina,Wilmington
Christopher Terry,Augusta State University
Diane Tesar, South Suburban College
Tommy Thompson, Brookhaven College
Martha K.Tietze, Shawnee Mission Northwest High School
Richard J.Tondra, Iowa State University
Suzanne Topp, Salt Lake Community College
Marilyn Toscano, University of Wisconsin, Superior
Marvel Townsend, University of Florida
Jim Trudnowski, Carroll College
Robert Tuskey, Joliet Junior College
Mihaela Vajiac, Chapman UniversityOrange
Richard G.Vinson, University of South Alabama
Jorge ViolaPrioli, Florida Atlantic University
Mary Voxman, University of Idaho
Jennifer Walsh, Daytona Beach Community College
Donna Wandke, Naperville North High School
Timothy L.Warkentin, Cloud County
Community College
Hayat Weiss, Middlesex Community College
Kathryn Wetzel,Amarillo College
Darlene Whitkenack, Northern Illinois University
Suzanne Williams, Central Piedmont
Community College
Larissa Williamson, University of Florida
Christine Wilson,West Virginia University
Brad Wind, Florida International University
Anna Wiodarczyk, Florida International University
Mary Wolyniak, Broome Community College
Canton Woods,Auburn University
Tamara S.Worner,Wayne State College
Terri Wright, New Hampshire Community Technical
College, Manchester
George Zazi, Chicago State University
Steve Zuro, Joliet Junior College
Michael Sullivan
Chicago State University
20.
xxi
Available to students are the following supplements:
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Fully worked solutions to oddnumbered exercises.
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STUDENT RESOURCES
21.
• Pearson Tutor Center (www.pearsontutorservices.com) access is automatically included with MyMathLab. The Tutor Center is
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xxii
22.
xxiii
Acoustics
amplifying sound, 344
loudness of sound, 295
loudspeaker, 548
tuning fork, 548–49
whispering galleries, 648–49, 693
Aerodynamics
modeling aircraft motion, 630
Aeronautics
Challenger disaster, 331
Agriculture
crop allocation, 788
farm management, 783
field enclosure, 768
grazing area for cow, 539–40
minimizing cost, 783
removing stump, 602–3
Air travel
bearing of aircraft, 515, 526
frequent flyer miles, 526
holding pattern, 462
parking at O’Hare International
Airport, 88
revising a flight plan, 533
speed and direction of aircraft, 597–98,
601–2, 628
Archaeology
age of ancient tools, 324–25
age of fossil, 330
age of tree, 330
date of prehistoric man’s death, 344
Architecture
brick staircase, 808, 832
Burj Khalifa building,A15
floor design, 806–7, 832
football stadium seating, 808
mosaic design, 808, 832
Norman window, 153,A21
parabolic arch, 153
racetrack design, 651
special window, 153, 161
stadium construction, 808
window design, 153
Area
of Bermuda Triangle, 539
under a curve, 447–48
of isosceles triangle, 493
of sector of circle, 360
of segment of circle, 551
Art
fine decorative pieces, 378
Astronomy
angle of elevation of Sun, 514
distances of planets from Sun, 802
planetary orbits, 648
Earth, 651
Jupiter, 651
Mars, 651
Mercury, 678
Neptune, 695
Pluto, 651, 695
Aviation
modeling aircraft motion, 630
orbital launches, 709
speed of plane,A71
Biology
alcohol and driving, 291, 296
bacterial growth, 323–24, 336–37
Ecoli, 79, 119
blood types, 840–41
bone length, 161–62
cricket chirp rate and temperature, 155
healing of wounds, 281, 295
maternal age versus Down
syndrome, 134
yeast biomass as function of time,
335–36
Business
advertising, 133, 162
automobile production, 253–54, 726
blending coffee,A69
car rentals, 125
checkout lines, 859
cigarette exports, 337
clothing store, 861
commissions, 161
cookie orders, 788
cost
of can, 210, 213
of commodity, 254
of manufacturing, 186, 219, 776–77,
A13,A69
marginal, 145, 161
minimizing, 161, 783
of production, 79, 254, 752, 789
of transporting goods, 89
cost equation, 32
cost function, 126
average, 63
demand
for jeans, 133
for PCs, 337
demand equation, 161, 244
depreciation, 246
discounts, 254
drivethru rate
at Burger King, 277
at Citibank, 281, 295
at McDonald’s, 281
equipment depreciation, 818
expense computation,A70
Jiffy Lube’s car arrival rate, 281, 295
managing a meat market, 783
mixing candy,A69
mixing nuts,A69
newcar markup,A81
orange juice production, 726
precision ball bearings,A13
price vs. quantity demanded, 127,
337–38
product design, 784
production scheduling, 783
product promotion, 33
profit, 752
cigar company, 102
on figurines, 790
maximizing, 781–82, 783–84
profit function, 59
rate of return on, 320
restaurant management, 710
revenue, 145, 158–59,A69
advertising and, 133
airline, 784
of clothing store, 741
daily, 145
from digital music, 102
instantaneous rate of change of, 891,
892, 901
maximizing, 145, 152
monthly, 145
theater, 711
RV rental, 163
salary, 808
gross, 58
increases in, 818, 832
sales
commission on,A80
of movie theater ticket, 697, 702, 709
net, 8
profit from,A71
salvage value, 344
straightline depreciation, 122–23, 126
supply and demand, 123–24, 126
tax, 219
toy truck manufacturing, 776–77
Applications Index
23.
transporting goods, 777
truck rentals, 32, 127
unemployment, 862
wages
of car salesperson, 32
Calculus
area under a curve, 103, 447–48
area under graph, 78
carrying a ladder around a corner,
462–63
maximizing rain gutter construction,
492–93
projectile motion, 462–63
Simpson’s rule, 153
Carpentry
pitch, 33
Chemistry
alpha particles, 664
decomposition reactions, 331
drug concentration, 212
pH, 294
purity of gold,A70
radioactive decay, 330, 337, 344
radioactivity from Chernobyl, 331
reactions, 153
salt solutions,A70
sugar molecules,A70–A71
volume of gas,A80
Combinatorics
airport codes, 842
binary codes, 862
birthday permutations, 844, 848, 849,
855–56, 860, 862
blouses and skirts combinations, 840
book arrangements, 848, 861
box stacking, 848
code formation, 848
combination locks, 849
committee formation, 846, 848–49, 862
Senate committees, 849
flag arrangement, 847, 862
letter codes, 842, 862
letter combinations, 861
license plate possibilities, 848, 861, 863
light bulb wattage, 862
lining up people, 843, 848
number formation, 840, 848, 849, 862
objects selection, 849
seating arrangements, 861
shirts and ties combinations, 840
telephone numbers, 861
twosymbol codewords, 839
word formation, 846–47, 849, 862
Communications
cell phone service, 88, 115
cell phone usage, 333–34, 339
death rates, 862
diversity index, 294–95
divorced population, 150–51
life expectancy,A80
marital status, 841
mosquito colony growth, 330
population of world, future of, 903
poverty rates, 242
rabbit colony growth, 801
Design
of awning, 527
of box with minimum surface area,
212–13
of fine decorative pieces, 378
of Little League Field, 362
of water sprinkler, 361
Direction
of aircraft, 597–98, 601, 628
compass heading, 602
for crossing a river, 602
of fireworks display, 663–64
of lightning strikes, 664
of motorboat, 602
of swimmer, 628
Distance
Bermuda Triangle,A21
bicycle riding, 68
from Chicago to Honolulu, 448
circumference of Earth, 362
between cities, 356–57, 361
between Earth and Mercury, 527
between Earth and Venus, 527
from Earth to a star, 514–15
of explosion, 664
height
of aircraft, 344, 525, 527
of bouncing ball, 818, 832
of bridge, 525
of building, 514, 515
of cloud, 510
of Eiffel Tower, 514
of embankment, 515
of Ferris Wheel rider, 462
of Great Pyramid of Cheops,
527,A21
of helicopter, 552
of hotair balloon, 515
of Lincoln’s caricature on
Mt. Rushmore, 515
of mountain, 344, 522, 525
of statue on a building, 510–11
of tower, 516
of tree, 525
of Washington Monument, 515
of Willis Tower, 515
from home, 68
from Honolulu to Melbourne,
Australia, 448
cellular telephone plan, 45
installing cable TV, 108
long distance, 127
comparing phone companies, 161
phone charges, 126
satellite dish, 638–39, 640
spreading of rumors, 281, 295
TouchTone phones, 497, 549
Computers and computing
Dell PCs, 337–38
graphics, 601
laser printers,A70
Construction
of box, 768,A71
closed, 113
open, 108–9
of brick staircase, 832
of can, 242
of coffee can,A71
of cylindrical tube, 768
of enclosures
around garden,A70
around pond,A70
maximizing area of, 148, 152, 161
of fencing, 148, 152, 161, 768
minimum cost for, 212
of flashlight, 640–41
of garden walk, 433
of headlight, 641
of highway, 515, 526, 552
home choices, 861
installing cable TV, 108
pitch of roof, 516
of playpen, 105–6
of rain gutter, 153, 371–72, 492–93, 506–7
of ramp, 525
access ramp, 32
of rectangular field enclosure, 152
of stadium, 153, 808
of steel drum, 213
of swimming pool,A21,A22
of swing set, 534
of tent, 538
TV dish, 640
vent pipe installation, 651
Crime
income vs. rate of, 339
Cryptography
matrices in, 752–53
Decorating
Christmas tree,A16
Demographics
age and high school diplomas, 155
birthrate of unmarried women, 145
xxiv Applications Index
24.
of hotair balloon
to airport, 554
from intersection, 8
from intersection, 107–8
length
of guy wire, 533
of mountain trail, 515
of ski lift, 525
limiting magnitude of
telescope, 344
to the Moon, 525
pendulum swings, 814, 818
to plateau, 514
across a pond, 514
range of airplane,A71
reach of ladder, 514
of rotating beacon, 415
at sea, 526, 552
to shore, 514, 526
between skyscrapers, 516, 517
surveying, 553
to tower, 527
traveled by wheel,A21
between two moving vehicles, 8
toward intersection, 107–8
between two objects, 514, 515
visibility of Gibb’s Hill Lighthouse beam,
511–12, 517,A22
visual,A22
walking, 68
width
of gorge, 513
of Mississippi River, 516
of river, 509
Economics
Consumer Price Index (CPI), 321
Dell personal computer price and
demand, 337–38
demand equations, 244
federal stimulus package of
2009, 320
inflation, 320
ISLM model in, 710
marginal propensity
to consume, 819
multiplier, 819
participation rate, 59
per capita federal debt, 320
poverty rates, 242
poverty threshold, 9
relative income of child, 753
unemployment, 862
Education
age distribution of community
college, 863
college costs, 320, 818–19
college tuition and fees, 752
degrees awarded, 838
doctorate degrees awarded, 859
faculty composition, 860
field trip, 220
funding a college education, 344
grade computation,A81
high school diploma, 155
IQ tests,A81
learning curve, 282, 295
maximum level achieved, 790–91
multiplechoice test, 848
Spring break, 783
student loan, 114
interest on, 752
test design, 861
true/false test, 848
video games and gradepoint
average, 133
Electricity
alternating current (ac), 431, 483
alternating current (ac) circuits,
406, 424
alternating current (ac) generators,
406–7
alternating voltage, 431
charging a capacitor, 549
cost of, 86–87
current in RC circuit, 282
current in RL circuit, 282, 295
impedance,A61
Kirchhoff’s Rules, 710–11, 726
light bulbs, 863
parallel circuits,A61
resistance in, 198
rates for, 32,A80
resistance, 198,A43
voltage
foreign,A13
U.S.,A13
wiring, 861
Electronics
loudspeakers, 548
microphones, 18
sawtooth curve, 493, 549
Energy
nuclear power plant, 664
solar, 18, 609
solar heat, 641
thermostat control, 101–2
Engineering
bridges
clearance, 407
Golden Gate, 149–50
parabolic arch, 161, 641, 693
semielliptical arch, 651, 693
suspension, 153, 641
drive wheel, 553
Gateway Arch (St. Louis), 641
grade of road, 33
lean of Leaning Tower of Pisa, 526
moment of inertia, 497
piston engines, 378
product of inertia, 493
road system, 566
robotic arm, 619
rods and pistons, 534
searchlight, 471, 641, 693
whispering galleries, 651, 693
Entertainment
cable subscribers, 338–39
Demon Roller Coaster customer rate,
281–82
movie theater, 447
theater revenues, 711
Environment
lake pollution control laws, 801
oil leakage, 254
Exercise and fitness
for weight loss,A80
Finance
balancing a checkbook,A13
bills in wallet, 862
clothes shopping, 790
college costs, 320, 818–19
computer system purchase, 320
cost
of car rental, 89
of driving car, 32
of electricity, 86–87
of fast food, 710
of land, 553
minimizing, 161, 212
of natural gas, 88
of transAtlantic travel, 59, 67
of triangular lot, 538
cost function, 126
cost minimization, 145
credit cards
debt, 801
interest on, 319
payment, 89, 801
depreciation, 281, 339
of car, 311, 347–48
discounts, 254
division of money,A64,A69
electricity rates, 32
federal income tax,A80
federal stimulus package of
2009, 320
financial planning, 709–10, 722–23, 725,
A64,A69
foreign exchange, 254
funding a college education, 344
future value of money, 186–87
gross salary, 58
income vs. crime rate, 339
life cycle hypothesis, 154
Applications Index xxv
25.
xxvi Applications Index
loans,A69
car, 801
interest on, 114, 752,A64
repayment of, 319
student, 752
mortgages
interest rates on, 320
second, 320
phone charges, long distance, 127
price appreciation of homes, 319
prices of fast food, 711
price vs. quantity demanded, 127
refunds, 710
revenue maximization, 145,
146–47, 152
rich man’s promise, 819
salary options, 819–20
saving
for a car, 319
for a home, 818
savings accounts interest, 320
sinking fund, 818–19
taxes, 126
efiling returns, 79
federal income, 89, 266
luxury, 126
usedcar purchase, 319
water bills,A80–A81
Food and nutrition
animal, 784
candy, 132
color mix of candy, 863
cooler contents, 863
cooling time of pizza, 330
fast food, 710, 711
Girl Scout cookies, 859
hospital diet, 711, 725
ice cream, 783
“light” foods,A81
number of possible meals, 838–39
pig roasts, 331
raisins, 132–33
warming time of Beer stein, 331
Forestry
wood product classification, 329
Games
die rolling, 863
grains of wheat on a chess board, 819
Powerball, 863
Gardens and gardening.See also
Landscaping
enclosure for,A70
Geography
area of Bermuda Triangle, 539
area of lake, 539, 553
inclination of hill, 597
inclination of mountain trail,
509, 552
Geology
earthquakes, 296
Geometry
angle between two lines, 483
balloon volume, 253
circle
area of, 539,A69
center of, 40
circumference of,A12,A69
inscribed in square, 107
length of chord of, 534
radius of, 40, 768
collinear points, 736
cone volume, 254
cube
length of edge of, 233
surface area of,A13
volume of,A13
cylinder
inscribing in cone, 108
inscribing in sphere, 108
volume of, 254
Descartes’s method of equal
roots, 768
equation of line, 736
ladder angle, 554
quadrilateral area, 555
rectangle
area of, 58, 104–5,A12
dimensions of, 768
inscribed in semicircle,
107, 493
perimeter of,A12
semicircle inscribed in, 107
semicircle area, 538, 555
sphere
surface area of,A13
volume of,A13
square
area of,A21,A69
perimeter of,A69
surface area
of balloon, 253
of cube,A13
of sphere,A13
triangle
area of, 538, 539, 555, 736,A13
circumscribing, 528
equilateral,A13
inscribed in circle, 107
isosceles, 58, 555, 768
Pascal’s, 801
perimeter of,A13
right, 513
sides of, 555
volume of parallelepiped, 625
Government
federal deficit, 344
federal income tax, 59, 89, 266,A80
efiling returns, 79
federal stimulus package of 2009, 320
firstclass mail, 89
per capita federal debt, 320
Health.See also Medicine
age vs. total cholesterol, 339
blood pressure, 462
cigarette use among teens, 32–33
expenditures on, 59
heartbeats during exercise, 120–21
ideal body weight, 266
life cycle hypothesis, 154
Home improvement.See also
Construction
painting a house, 711
Housing
apartment rental, 154
number of rooms in, 58
price appreciation of homes, 319
Investment(s)
annuity, 815–16, 818
in bonds, 784
Treasuries, 725, 726, 774, 776, 778
zerocoupon, 317, 320–21
in CDs, 784
compound interest on, 312–13, 314,
315–16, 319–20
diversified, 711
division among instruments,A69
doubling of, 317–18, 321
finance charges, 319
in fixedincome securities,
320–21, 784
401K, 818, 832
growth rate for, 319–20
IRA, 320, 344, 815–16, 818, 832
money market account, 316
return on, 319–20, 783, 784
savings account, 315–16
in stock
analyzing, 164
appreciation, 319–20
NASDAQ stocks, 848
NYSE stocks, 848
portfolios of, 841
price of, 819
time to reach goal, 319, 321
tripling of, 318, 321
Landscaping.See also Gardens
and gardening
building a walk, 433
height of tree, 525
26.
Applications Index xxvii
pond enclosure, 161
rectangular pond border, 161
removing stump, 602–3
tree planting, 725
watering lawn, 361
Law and law enforcement
motor vehicle thefts, 859
violent crimes, 59
Leisure and recreation
cable TV, 108
community skating rink, 114
Ferris wheel, 39, 462, 527, 548
field trip, 220
video games and gradepoint
average, 133
Marketing
Dell personal computer price and
demand, 337–38
Measurement
optical methods of, 471
of rainfall, 609
Mechanics.See Physics
Medicine.See also Health
age vs. total cholesterol, 339
blood pressure, 462
drug concentration, 79, 212
drug medication, 281, 295
healing of wounds, 281, 295
spreading of disease, 345
Meteorology
weather balloon height and atmospheric
pressure, 334–35
Miscellaneous
banquet seating, 783
bending wire, 768
biorhythms, 407
carrying a ladder around a
corner, 415, 462–63
citrus ladders, 808
coffee container, 348
crosssectional area of beam, 59, 66
curve fitting, 710, 725, 788
drafting error, 8
Mandelbrot sets, 589
pet ownership, 859
rescue at sea, 522–23, 525
surface area of balloon, 253
surveillance satellites, 516–17
volume of balloon, 253
wire enclosure area, 107
working together on a job,A70
Mixtures.See also Chemistry
blending coffees, 777, 788,
A64–A65,A69
blending teas,A69
cement,A71
mixed nuts, 709, 777, 788,A69
mixing candy,A69
water and antifreeze,A70
Motion,549.See also Physics
catching a train, 694
on a circle, 361
of Ferris Wheel rider, 462
of golf ball, 66
minute hand of clock, 360, 431
objects approaching intersection, 690
of pendulum, 549
revolutions of circular disk,A21
simulating, 684
tortoise and the hare race, 768
uniform, 690,A65–A67,A69–A70
Motor vehicles
alcohol and driving, 291, 296
approaching intersection, 690
automobile production, 253–54, 726
average car speed,A71
brake repair with tuneup, 862
braking load, 609–10, 628
crankshafts, 526
depreciation of, 246, 311, 339, 347–48
gas mileage and speed and, 154–55
with Global Positioning System
(GPS), 345
loans for, 801
newcar markup,A81
runaway car, 159
spin balancing tires, 362, 431
stopping distance, 145, 266
usedcar purchase, 319
Music
revenues from, 102
Navigation
avoiding a tropical storm, 533
bearing, 511–12, 532
of aircraft, 515, 526
of ship, 515, 553
commercial, 525–26
compass heading, 602
crossing a river, 602
error in
correcting, 530–31, 552
time lost due to, 526
rescue at sea, 522–23, 525
revising a flight plan, 533
Oceanography
tides, 425–26
Optics
angle of refraction, 463–64
bending light, 464
index of refraction, 463–64
laser beam, 514
laser projection, 493
lensmaker’s equation,A43
light obliterated through glass, 281
mirrors, 664
reflecting telescope, 641
Pediatrics
height vs. head circumference,
133, 266
Pharmacy
vitamin intake, 710, 726
Photography
camera distance, 515
Physics
angle of elevation of Sun, 514
bouncing balls, 832
braking load, 610
damped motion, 544
Doppler effect, 212
effect of elevation on weight, 67
force, 601,A69
to hold a wagon on a hill, 607
resultant, 601
gravity, 198, 220
on Earth, 58, 266
on Jupiter, 58
harmonic motion, 543
heat transfer, 462
inclination of mountain trail, 509
kinetic energy,A69
missile trajectory, 164
moment of inertia, 497
motion of object, 543
pendulum motion, 360, 549, 814,A90
period, 102, 267
pressure,A69
product of inertia, 493
projectile motion, 148–49, 152, 377–79,
462, 463, 487–88, 493, 497, 682–83,
689–91, 694
artillery, 159, 453
thrown object, 689
simulating motion, 684
static equilibrium, 598–99, 602,
603, 628
tension, 598–99, 602, 628
thrown object
ball, 154, 158, 889–90, 891
truck pulls, 602
uniform motion, 107–8, 690, 694,
A65–A67,A69–A70
velocity down inclined planes,A89
27.
xxviii Applications Index
vertically propelled object, 158
weight
of a boat, 602
of a car, 602
of a piano, 598
work, 619,A69
Play
swinging, 555
wagon pulling, 601, 608
Population.See also Demographics
bacterial, 330, 331, 336–37
decline in, 330
Ecoli growth, 79, 119
of endangered species, 331
of fruit fly, 328
as function of age, 58
growth in, 330, 331
insect, 198, 330
of trout, 801
of United States, 311, 338, 834
of world, 311, 338, 344, 792
Probability
checkout lines, 859
classroom composition, 859
“Deal or No Deal”TV show, 835
exponential, 277, 281, 295
household annual income, 859
Monty Hall Game, 863–64
Poisson, 281–82
“Price is Right” games, 859
of winning a lottery, 860
Publishing
page design, 113
Pyrotechnics
fireworks display, 663–64
Rate.See also Speed
of car, 361
catching a bus, 689
catching a train, 689
current of stream, 710
of emptying
oil tankers,A71
a pool,A71
a tub,A71
to keep up with the Sun, 362
revolutions per minute
of bicycle wheels, 361
of pulleys, 363
speed
average,A71
of current,A69–A70
of motorboat,A69–A70
of moving walkways,A70
per gallon rate and, 154–55
of plane,A71
golf, 66, 682–83, 690
distance to the green, 532
sand bunkers, 453
hammer throw, 433
Olympic heroes,A71
Olympic performance, 43–44
races, 765–66, 768,A71
relay runners, 861
swimming, 554, 628
tennis,A70
Statistics.See Probability
Surveying
distance between houses, 553
Surveys
of appliance purchases, 840
data analysis, 837–38, 840
stock portfolios, 841
of summer session attendance, 840
of TV sets in a house, 859
Temperature
of air parcel, 808
body,A13
conversion of, 254, 266
cooling time of pizza, 330
cricket chirp rate and, 155
measuring, 32
after midnight, 186
monthly, 424–25, 432
of portable heater, 345
relationship between scales, 102
sinusoidal function from,
420–21
of skillet, 344
warming time of Beer stein, 331
wind chill factor, 345
Tests and testing
IQ,A81
Time
for Beer stein to warm, 331
for block to slide down inclined
plane, 378
Ferris Wheel rider height as
function of, 462
to go from an island to a town, 108
hours of daylight, 244, 349, 422–23, 426,
432, 434, 447
for pizza to cool, 330
of sunrise, 362, 447
of trip, 378, 392
Transportation
deicing salt, 453
Niagara Falls Incline Railway, 515
Real estate
commission schedule,A80
cost of land, 553
cost of triangular lot, 538
Recreation
bungee jumping, 219
Demon Roller Coaster customer rate,
281–82
online gambling, 859
Security
security cameras, 514
Seismology
calibrating instruments, 694
Sequences.See also Combinatorics
ceramic tile floor design, 806–7
Drury Lane Theater, 808
football stadium seating, 808
seats in amphitheater, 808
Speed
of aircraft, 601, 628
angular, 361, 431
of current, 361–62, 788
as function of time, 68, 107–8
instantaneous
of ball, 891, 901
on the Moon, 892
of parachutist, 901
of jet stream, 788
linear, 358–59
on Earth, 361
of merrygorounds, 431
of Moon, 361
revolutions per minute of pulley, 361
of rotation of lighthouse beacons, 431
of swimmer, 628
of truck, 514
of wheel pulling cable cars, 362
wind, 710
Sports
baseball, 690, 849, 861
diamond, 8
dimensions of home plate, 538
field, 533, 534
Little League, 8, 362–63
onbase percentage, 128–29
stadium, 533
World Series, 849
basketball, 849
free throws, 66, 516
granny shots, 66
biathlon,A71
bungee jumping, 219
exacta betting, 863
football, 651,A70
28.
Applications Index xxix
Travel.See also Air travel; Navigation
bearing, 526, 553
drivers stopped by the
police, 347
parking at O’Hare International
Airport, 88
Volume
of gasoline in tank,A89
of ice in skating rink, 114
of water in cone, 108
rainfall measurement, 609
relative humidity, 282
weather satellites, 39
wind chill, 89, 345
Work,608
computing, 608, 609, 628
constant rate jobs, 788
pulling a wagon, 608
ramp angle, 610
wheel barrow push, 601
Weapons
artillery, 159, 453
cannons, 164
Weather
atmospheric pressure,
281, 295
avoiding a tropical storm, 533
cooling air, 808
hurricanes, 186
lightning strikes, 660–61, 664
29.
Photo Credits
xxx
Chapter 1 Pages 1 and 43, Getty Images; Page 18, DOE Digital Photo Archive; Page 32
Tetra Images/Alamy; Page 39, Jasonleehl/Shutterstock.
Chapter 2 Pages 45 and 115, Stephen Coburn/Shutterstock; Page 58, JPLCaltech/NASA;
Page 66, Exactostock/SuperStock; Page 102, Kg Kua/Dreamstime.
Chapter 3 Pages 117 and 164, Peter Morgan/AP Images; Page 153, Sajko/Shutterstock.
Chapter 4 Pages 165 and 244, Ivanova Inga/Shutterstock; Page 213, Oonai/Stockphoto.
Chapter 5 Pages 246 and 347,Thinkstock; Page 302, Getty Images; Page 311 col. 1,
Stockbyte/Thinkstock; Page 311 col. 2,Transtock/SuperStock; Page 316,
iStockphoto/Thinkstock; Page 331, JuniperImages/Thinkstock.
Chapter 6 Pages 349 and 434, Nova for Widows, http://www.nisa.com; Page 361, Ryan
McVay/Thinkstock; Page 407, Srdjan Draskovic/Dreamstime.
Chapter 7 Pages 435 and 503, Exaxion/iStockphoto Image.
Chapter 8 Pages 504 and 556, PhotoDisc/Getty Images; Page 516, Sergey Karpov/Shutterstock;
Page 539,Alexandre Fagundes De Fagundes/Dreamstime; Page 541, iStockphoto/
Thinkstock.
Chapter 9 Pages 557 and 630, Igmax/iStockphoto; Page 579, North Wind Picture
Archive/Alamy; Page 587, SSPL/Getty Images; Page 599, Hulton Archive/Getty
Images.
Chapter 10 Pages 631 and 695, JPL/Caltech/NASA; Page 649,Thomas Barrat/Shutterstock.
Chapter 11 Pages 696 and 790, Rob Crandall/Stock Connection/Alamy; Page 750, SSPL/Getty
Images.
Chapter 12 Pages 792 and 834,Albo/Shutterstock Image.
Chapter 13 Pages 835 and 863,Trae Patton/NBCU Photo Bank/AP Images; Page 856,
Thinkstock.
Chapter 14 Pages 865 and 903, Rafael Macia/Photo Researchers, Inc.
Appendix A Page A15, Hainaultphoto/Shutterstock; Page A21, Problem 51,Whole Spirit Press;
Page A21, Problem 52, Red River Press, Inc.
33.
2 CHAPTER 1 Graphs
x
y
–4 –2 42
2
–2
4
–4
O
Figure 1
Rectangular Coordinates
We locate a point on the real number line by assigning it a single real number, called
the coordinate of the point. For work in a twodimensional plane, we locate points by
using two numbers.
We begin with two real number lines located in the same plane: one horizontal
and the other vertical. The horizontal line is called the xaxis, the vertical line the
yaxis, and the point of intersection the origin O. See Figure 1.We assign coordinates
to every point on these number lines using a convenient scale. We usually use the
same scale on each axis, but in applications, different scales appropriate to the
application may be used.
The origin O has a value of 0 on both the xaxis and yaxis. Points on the xaxis
to the right of O are associated with positive real numbers, and those to the left of
O are associated with negative real numbers. Points on the yaxis above O are
associated with positive real numbers, and those below O are associated with negative
real numbers. In Figure 1, the xaxis and yaxis are labeled as x and y, respectively,
and we have used an arrow at the end of each axis to denote the positive direction.
The coordinate system described here is called a rectangular or Cartesian*
coordinate system. The plane formed by the xaxis and yaxis is sometimes called
the xyplane, and the xaxis and yaxis are referred to as the coordinate axes.
Any point P in the xyplane can be located by using an ordered pair of
real numbers. Let x denote the signed distance of P from the yaxis (signed means
that, if P is to the right of the yaxis, then and if P is to the left of the yaxis,
then ); and let y denote the signed distance of P from the xaxis. The ordered
pair also called the coordinates of P, then gives us enough information to
locate the point P in the plane.
For example, to locate the point whose coordinates are go 3 units along
the xaxis to the left of O and then go straight up 1 unit.We plot this point by placing
a dot at this location. See Figure 2, in which the points with coordinates
, and are plotted.
The origin has coordinates .Any point on the xaxis has coordinates of the
form and any point on the yaxis has coordinates of the form
If are the coordinates of a point P, then x is called the xcoordinate, or
abscissa, of P and y is the ycoordinate, or ordinate, of P.We identify the point P by
its coordinates by writing . Usually, we will simply say “the point
” rather than “the point whose coordinates are .”
The coordinate axes divide the xyplane into four sections called quadrants, as
shown in Figure 3. In quadrant I, both the xcoordinate and the ycoordinate of all
points are positive; in quadrant II, x is negative and y is positive; in quadrant III,
both x and y are negative; and in quadrant IV, x is positive and y is negative. Points
on the coordinate axes belong to no quadrant.
Now Work P R O B L E M 1 3
1x, y21x, y2
P = 1x, y21x, y2
1x, y2
10, y2.1x, 02,
10, 02
13, 2212, 32, 13, 22
13, 12,
13, 12,
1x, y2,
x 6 0
x 7 0,
1x, y2
*Named after René Descartes (1596–1650), a French mathematician, philosopher, and theologian.
x
y
–4 4
1
3
2
2
3
3
2
4
(3, 2)
3
(–3, 1)
(–2, –3)
(3, –2)
O
Figure 2
x
y
Quadrant I
x > 0, y > 0
Quadrant IV
x > 0, y < 0
Quadrant II
x < 0, y > 0
Quadrant III
x < 0, y < 0
Figure 3
1.1 The Distance and Midpoint Formulas
• Algebra Essentials (Appendix A, Section A.1,
pp.A1–A10)
• Geometry Essentials (Appendix A, Section A.2,
pp.A14–A19)
PREPARING FOR THIS SECTION Before getting started, review the following:
Now Work the ‘Are You Prepared?’ problems on page 6.
OBJECTIVES 1 Use the Distance Formula (p.3)
2 Use the Midpoint Formula (p.5)
34.
SECTION 1.1 The Distance and Midpoint Formulas 3
COMMENT On a graphing calculator, you can set the scale on each axis. Once this has been done,
you obtain the viewing rectangle. See Figure 4 for a typical viewing rectangle. You should now read
Section B.1, The Viewing Rectangle, in Appendix B.
Figure 4
1 Use the Distance Formula
If the same units of measurement, such as inches, centimeters, and so on, are used
for both the xaxis and yaxis, then all distances in the xyplane can be measured
using this unit of measurement.
Finding the Distance between Two Points
Find the distance d between the points and .15, 6211, 32
EXAMPLE 1
Solution First plot the points and and connect them with a straight line. See
Figure 5(a). We are looking for the length d. We begin by drawing a horizontal line
from to and a vertical line from to forming a right triangle,
as shown in Figure 5(b). One leg of the triangle is of length 4 (since ),
and the other is of length 3 (since ). By the Pythagorean Theorem, the
square of the distance d that we seek is
d = 225 = 5
d2
= 42
+ 32
= 16 + 9 = 25
ƒ6  3ƒ = 3
ƒ5  1ƒ = 4
15, 62,15, 3215, 3211, 32
15, 6211, 32
THEOREM Distance Formula
The distance between two points and denoted by
is
(1)d1P1, P22 = 41x2  x122
+ 1y2  y122
d1P1, P22,
P2 = 1x2, y22,P1 = 1x1, y12
Proof of the Distance Formula Let denote the coordinates of point
and let denote the coordinates of point . Assume that the line joining
and is neither horizontal nor vertical. Refer to Figure 6(a) on page 4. The
coordinates of are .The horizontal distance from to is the absoluteP3P11x2, y12P3
P2P1
P21x2, y22
P11x1, y12
x
y
3
(1, 3)
(5, 6)
3
4
d
6
(b)
6
3
(5, 3)
x
y
3
(1, 3)
(5, 6)
d
6
(a)
6
3
Figure 5
In Words
To compute the distance between
two points, find the difference of
the xcoordinates, square it, and
add this to the square of the
difference of the ycoordinates.
The square root of this sum is
the distance.
The distance formula provides a straightforward method for computing the
distance between two points.
᭹
35.
4 CHAPTER 1 Graphs
value of the difference of the xcoordinates, . The vertical distance from
to is the absolute value of the difference of the ycoordinates, . See
Figure 6(b). The distance that we seek is the length of the hypotenuse of
the right triangle, so, by the Pythagorean Theorem, it follows that
d1P1, P22 = 41x2  x122
+ 1y2  y122
= 1x2  x122
+ 1y2  y122
3d1P1, P2242
= ƒx2  x1ƒ2
+ ƒy2  y1ƒ2
d1P1, P22
ƒy2  y1ƒP2P3
ƒx2  x1ƒ
P1 ϭ (x1, y1)
x1
(a)
x2
P3 ϭ (x2, y1)
y y
x x
P2
ϭ (x2
, y2
)y2
y1
(b)
x1 x2
P2 ϭ (x2, y2)
y2
y1
P1
ϭ (x1
, y1
)
d(P1
, P2
)
P3 ϭ (x2, y1)⏐x2
Ϫ x1⏐
⏐y2 Ϫ y1⏐
Figure 6
Now, if the line joining and is horizontal, then the ycoordinate of equals
the ycoordinate of that is, Refer to Figure 7(a). In this case, the distance
formula (1) still works, because, for it reduces to
d1P1, P22 = 41x2  x122
+ 02
= 41x2  x122
= ƒx2  x1ƒ
y1 = y2,
y1 = y2.P2;
P1P2P1
⏐x2 Ϫ x1⏐
x1 x2
y1
x
y
P2 ϭ (x2, y1)P1 ϭ (x1, y1) d(P1, P2)
(a)
⏐y2 Ϫ y1⏐
x1
y1
y2 P2
ϭ (x1
, y2
)
P1 ϭ (x1, y1)
d(P1, P2)
x
y
(b)
Figure 7
A similar argument holds if the line joining and is vertical. See
Figure 7(b).
P2P1
Using the Distance Formula
Find the distance d between the points and (3, 2).14, 52
EXAMPLE 2
Solution Using the distance formula, equation (1), the distance d is
Now Work P R O B L E M S 1 7 A N D 2 1
The distance between two points and is never a
negative number. Furthermore, the distance between two points is 0 only when the
points are identical, that is, when and . Also, because
and , it makes no difference whether the distance
is computed from or from ; that is, .
The introduction to this chapter mentioned that rectangular coordinates enable
us to translate geometry problems into algebra problems, and vice versa. The next
example shows how algebra (the distance formula) can be used to solve geometry
problems.
d1P2, P12d1P1, P22 =P2 to P1P1 to P2
1y1  y222
1y2  y122
=1x1  x222
1x2  x122
=y1 = y2x1 = x2
P2 = 1x2, y22P1 = 1x1, y12
= 249 + 9 = 258 L 7.62
d = 233  14242
+ 12  522
= 272
+ 1322
᭹
36.
SECTION 1.1 The Distance and Midpoint Formulas 5
Using Algebra to Solve Geometry Problems
Consider the three points and .
(a) Plot each point and form the triangle ABC.
(b) Find the length of each side of the triangle.
(c) Verify that the triangle is a right triangle.
(d) Find the area of the triangle.
C = 13, 12A = 12, 12, B = 12, 32,
EXAMPLE 3
Solution (a) Figure 8 shows the points A, B, C and the triangle ABC.
(b) To find the length of each side of the triangle, use the distance formula,
equation (1).
(c) To show that the triangle is a right triangle, we need to show that the sum of the
squares of the lengths of two of the sides equals the square of the length of the
third side. (Why is this sufficient?) Looking at Figure 8, it seems reasonable to
conjecture that the right angle is at vertex B. We shall check to see whether
Using the results in part (b),
It follows from the converse of the Pythagorean Theorem that triangle ABC is a
right triangle.
(d) Because the right angle is at vertex B, the sides AB and BC form the base and
height of the triangle. Its area is
Now Work P R O B L E M 2 9
Area =
1
2
1Base21Height2 =
1
2
A225B A 25B = 5 square units
= 20 + 5 = 25 = 3d1A, C242
3d1A, B242
+ 3d1B, C242
= A225B2
+ A 25B2
3d1A, B242
+ 3d1B, C242
= 3d1A, C242
d1A, C2 = 433  12242
+ 11  122
= 225 + 0 = 5
d1B, C2 = 413  222
+ 11  322
= 21 + 4 = 25
d1A, B2 = 432  12242
+ 13  122
= 216 + 4 = 220 = 225
2 Use the Midpoint Formula
We now derive a formula for the coordinates of the midpoint of a line segment. Let
and be the endpoints of a line segment, and let
be the point on the line segment that is the same distance from as
it is from See Figure 9. The triangles and are congruent. Do
you see why? is given; * and
. So, we have angle–side–angle. Because triangles and are
congruent, corresponding sides are equal in length.That is,
x =
x1 + x2
2
y =
y1 + y2
2
2x = x1 + x2 2y = y1 + y2
x  x1 = x2  x and y  y1 = y2  y
MBP2P1AM∠MP2B
∠P1MA =∠AP1M = ∠BMP2d1P1, M2 = d1M, P22
MBP2P1AMP2.
P1M = 1x, y2
P2 = 1x2, y22P1 = 1x1, y12
*A postulate from geometry states that the transversal forms congruent corresponding angles with
the parallel line segments and .MBP1A
P1P2
x
y
–3
3
3
C = (3, 1)
B = (2, 3)
A = (–2, 1)
Figure 8
x1 x2x
y1
x
y
A = (x, y1
)
M = (x, y)
P1
= (x1
, y1
)
x – x1
x2 – x
y2
– y
y – y1
P2 = (x2, y2)
B = (x2
, y)
y
y2
Figure 9
᭹
37.
6 CHAPTER 1 Graphs
THEOREM Midpoint Formula
Themidpoint ofthelinesegmentfrom to is
(2)M = 1x, y2 = ¢
x1 + x2
2
,
y1 + y2
2
≤
P2 = 1x2, y22P1 = 1x1, y12M = 1x, y2
Finding the Midpoint of a Line Segment
Find the midpoint of the line segment from to . Plot the
points and and their midpoint.P2P1
P2 = 13, 12P1 = 15, 52
EXAMPLE 4
Solution Apply the midpoint formula (2) using and . Then
the coordinates of the midpoint M are
That is, . See Figure 10.
Now Work P R O B L E M 3 5
M = 11, 32
x =
x1 + x2
2
=
5 + 3
2
= 1 and y =
y1 + y2
2
=
5 + 1
2
= 3
1x, y2
y2 = 1x1 = 5, y1 = 5, x2 = 3,
x
y
5
5–5
P2 ϭ (3, 1)
P1 ϭ (–5, 5)
M ϭ (–1, 3)
Figure 10
᭹
7. If are the coordinates of a point P in the xyplane,
then x is called the of P and y is the
of P.
8. The coordinate axes divide the xyplane into four sections
called .
9. If three distinct points P, Q, and R all lie on a line and if
then Q is called the
of the line segment from P to R.
d1P, Q2 = d1Q, R2,
1x, y2 10. True or False The distance between two points is some
times a negative number.
11. True or False The point lies in quadrant IV of the
Cartesian plane.
12. True or False The midpoint of a line segment is found by
averaging the xcoordinates and averaging the ycoordinates
of the endpoints.
11, 42
Concepts and Vocabulary
‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.
1. On the real number line the origin is assigned the number
. (p.A4)
2. If and 5 are the coordinates of two points on the real
number line, the distance between these points is .
(pp.A5–A6)
3. If 3 and 4 are the legs of a right triangle, the hypotenuse is
. (p.A14)
4. Use the converse of the Pythagorean Theorem to show that
a triangle whose sides are of lengths 11, 60, and 61 is a right
triangle. (pp.A14–A15)
3
5. The area A of a triangle whose base is b and whose altitude is
h is A ϭ . (p.A15)
6. True or False Two triangles are congruent if two angles
and the included side of one equals two angles and the
included side of the other. (pp.A16–A17)
1.1 Assess Your Understanding
In Problems 13 and 14, plot each point in the xyplane.Tell in which quadrant or on what coordinate axis each point lies.
13. (a)
(b)
(c) C = 12, 22
B = 16, 02
A = 13, 22 14. (a)
(b)
(c) C = 13, 42
B = 13, 42
A = 11, 42
Skill Building
(d)
(e)
(f) F = 16, 32
E = 10, 32
D = 16, 52 (d)
(e)
(f) F = 13, 02
E = 10, 12
D = 14, 12
In Words
To find the midpoint of a line
segment, average the xcoordinates
and average the ycoordinates of
the endpoints.
38.
SECTION 1.1 The Distance and Midpoint Formulas 7
15. Plot the points and . Describe the set of all points of the form where y is a real number.
16. Plot the points , and . Describe the set of all points of the form where x is a real number.
In Problems 17–28, find the distance between the points and P2.P1d1P1, P22
1x, 32,14, 3215, 3212, 32,11, 32,10, 32,
12, y2,12, 1212, 12,12, 42,12, 32,12, 02,
17. 18. 19. 20.
P1
= (0, 0)
P2
= (2, 1)
x
y
–2
–1
2
2
P1
= (0, 0)P2 = (–2, 1)
x
y
–2
–1
2
2
–1
x
y
2
2–2
P2 ϭ (–2, 2)
P1 ϭ (1, 1)
x
y
–2
–1
2
2P1
= (–1, 1)
P2 = (2, 2)
21. P1 = 13, 42; P2 = 15, 42 22. P1 = 11, 02; P2 = 12, 42
23. P1 = 13, 22; P2 = 16, 02 24. P1 = 12, 32; P2 = 14, 22
25. P1 = 14, 32; P2 = 16, 42 26. P1 = 14, 32; P2 = 16, 22
27. P1 = 1a, b2; P2 = 10, 02 28. P1 = 1a, a2; P2 = 10, 02
29. A = 12, 52; B = 11, 32; C = 11, 02 30. A = 12, 52; B = 112, 32; C = 110, 112
31. A = 15, 32; B = 16, 02; C = 15, 52 32. A = 16, 32; B = 13, 52; C = 11, 52
33. A = 14, 32; B = 10, 32; C = 14, 22 34. A = 14, 32; B = 14, 12; C = 12, 12
35. P1 = 13, 42; P2 = 15, 42 36. P1 = 12, 02; P2 = 12, 42
37. P1 = 13, 22; P2 = 16, 02 38. P1 = 12, 32; P2 = 14, 22
39. P1 = 14, 32; P2 = 16, 12 40. P1 = 14, 32; P2 = 12, 22
41. P1 = 1a, b2; P2 = 10, 02 42. P1 = 1a, a2; P2 = 10, 02
In Problems 29–34, plot each point and form the triangle ABC.Verify that the triangle is a right triangle. Find its area.
In Problems 35–42, find the midpoint of the line segment joining the points and P2.P1
43. If the point is shifted 3 units to the right and 2 units
down, what are its new coordinates?
44. If the point is shifted 2 units to the left and 4 units
up, what are its new coordinates?
45. Find all points having an xcoordinate of 3 whose distance
from the point is 13.
(a) By using the Pythagorean Theorem.
(b) By using the distance formula.
46. Find all points having a ycoordinate of whose distance
from the point is 17.
(a) By using the Pythagorean Theorem.
(b) By using the distance formula.
47. Find all points on the xaxis that are 6 units from the point
.
48. Find all points on the yaxis that are 6 units from the point
.
49. The midpoint of the line segment from P1 to P2 is . If
, what is ?
50. The midpoint of the line segment from to is . If
what is ?
51. Geometry The medians of a triangle are the line segments
from each vertex to the midpoint of the opposite side (see
the figure). Find the lengths of the medians of the triangle
with vertices at , and .C = 14, 42A = 10, 02, B = 16, 02
P1P2 = 17, 22,
15, 42P2P1
P2P1 = 13, 62
11, 42
14, 32
14, 32
11, 22
6
12, 12
11, 62
12, 52
52. Geometry An equilateral triangle is one in which all three
sides are of equal length. If two vertices of an equilateral
triangle are and find the third vertex. How
many of these triangles are possible?
10, 02,10, 42
Applications and Extensions
Median
A B
C
Midpoint
53. Geometry Find the midpoint of each diagonal of a square
with side of length s. Draw the conclusion that the diagonals
of a square intersect at their midpoints.
[Hint: Use (0, 0), (0, s), (s, 0), and (s, s) as the vertices of the
square.]
54. Geometry Verify that the points (0, 0), (a, 0), and
are the vertices of an equilateral triangle. Then
show that the midpoints of the three sides are the vertices of
a second equilateral triangle (refer to Problem 52).
a
a
2
,
23a
2
b
s s
s
39.
8 CHAPTER 1 Graphs
65. Drafting Error When a draftsman draws three lines that
are to intersect at one point, the lines may not intersect as
intended and subsequently will form an error triangle. If
this error triangle is long and thin, one estimate for the
location of the desired point is the midpoint of the shortest
side.The figure shows one such error triangle.
Source: www.uwgb.edu/dutchs/STRUCTGE/sl00.htm
In Problems 55–58, find the length of each side of the triangle determined by the three points and . State whether the triangle
is an isosceles triangle, a right triangle, neither of these, or both. (An isosceles triangle is one in which at least two of the sides are of
equal length.)
P3P1, P2,
55.
56.
57.
58.
59. Baseball A major league baseball “diamond” is actually a
square, 90 feet on a side (see the figure).What is the distance
directly from home plate to second base (the diagonal of the
square)?
P1 = 17, 22; P2 = 14, 02; P3 = 14, 62
P1 = 12, 12; P2 = 10, 72; P3 = 13, 22
P1 = 11, 42; P2 = 16, 22; P3 = 14, 52
P1 = 12, 12; P2 = 14, 12; P3 = 14, 32
Pitching
rubber
Home plate
1st base
2nd base
3rd base
90 ft
90 ft
60. Little League Baseball The layout of a Little League
playing field is a square, 60 feet on a side. How far is it
directly from home plate to second base (the diagonal of the
square)?
Source: Little League Baseball, Official Regulations and
Playing Rules, 2010.
61. Baseball Refer to Problem 59. Overlay a rectangular
coordinate system on a major league baseball diamond so
that the origin is at home plate, the positive xaxis lies
in the direction from home plate to first base, and the
positive yaxis lies in the direction from home plate to
third base.
(a) What are the coordinates of first base, second base, and
third base? Use feet as the unit of measurement.
(b) If the right fielder is located at how far is it
from the right fielder to second base?
(c) If the center fielder is located at how far is it
from the center fielder to third base?
62. Little League Baseball Refer to Problem 60. Overlay a
rectangular coordinate system on a Little League baseball
diamond so that the origin is at home plate, the positive
xaxis lies in the direction from home plate to first base, and
the positive yaxis lies in the direction from home plate to
third base.
(a) What are the coordinates of first base, second base, and
third base? Use feet as the unit of measurement.
(b) If the right fielder is located at how far is it
from the right fielder to second base?
(c) If the center fielder is located at , how far is it
from the center fielder to third base?
63. Distance between Moving Objects A Dodge Neon and a
Mack truck leave an intersection at the same time.The Neon
heads east at an average speed of 30 miles per hour, while
the truck heads south at an average speed of 40 miles per
hour. Find an expression for their distance apart d (in miles)
at the end of t hours.
1220, 2202
1180, 202,
1300, 3002,
1310, 152,
15 mph
East
100 ft
x
y
(1.4, 1.3)
(2.7, 1.7)
(2.6, 1.5)
1.7
1.5
1.3
2.6 2.71.4
WalMart Stores, Inc.
Net sales (in $ billions)
2002 2004
375
204
2003
Year
2005 2006 2007 2008
350
300
250
200
Netsales($billions)
150
50
0
100
64. Distance of a Moving Object from a Fixed Point A hotair
balloon, headed due east at an average speed of 15 miles per
hour and at a constant altitude of 100 feet, passes over an
intersection (see the figure).Find an expression for the distance
d (measured in feet) from the balloon to the intersection
t seconds later.
(a) Find an estimate for the desired intersection point.
(b) Find the length of the median for the midpoint found in
part (a). See Problem 51.
66. Net Sales The figure illustrates how net sales of WalMart
Stores, Inc., have grown from 2002 through 2008. Use the
midpoint formula to estimate the net sales of WalMart
Stores, Inc., in 2005. How does your result compare to the
reported value of $282 billion?
Source: WalMart Stores, Inc., 2008 Annual Report
40.
SECTION 1.2 Graphs of Equations in Two Variables;Intercepts;Symmetry 9
68. Write a paragraph that describes a Cartesian plane. Then
write a second paragraph that describes how to plot points
in the Cartesian plane. Your paragraphs should include the
terms “coordinate axes,” “ordered pair,” “coordinates,”
“plot,”“xcoordinate,” and “ycoordinate.”
Explaining Concepts: Discussion and Writing
1. 0 2. 8 3. 5 4. 5. 6. TrueA =
1
2
bh112
+ 602
= 121 + 3600 = 3721 = 612
‘Are You Prepared?’Answers
Now Work the ‘Are You Prepared?’ problems on page 16.
OBJECTIVES 1 Graph Equations by Plotting Points (p.9)
2 Find Intercepts from a Graph (p.11)
3 Find Intercepts from an Equation (p.12)
4 Test an Equation for Symmetry with Respect to the xAxis,the yAxis,
and the Origin (p.12)
5 Know How to Graph Key Equations (p.15)
1.2 Graphs of Equations in Two Variables; Intercepts; Symmetry
• Solving Equations (Appendix A, Section A.6, pp.A44–A48)
PREPARING FOR THIS SECTION Before getting started, review the following:
1 Graph Equations by Plotting Points
An equation in two variables, say x and y, is a statement in which two expressions
involving x and y are equal. The expressions are called the sides of the equation.
Since an equation is a statement, it may be true or false, depending on the value of
the variables.Any values of x and y that result in a true statement are said to satisfy
the equation.
For example, the following are all equations in two variables x and y:
The first of these, is satisfied for since
Other choices of x and y, such as , also satisfy this
equation. It is not satisfied for and , since .
The graph of an equation in two variables x and y consists of the set of points in
the xyplane whose coordinates satisfy the equation.
Graphs play an important role in helping us to visualize the relationships that
exist between two variables or quantities. Figure 11 on page 10 shows the relation
between the level of risk in a stock portfolio and the average annual rate of return.
From the graph, we can see that, when 30% of a portfolio of stocks is invested in
foreign companies, risk is minimized.
1x, y2
22
+ 32
= 4 + 9 = 13 Z 5y = 3x = 2
y = 2x = 1,1 + 4 = 5.
12
+ 22
=x = 1, y = 2,x2
+ y2
= 5,
x2
+ y2
= 5 2x  y = 6 y = 2x + 5 x2
= y
67. Poverty Threshold Poverty thresholds are determined by
the U.S. Census Bureau.A poverty threshold represents the
minimum annual household income for a family not to be
considered poor. In 1998, the poverty threshold for a family
of four with two children under the age of 18 years was
$16,530. In 2008, the poverty threshold for a family of four
with two children under the age of 18 years was $21,834.
Assuming poverty thresholds increase in a straightline
fashion, use the midpoint formula to estimate the poverty
threshold of a family of four with two children under the
age of 18 in 2003. How does your result compare to the
actual poverty threshold in 2003 of $18,660?
Source: U.S. Census Bureau
41.
10 CHAPTER 1 Graphs
18.5
18.5 19.5
17.5
17.5
16.5
16.5
15.5
15.5
14.5
14.5
13.5
13.5
18
18 19 20
17
17
16
16
15
15
14
14
AverageAnnualReturns(%)
Level of Risk (%)
10%
40%
50%
30%
60%
70%
(30% foreign/70% U.S.)
(100% U.S.)0%
80%
90%
100%
(100% foreign)
20%
Figure 11
Source: T. Rowe Price
Determining Whether a Point Is on the Graph of an Equation
Determine if the following points are on the graph of the equation .
(a) (b) 12, 2212, 32
2x  y = 6
EXAMPLE 1
Solution (a) For the point , check to see if satisfies the equation
.
The equation is not satisfied,so the point is not on the graph of
(b) For the point ,
The equation is satisfied, so the point is on the graph of .
Now Work P R O B L E M 1 1
2x  y = 612, 22
2x  y = 2122  122 = 4 + 2 = 6
12, 22
2x  y = 6.12, 32
2x  y = 2122  3 = 4  3 = 1 Z 6
2x  y = 6
x = 2, y = 312, 32
᭹
Graphing an Equation by Plotting Points
Graph the equation: y = 2x + 5
EXAMPLE 2
Solution We want to find all points that satisfy the equation. To locate some of these
points (and get an idea of the pattern of the graph), assign some numbers to x and
find corresponding values for y.
1x, y2
x
y
(10, 25)25
(1, 7)
(0, 5)
(–5, –5) 25–25
–25
Figure 12
By plotting these points and then connecting them, we obtain the graph of the
equation (a line), as shown in Figure 12.
᭹
Graphing an Equation by Plotting Points
Graph the equation: y = x2
EXAMPLE 3
Solution Table 1 provides several points on the graph. In Figure 13 we plot these points and
connect them with a smooth curve to obtain the graph (a parabola).
If Then Point on Graph
110, 252y = 21102 + 5 = 25x = 10
15, 52y = 2152 + 5 = 5x = 5
11, 72y = 2112 + 5 = 7x = 1
10, 52y = 2102 + 5 = 5x = 0
42.
SECTION 1.2 Graphs of Equations in Two Variables;Intercepts; Symmetry 11
The graphs of the equations shown in Figures 12 and 13 do not show all points.
For example, in Figure 12, the point is a part of the graph of , but
it is not shown. Since the graph of could be extended out as far as we
please, we use arrows to indicate that the pattern shown continues. It is important
when illustrating a graph to present enough of the graph so that any viewer of the
illustration will “see” the rest of it as an obvious continuation of what is actually
there.This is referred to as a complete graph.
One way to obtain a complete graph of an equation is to plot a sufficient number
of points on the graph until a pattern becomes evident. Then these points are
connected with a smooth curve following the suggested pattern. But how many
points are sufficient? Sometimes knowledge about the equation tells us. For example,
we will learn in the next section that, if an equation is of the form , then
its graph is a line. In this case, only two points are needed to obtain the graph.
One purpose of this book is to investigate the properties of equations in order
to decide whether a graph is complete. Sometimes we shall graph equations by
plotting points. Shortly, we shall investigate various techniques that will enable us to
graph an equation without plotting so many points.
y = mx + b
y = 2x + 5
y = 2x + 5120, 452
Table 1
x
y
–4 4
(4, 16)(–4, 16)
(3, 9)(–3, 9)
(2, 4)(–2, 4)
(1, 1)
(0, 0)
(–1, 1)
5
10
15
20
Figure 13
x
16
9
4
1
0 0 (0,0)
1 1 (1,1)
2 4 (2,4)
3 9 (3,9)
4 16 (4,16)
(1, 1)1
(2, 4)2
(3, 9)3
(4, 16)4
(x, y)y ؍ x2
᭹
COMMENT Another way to obtain
the graph of an equation is to use a
graphing utility. Read Section B.2,
Using a Graphing Utility to Graph
Equations, in Appendix B.
Two techniques that sometimes reduce the number of points required to graph
an equation involve finding intercepts and checking for symmetry.
2 Find Intercepts from a Graph
The points, if any, at which a graph crosses or touches the coordinate axes are called
the intercepts. See Figure 14.The xcoordinate of a point at which the graph crosses
or touches the xaxis is an xintercept, and the ycoordinate of a point at which the
graph crosses or touches the yaxis is a yintercept. For a graph to be complete, all its
intercepts must be displayed.
x
y
Intercepts
Graph
touches
xaxis
Graph
crosses
xaxis
Graph
crosses
yaxis
Figure 14
In Example 4,you should notice the following usage:If we do not specify the type
of intercept (x versus y), then report the intercept as an ordered pair. However,
if the type of intercept is specified, then report the coordinate of the specified
intercept. For xintercepts, report the xcoordinate of the intercept; for yintercepts,
report the ycoordinate of the intercept.
Now Work P R O B L E M 3 9 ( a )
Solution The intercepts of the graph are the points
The xintercepts are , and 4.5; the yintercepts are , and 3.
4
3
,3.5
3
2
,3
13, 02, 10, 32, a
3
2
, 0b, a0, 
4
3
b, 10, 3.52, 14.5, 02
Finding Intercepts from a Graph
Find the intercepts of the graph in Figure 15.What are its xintercepts? What are its
yintercepts?
EXAMPLE 4
᭹
x
y
(0, 3)
(Ϫ3, 0)
, 0( )
(0, Ϫ )4
–
3
3–
2
(4.5, 0)
(0, Ϫ3.5)
4
Ϫ4 5
Figure 15
43.
12 CHAPTER 1 Graphs
3 Find Intercepts from an Equation
The intercepts of a graph can be found from its equation by using the fact that
points on the xaxis have ycoordinates equal to 0 and points on the yaxis have
xcoordinates equal to 0.
Procedure for Finding Intercepts
1. To find the xintercept(s), if any, of the graph of an equation, let in
the equation and solve for x, where x is a real number.
2. To find the yintercept(s), if any, of the graph of an equation, let in
the equation and solve for y, where y is a real number.
x = 0
y = 0
Finding Intercepts from an Equation
Find the xintercept(s) and the yintercept(s) of the graph of . Then
graph by plotting points.y = x2
 4
y = x2
 4
EXAMPLE 5
COMMENT For many equations, finding
intercepts may not be so easy. In such
cases, a graphing utility can be used.
Read the first part of Section B.3,
Using a Graphing Utility to Locate In
tercepts and Check for Symmetry, in
Appendix B, to find out how to locate
intercepts using a graphing utility.
Solution To find the xintercept(s), let and obtain the equation
or
or
The equation has two solutions, and 2.The xintercepts are and 2.
To find the yintercept(s), let in the equation.
The yintercept is .
Since for all x, we deduce from the equation that for
all x.This information, the intercepts, and the points from Table 2 enable us to graph
. See Figure 16.y = x2
 4
y Ú 4y = x2
 4x2
Ú 0
4
= 02
 4 = 4
y = x2
 4
x = 0
22
Solve.x = 2x = 2
ZeroProduct Propertyx  2 = 0x + 2 = 0
Factor.1x + 221x  2) = 0
y = x2
 4 with y = 0x2
 4 = 0
y = 0
x
y
(3, 5)5
(2, 0)
(–1, –3) (1, –3)
(0, –4)
(–3, 5)
(–2, 0)
5–5
–5
Figure 16Table 2 x
5
1
3 5 (3, 5)
(1, 3)3
(1, 3)31
(3, 5)3
(x, y)y ؍ x2
؊ 4
Now Work P R O B L E M 2 1
᭹
4 Test an Equation for Symmetry with Respect
to the xAxis,the yAxis,and the Origin
We just saw the role that intercepts play in obtaining key points on the graph of an
equation.Another helpful tool for graphing equations involves symmetry, particularly
symmetry with respect to the xaxis, the yaxis, and the origin.
DEFINITION A graph is said to be symmetric with respect to the xaxis if, for every point
on the graph, the point is also on the graph.1x, y21x, y2
44.
SECTION 1.2 Graphs of Equations in Two Variables;Intercepts; Symmetry 13
Figure 17 illustrates the definition. When a graph is symmetric with respect to
the xaxis, notice that the part of the graph above the xaxis is a reflection or mirror
image of the part below it, and vice versa.
Points Symmetric with Respect to the xAxis
If a graph is symmetric with respect to the xaxis and the point is on the graph,
then the point is also on the graph.13, 22
13, 22
EXAMPLE 6
᭹
DEFINITION A graph is said to be symmetric with respect to the yaxis if, for every point
on the graph, the point is also on the graph.1x, y21x, y2
Figure 18 illustrates the definition. When a graph is symmetric with respect to
the yaxis, notice that the part of the graph to the right of the yaxis is a reflection of
the part to the left of it, and vice versa.
Points Symmetric with Respect to the yAxis
If a graph is symmetric with respect to the yaxis and the point is on the graph,
then the point is also on the graph.15, 82
15, 82
EXAMPLE 7
᭹
(x, –y)
(x, y)
(x, –y)
(x, y)
(x, y)
(x, –y)
x
yFigure 17
Symmetry with respect to the xaxis
(–x, y)
(–x, y) (x, y)
(x, y)
x
y
Figure 18
Symmetry with respect to the yaxis
DEFINITION A graph is said to be symmetric with respect to the origin if, for every point
on the graph, the point is also on the graph.1x, y21x, y2
Figure 19 illustrates the definition. Notice that symmetry with respect to the
origin may be viewed in three ways:
1. As a reflection about the yaxis, followed by a reflection about the xaxis
2. As a projection along a line through the origin so that the distances from the
origin are equal
3. As half of a complete revolution about the origin
(x, y)
(x, y)
(–x, –y)
(–x, –y) x
y
Figure 19
Symmetry with respect to the origin
Points Symmetric with Respect to the Origin
If a graph is symmetric with respect to the origin and the point 4, 2 is on the graph,
then the point is also on the graph.
Now Work P R O B L E M S 2 9 A N D 3 9 ( b )
When the graph of an equation is symmetric with respect to a coordinate axis or
the origin, the number of points that you need to plot in order to see the pattern is
reduced. For example, if the graph of an equation is symmetric with respect to the
yaxis, then, once points to the right of the yaxis are plotted, an equal number of
points on the graph can be obtained by reflecting them about the yaxis. Because of
this, before we graph an equation, we first want to determine whether it has any
symmetry.The following tests are used for this purpose.
14, 22
21
EXAMPLE 8
᭹
45.
14 CHAPTER 1 Graphs
Testing an Equation for Symmetry
Test for symmetry.y =
4x2
x2
+ 1
EXAMPLE 9
Solution xAxis: To test for symmetry with respect to the xaxis, replace y by . Since
is not equivalent to the graph of the equation is
not symmetric with respect to the xaxis.
yAxis: To test for symmetry with respect to the yaxis, replace x by . Since
is equivalent to the graph of the
equation is symmetric with respect to the yaxis.
Origin: To test for symmetry with respect to the origin, replace x by and y by .
Replace x by and y by
Simplify.
Multiply both sides by
Since the result is not equivalent to the original equation, the graph of the
equation is not symmetric with respect to the origin.
Seeing the Concept
Figure 20 shows the graph of using a graphing utility. Do you see the symmetry with
respect to the yaxis?
y =
4x2
x2
+ 1
y =
4x2
x2
+ 1
1.y = 
4x2
x2
+ 1
y =
4x2
x2
+ 1
y.xy =
41x22
1x22
+ 1
yx
y =
4x2
x2
+ 1
,y =
41x22
1x22
+ 1
=
4x2
x2
+ 1
x
y =
4x2
x2
+ 1
,y =
4x2
x2
+ 1
y
᭹
Tests for Symmetry
To test the graph of an equation for symmetry with respect to the
xAxis Replace y by Ϫy in the equation and simplify. If an equivalent
equation results, the graph of the equation is symmetric with respect
to the xaxis.
yAxis Replace x by Ϫx in the equation and simplify. If an equivalent
equation results, the graph of the equation is symmetric with respect
to the yaxis.
Origin Replace x by Ϫx and y by Ϫy in the equation and simplify. If an
equivalent equation results, the graph of the equation is symmetric
with respect to the origin.
5
Ϫ5
Ϫ5 5
Figure 20
Now Work P R O B L E M 5 9
46.
SECTION 1.2 Graphs of Equations in Two Variables;Intercepts; Symmetry 15
xAxis: Replace y by . Since is not equivalent to , the
graph is not symmetric with respect to the xaxis.
yAxis: Replace x by . Since is not equivalent to
, the graph is not symmetric with respect to the yaxis.
Origin: Replace x by and y by . Since is equivalent
to (multiply both sides by ), the graph is symmetric with
respect to the origin.
1y = x3
y = 1x23
= x3
yx
y = x3
y = 1x23
= x3
x
y = x3
y = x3
y
5 Know How to Graph Key Equations
The next three examples use intercepts, symmetry, and point plotting to obtain the
graphs of key equations. It is important to know the graphs of these key equations
because we use them later.The first of these is .y = x3
Graphing the Equation y ؍ x3
by Finding Intercepts,
Checking for Symmetry, and Plotting Points
Graph the equation by plotting points. Find any intercepts and check for
symmetry first.
y = x3
EXAMPLE 10
Solution First, find the intercepts. When , then ; and when , then .
The origin is the only intercept. Now test for symmetry.10, 02
x = 0y = 0y = 0x = 0
To graph , we use the equation to obtain several points on the graph.
Because of the symmetry, we only need to locate points on the graph for which
. See Table 3. Since 1, 1 is on the graph, and the graph is symmetric with
respect to the origin, the point is also on the graph. Plot the points from
Table 3 and use the symmetry. Figure 21 shows the graph.
11, 12
21x Ú 0
y = x3
x
y
(2, 8)8
(1, 1)(0, 0)
(–1, –1) 6–6
–8(–2, –8)
Figure 21
Table 3 x
0 0
1 1
2 8 (2,8)
3 27 (3,27)
(1, 1)
(0, 0)
(x, y)y ؍ x3
᭹
Graphing the Equation x ؍ y2
(a) Graph the equation . Find any intercepts and check for symmetry first.
(b) Graph x = y2
, y Ú 0.
x = y2
EXAMPLE 11
x
y
(9, 3)
6
(4, 2)
(0, 0)
(1, –1)
10–2
(9, –3)
5
(1, 1)
(4, –2)
Figure 22
x = y2
x
y
(9, 3)
6
(4, 2)
(0, 0)
10–2 5
(1, 1)
Figure 23
y = 2x
Y1 ϭ
6
Ϫ6
Ϫ2 10
x
Y2 ϭ Ϫ x
Figure 24
Solution (a) The lone intercept is . The graph is symmetric with respect to the xaxis.
(Do you see why? Replace y by .) Figure 22 shows the graph.
(b) If we restrict y so that , the equation , , may be written
equivalently as . The portion of the graph of in quadrant I is
therefore the graph of See Figure 23.y = 1x.
x = y2
y = 1x
y Ú 0x = y2
y Ú 0
y
10, 02
COMMENT To see the graph of the equation on a graphing calculator, you will need to graph
two equations: and We discuss why in Chapter 2. See Figure 24. Y2 =  1x.Y1 = 1x
x = y2
᭹
47.
16 CHAPTER 1 Graphs
Graphing the Equation
Graph the equation Find any intercepts and check for symmetry first.y =
1
x
.
y ؍
1
x
EXAMPLE 12
Solution Check for intercepts first. If we let , we obtain 0 in the denominator, which
makes y undefined.We conclude that there is no yintercept. If we let , we get
the equation , which has no solution.We conclude that there is no xintercept.
The graph of does not cross or touch the coordinate axes.
Next check for symmetry:
xAxis: Replacing y by yields which is not equivalent to
yAxis: Replacing x by yields , which is not equivalent to
Origin: Replacing x by and y by yields which is equivalent to
The graph is symmetric with respect to the origin.
Now, set up Table 4, listing several points on the graph. Because of the symmetry
with respect to the origin, we use only positive values of x. From Table 4 we
infer that if x is a large and positive number, then is a positive number close
to 0.We also infer that if x is a positive number close to 0, then is a large and
positive number.Armed with this information, we can graph the equation.
Figure 25 illustrates some of these points and the graph of Observe how
the absence of intercepts and the existence of symmetry with respect to the origin
were utilized.
y =
1
x
.
y =
1
x
y =
1
x
y =
1
x
.
y = 
1
x
,yx
y =
1
x
.
y =
1
x
= 
1
x
x
y =
1
x
.y =
1
x
,y
y =
1
x
1
x
= 0
y = 0
x = 0Table 4
x
10
3
2
1 1
2
3
10 a10,
1
10
b
1
10
a3,
1
3
b
1
3
a2,
1
2
b
1
2
(1, 1)
a
1
2
, 2b
1
2
a
1
3
, 3b
1
3
a
1
10
, 10b
1
10
(x, y)y ؍
1
x
x
y
3
3
–3
–3
, 2
–
1––
2
(1, 1)
2, 1––
2
–2,( )
( )
( )
( )
1––
2
(–1, –1)
, –21––
2
–
Figure 25
COMMENT Refer to Example 2 in Appendix B, Section B.3, for the graph of using a graphing
utility.
y =
1
x
᭹
‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.
1. Solve the equation . (pp.A44–A48)21x + 32  1 = 7 2. Solve the equation . (pp.A44–A48)x2
 9 = 0
1.2 Assess Your Understanding
3. The points, if any, at which a graph crosses or touches the
coordinate axes are called .
4. The xintercepts of the graph of an equation are those
xvalues for which .
5. If for every point on the graph of an equation the
point is also on the graph, then the graph is
symmetric with respect to the .
6. If the graph of an equation is symmetric with respect to the
yaxis and is an xintercept of this graph, then is
also an xintercept.
4
1x, y2
1x, y2
7. If the graph of an equation is symmetric with respect to the
origin and is a point on the graph, then
is also a point on the graph.
8. True or False To find the yintercepts of the graph of an
equation, let and solve for y.
9. True or False The ycoordinate of a point at which the
graph crosses or touches the xaxis is an xintercept.
10. True or False If a graph is symmetric with respect to the
xaxis, then it cannot be symmetric with respect to the
yaxis.
x = 0
13, 42
Concepts and Vocabulary
48.
SECTION 1.2 Graphs of Equations in Two Variables;Intercepts; Symmetry 17
In Problems 17–28, find the intercepts and graph each equation by plotting points. Be sure to label the intercepts.
Skill Building
In Problems 11–16, determine which of the given points are on the graph of the equation.
11. Equation:
Points: 10, 02; 11, 12; 11, 02
y = x4
 1x 12. Equation:
Points: 10, 02; 11, 12; 11, 12
y = x3
 21x 13. Equation:
Points: 10, 32; 13, 02; 13, 02
y2
= x2
+ 9
14. Equation:
Points: 11, 22; 10, 12; 11, 02
y3
= x + 1 15. Equation:
Points: 10, 22; 12, 22; A 22, 22B
x2
+ y2
= 4 16. Equation:
Points: 10, 12; 12, 02; a2,
1
2
b
x2
+ 4y2
= 4
17. y = x + 2 18. y = x  6 19. y = 2x + 8 20. y = 3x  9
21. y = x2
 1 22. y = x2
 9 23. y = x2
+ 4 24. y = x2
+ 1
25. 2x + 3y = 6 26. 5x + 2y = 10 27. 9x2
+ 4y = 36 28. 4x2
+ y = 4
In Problems 29–38, plot each point.Then plot the point that is symmetric to it with respect to (a) the xaxis; (b) the yaxis; (c) the origin.
29. 13, 42 30. 15, 32 31. 12, 12 32. 14, 22 33. 15, 22
34. 11, 12 35. 13, 42 36. 14, 02 37. 10, 32 38. 13, 02
In Problems 39–50, the graph of an equation is given. (a) Find the intercepts. (b) Indicate whether the graph is symmetric with respect to
the xaxis, the yaxis, or the origin.
39. 40. 41. 42.
43. 44. 45. 46.
47. 48. 49. 50.
In Problems 51–54, draw a complete graph so that it has the type of symmetry indicated.
51. yaxis 52. xaxis 53. Origin 54. yaxis
x
y
3
3–3
–3
x
y
3
3
–3
–3
y
6
Ϫ6
x3Ϫ3
40
Ϫ40
x
6Ϫ6
y
3
–3 x
y
3
–3
3
Ϫ3 x
y
3
Ϫ3
x
y
1
Ϫ1
Ϫ ––
2
Ϫ ––
2 x
y
3
4
Ϫ4
Ϫ3
y
3
Ϫ3
x3Ϫ3
y
3
Ϫ3
x3Ϫ3
Ϫ8
8
Ϫ2 2
Ϫ4
4
Ϫ4 4
9–9
–9
9
x
y
(0, –9)
(2, –5)
x
y
–5 5
–5
5
(–4, 0)
(0, 2)
(5, 3)
x
y
–4
4
––
2
(, 0)
(0, 0)
( , 2)
y
–3 3
–2
4
(0, 4)
(0, 0)
(2, 2)
x
49.
18 CHAPTER 1 Graphs
In Problems 55–70, list the intercepts and test for symmetry.
55. y2
= x + 4 56. y2
= x + 9 57. y = 1
3
x 58. y = 15
x
59. x2
+ y  9 = 0 60. x2
 y  4 = 0 61. 9x2
+ 4y2
= 36 62. 4x2
+ y2
= 4
63. y = x3
 27 64. y = x4
 1 65. y = x2
 3x  4 66. y = x2
+ 4
67. y =
3x
x2
+ 9
68. y =
x2
 4
2x
69. y =
x3
x2
 9
70. y =
x4
+ 1
2x5
In Problems 71–74, draw a quick sketch of each equation.
71. y = x3
72. x = y2
73. y = 1x 74. y =
1
x
75. If is a point on the graph of , what is b?y = 4x + 113, b2 76. If is a point on the graph of , what is b?2x + 3y = 212, b2
77. If is a point on the graph of , what is a?y = x2
+ 3x1a, 42 78. If is a point on the graph of , what is a?y = x2
+ 6x1a, 52
79. Given that the point (1, 2) is on the graph of an equation
that is symmetric with respect to the origin, what other point
is on the graph?
80. If the graph of an equation is symmetric with respect to the
yaxis and 6 is an xintercept of this graph, name another
xintercept.
81. If the graph of an equation is symmetric with respect to the
origin and is an xintercept of this graph, name another
xintercept.
82. If the graph of an equation is symmetric with respect to the
xaxis and 2 is a yintercept, name another yintercept.
83. Microphones In studios and on stages, cardioid micro
phones are often preferred for the richness they add to voices
and for their ability to reduce the level of sound from the
sides and rear of the microphone. Suppose one such cardioid
pattern is given by the equation
(a) Find the intercepts of the graph of the equation.
(b) Test for symmetry with respect to the xaxis, yaxis, and
origin.
Source: www.notaviva.com
1x2
+ y2
 x22
= x2
+ y2
.
4
84. Solar Energy The solar electric generating systems at
Kramer Junction, California, use parabolic troughs to heat a
heattransfer fluid to a high temperature. This fluid is used
to generate steam that drives a power conversion system to
produce electricity. For troughs 7.5 feet wide, an equation
for the crosssection is .16y2
= 120x  225
Applications and Extensions
(a) Find the intercepts of the graph of the equation.
(b) Test for symmetry with respect to the xaxis, yaxis, and
origin.
Source: U.S. Department of Energy
85. (a) Graph , and ,
noting which graphs are the same.
(b) Explain why the graphs of and are the
same.
(c) Explain why the graphs of and are
not the same.
(d) Explain why the graphs of and are not
the same.
86. Explain what is meant by a complete graph.
87. Draw a graph of an equation that contains two xintercepts;
at one the graph crosses the xaxis, and at the other the
graph touches the xaxis.
88. Make up an equation with the intercepts , and
. Compare your equation with a friend’s equation.
Comment on any similarities.
10, 12
12, 02, 14, 02
y = xy = 3x2
y = 11x22
y = x
y = ƒxƒy = 3x2
y = 11x22
y = 3x2
, y = x, y = ƒxƒ 89. Draw a graph that contains the points
, and . Compare your graph with those of other
students.Are most of the graphs almost straight lines? How
many are “curved”? Discuss the various ways that these
points might be connected.
90. An equation is being tested for symmetry with respect to
the xaxis, the yaxis, and the origin. Explain why, if two of
these symmetries are present, the remaining one must also
be present.
91. Draw a graph that contains the points , ,
and (0, 2) that is symmetric with respect to the yaxis.
Compare your graph with those of other students; comment
on any similarities. Can a graph contain these points and
be symmetric with respect to the xaxis? the origin? Why
or why not?
(1, 3)(2, 5)
13, 5211, 32
10, 12,12, 12,
Explaining Concepts: Discussion and Writing
50.
SECTION 1.3 Lines 19
In this section we study a certain type of equation that contains two variables, called
a linear equation, and its graph, a line.
Rise
Line
Run
Figure 26
1.3 Lines
OBJECTIVES 1 Calculate and Interpret the Slope of a Line (p.19)
2 Graph Lines Given a Point and the Slope (p.22)
3 Find the Equation of a Vertical Line (p.22)
4 Use the Point–Slope Form of a Line;Identify Horizontal Lines (p.23)
5 Find the Equation of a Line Given Two Points (p.24)
6 Write the Equation of a Line in Slope–Intercept Form (p.24)
7 Identify the Slope and yIntercept of a Line from Its Equation (p.25)
8 Graph Lines Written in General Form Using Intercepts (p.26)
9 Find Equations of Parallel Lines (p.27)
10 Find Equations of Perpendicular Lines (p.28)
1 Calculate and Interpret the Slope of a Line
Consider the staircase illustrated in Figure 26. Each step contains exactly the same
horizontal run and the same vertical rise. The ratio of the rise to the run, called
the slope, is a numerical measure of the steepness of the staircase. For example, if the
run is increased and the rise remains the same, the staircase becomes less steep. If the
run is kept the same, but the rise is increased, the staircase becomes more steep.This
important characteristic of a line is best defined using rectangular coordinates.
92. yaxis Symmetry Open the yaxis symmetry applet. Move
point A around the Cartesian plane with your mouse. How
are the coordinates of pointA and the coordinates of point B
related?
93. xaxis Symmetry Open the xaxis symmetry applet. Move
point A around the Cartesian plane with your mouse. How
are the coordinates of pointA and the coordinates of point B
related?
94. Origin Symmetry Open the origin symmetry applet. Move
point A around the Cartesian plane with your mouse. How
are the coordinates of pointA and the coordinates of point B
related?
Interactive Exercises
Ask your instructor if the applets below are of interest to you.
1. 2. 5 3, 36566
‘Are You Prepared?’Answers
DEFINITION Let and be two distinct points.If the slope m
of the nonvertical line L containing P and Q is defined by the formula
(1)
If L is a vertical line and the slope m of L is undefined (since this
results in division by 0).
x1 = x2,
m =
y2  y1
x2  x1
x1 Z x2
x1 Z x2,Q = 1x2, y22P = 1x1, y12
Figure 27(a) on page 20 provides an illustration of the slope of a nonvertical line;
Figure 27(b) illustrates a vertical line.
51.
Finding and Interpreting the Slope of a Line Given Two Points
The slope m of the line containing the points and may be computed as
For every 4unit change in x, y will change by . That is, if x increases by
4 units, then y will decrease by 5 units. The average rate of change of y with respect
to x is
Now Work P R O B L E M S 1 1 A N D 1 7

5
4
.
5 units
m =
3  2
5  1
=
5
4
= 
5
4
or as m =
2  132
1  5
=
5
4
= 
5
4
15, 3211, 22
EXAMPLE 1
᭹
20 CHAPTER 1 Graphs
Rise = y2
– y1
Q = (x2, y2)
P = (x1
, y1
)
L
Run = x2 – x1
Slope of L is m =(a)
x1
x2
y2
y1
y2
– y1_______
x2 – x1
Q = (x1, y2)
P = (x1, y1)
L
Slope is undefined; L is vertical(b)
x1
y2
y1
Figure 27
As Figure 27(a) illustrates, the slope m of a nonvertical line may be viewed as
m =
y2  y1
x2  x1
=
Rise
Run
or m =
y2  y1
x2  x1
=
Change in y
Change in x
=
¢y
¢x
That is, the slope m of a nonvertical line measures the amount y changes when x
changes from to . The expression is called the average rate of change of y,
with respect to x.
Two comments about computing the slope of a nonvertical line may prove
helpful:
1. Any two distinct points on the line can be used to compute the slope of the line.
(See Figure 28 for justification.)
¢y
¢x
x2x1
x
y
R
A
C
B
y2 – y1
Q = (x2, y2)
P = (x1
, y1
)
x2 – x1
Figure 28
Triangles ABC and PQR are similar (equal
angles),so ratios of corresponding sides
are proportional. Then
d(B, C)
d(A, C)
= Slope using A and B
Slope using P and Q =
y2  y1
x2  x1
=
Since any two distinct points can be used to compute the slope of a line, the
average rate of change of a line is always the same number.
2. The slope of a line may be computed from to or from
Q to P because
y2  y1
x2  x1
=
y1  y2
x1  x2
Q = 1x2, y22P = 1x1, y12
52.
SECTION 1.3 Lines 21
Finding the Slopes of Various Lines Containing the Same Point (2, 3)
Compute the slopes of the lines and containing the following pairs of
points. Graph all four lines on the same set of coordinate axes.
L4: P = 12, 32 Q4 = 12, 52
L3: P = 12, 32 Q3 = 15, 32
L2: P = 12, 32 Q2 = 13, 12
L1: P = 12, 32 Q1 = 11, 22
L4L3,L2,L1,
EXAMPLE 2
Solution Let and denote the slopes of the lines and
respectively. Then
A rise of 5 divided by a run of 3
The graphs of these lines are given in Figure 29.
Figure 29 illustrates the following facts:
m4 is undefined because x1 = x2 = 2
m3 =
3  3
5  2
=
0
3
= 0
m2 =
1  3
3  2
=
4
1
= 4
m1 =
2  3
1  2
=
5
3
=
5
3
L4,L3,L2,L1,m4m1, m2, m3,
x
y
Ϫ5 5
5
Ϫ3
Q4 ϭ (2, 5)
Q1 ϭ (Ϫ1, Ϫ2)
m1 ϭ
m2 ϭ Ϫ4
m4 undefined
m3 ϭ 0m3 ϭ 0
L4
Q3
ϭ (5, 3)
Q2 ϭ (3, Ϫ1)
P ϭ (2, 3)
5–
3
L3L3
L2 L1
Figure 29
᭹
1. When the slope of a line is positive, the line slants upward from left to
right .
2. When the slope of a line is negative, the line slants downward from left to
right .
3. When the slope is 0, the line is horizontal .
4. When the slope is undefined, the line is vertical .1L42
1L32
1L22
1L12
Seeing the Concept
On the same screen,graph the following equations:
Slope of line is 0.
Slope of line is
Slope of line is
Slope of line is 1.
Slope of line is 2.
Slope of line is 6.
See Figure 30.
Y6 = 6x
Y5 = 2x
Y4 = x
1
2
.Y3 =
1
2
x
1
4
.Y2 =
1
4
x
Y1 = 0
Ϫ2
2
1
4
Y6 ϭ 6x
Y5 ϭ 2x
Y4 ϭ x
Y3 ϭ x
Y2 ϭ x
Y1 ϭ 0
Ϫ3 3
1
2
Figure 30
Ϫ2
2
1
4
Y6 ϭ Ϫ6x
Y5 ϭ Ϫ2x
Y4 ϭ Ϫx
Y3 ϭ Ϫ x
Y2 ϭ Ϫ x
Y1 ϭ 0
Ϫ3 3
1
2
Figure 31
Seeing the Concept
On the same screen,graph the following equations:
Slope of line is 0.
Slope of line is
Slope of line is
Slope of line is 1.
Slope of line is 2.
Slope of line is 6.
See Figure 31.
Y6 = 6x
Y5 = 2x
Y4 = x

1
2
.Y3 = 
1
2
x

1
4
.Y2 = 
1
4
x
Y1 = 0
53.
3 Find the Equation of a Vertical Line
Graphing a Line
Graph the equation: x = 3
EXAMPLE 4
Solution To graph , we find all points in the plane for which . No matter
what ycoordinate is used, the corresponding xcoordinate always equals 3.
Consequently,the graph of the equation is a vertical line with xintercept 3 and
undefined slope. See Figure 34.
x = 3
x = 31x, y2x = 3
x
y
5Ϫ1
(3, 3)
(3, 2)
(3, 1)
(3, 0)
(3, Ϫ1)
4
Ϫ1
Figure 34
᭹
22 CHAPTER 1 Graphs
2 Graph Lines Given a Point and the Slope
Graphing a Line Given a Point and a Slope
Draw a graph of the line that contains the point and has a slope of:
(a) (b) 
4
5
3
4
13, 22
EXAMPLE 3
Solution (a) The fact that the slope is means that for every horizontal
movement (run) of 4 units to the right there will be a vertical movement (rise)
of 3 units. Look at Figure 32. If we start at the given point and move
4 units to the right and 3 units up, we reach the point . By drawing the line
through this point and the point , we have the graph.
(b) The fact that the slope is
means that for every horizontal movement of 5 units to the right there will be a
corresponding vertical movement of units (a downward movement). If we
start at the given point and move 5 units to the right and then 4 units
down, we arrive at the point . By drawing the line through these points,
we have the graph. See Figure 33.
Alternatively, we can set
so that for every horizontal movement of units (a movement to the left) there
will be a corresponding vertical movement of 4 units (upward). This approach
brings us to the point , which is also on the graph shown in Figure 33.
Now Work P R O B L E M 2 3
12, 62
5

4
5
=
4
5
=
Rise
Run
18, 22
13, 22
4

4
5
=
4
5
=
Rise
Run
13, 22
17, 52
13, 22
3
4
Slope =
Rise
Run
.
x
y
–2 10
Rise = 3
Run = 4
(7, 5)
(3, 2)
5
6
Figure 32
x
y
–2 10
(8, –2)
(–2, 6)
(3, 2)
6
–2
Run = 5
Run = –5
Rise=4
Rise = –4
Figure 33
᭹
Figures 30 and 31 on page 21 illustrate that the closer the line is to the vertical
position, the greater the magnitude of the slope.
54.
SECTION 1.3 Lines 23
As suggested by Example 4, we have the following result:
THEOREM Equation of a Vertical Line
A vertical line is given by an equation of the form
where a is the xintercept.
x = a
COMMENT To graph an equation using a graphing utility, we need to express the equation in the
form in But cannot be put in this form. To overcome this, most
graphing utilities have special commands for drawing vertical lines. DRAW, LINE, PLOT, and VERT
are among the more common ones. Consult your manual to determine the correct methodology
for your graphing utility.
x = 3x6.y = 5expression
x
y
(x, y)
L
(x1, y1)
x – x1
y – y1
4 Use the Point–Slope Form of a Line; Identify Horizontal Lines
Let L be a nonvertical line with slope m and containing the point . See
Figure 35. For any other point on L, we have
m =
y  y1
x  x1
or y  y1 = m1x  x12
1x, y2
1x1, y12
THEOREM Point–Slope Form of an Equation of a Line
An equation of a nonvertical line with slope m that contains the point is
(2)y  y1 = m1x  x12
1x1, y12
Figure 35
Using the Point–Slope Form of a Line
An equation of the line with slope 4 and containing the point can be found by
using the point–slope form with , and .
Solve for y.
See Figure 36 for the graph.
Now Work P R O B L E M 4 5
y = 4x  2
m = 4, x1 = 1, y1 = 2y  2 = 41x  12
y  y1 = m1x  x12
y1 = 2m = 4, x1 = 1
11, 22
EXAMPLE 5
x
y
Ϫ2 105
(1, 2)
(2, 6)
6
Run ϭ 1
Rise ϭ 4
Figure 36
᭹
Finding the Equation of a Horizontal Line
Find an equation of the horizontal line containing the point .13, 22
EXAMPLE 6
Solution Because all the yvalues are equal on a horizontal line, the slope of a horizontal line
is 0.To get an equation, we use the point–slope form with , and .
See Figure 37 for the graph.
y = 2
y  2 = 0
m = 0, x1 = 3, and y1 = 2y  2 = 0 # 1x  32
y  y1 = m1x  x12
y1 = 2m = 0, x1 = 3
(3, 2)
x
y
–1 531
4
Figure 37
᭹
55.
6 Write the Equation of a Line in Slope–Intercept Form
Another useful equation of a line is obtained when the slope m and yintercept b
are known. In this event, we know both the slope m of the line and a point on
the line; then we use the point–slope form, equation (2), to obtain the following
equation:
y  b = m1x  02 or y = mx + b
10, b2
THEOREM Slope–Intercept Form of an Equation of a Line
An equation of a line with slope m and yintercept b is
(3)
Now Work P R O B L E M 5 1 ( E X P R E S S A N S W E R
I N S L O P E – I N T E R C E P T F O R M )
y = mx + b
In the solution to Example 7, we could have used the other point, ,
instead of the point . The equation that results, although it looks different, is
equivalent to the equation that we obtained in the example. (Try it for yourself.)
Now Work P R O B L E M 3 7
12, 32
14, 52
24 CHAPTER 1 Graphs
THEOREM Equation of a Horizontal Line
A horizontal line is given by an equation of the form
where b is the yintercept.
y = b
5 Find the Equation of a Line Given Two Points
Finding an Equation of a Line Given Two Points
Find an equation of the line containing the points and . Graph the line.14, 5212, 32
EXAMPLE 7
Solution First compute the slope of the line.
Use the point and the slope to get the point–slope form of the
equation of the line.
See Figure 38 for the graph.
y  3 = 
1
3
1x  22
m = 
1
3
12, 32
m =
5  3
4  2
=
2
6
= 
1
3
᭹
x
y
–4 10
(–4, 5)
(2, 3)
2
–2
Figure 38
As suggested by Example 6, we have the following result:
56.
SECTION 1.3 Lines 25
Ϫ4
4
Y4 ϭ 3x ϩ 2Y5 ϭ Ϫ3x ϩ 2
Y3 ϭ Ϫx ϩ 2 Y2 ϭ x ϩ 2
Y1 ϭ 2
Ϫ6 6
Figure 39 y = mx + 2
Seeing the Concept
To see the role of the yintercept b,graph the following lines on the same screen.
See Figure 40. What do you conclude about the lines y = 2x + b?
Y5 = 2x  4
Y4 = 2x + 4
Y3 = 2x  1
Y2 = 2x + 1
Y1 = 2x
Ϫ4
4
Y4 ϭ 2x ϩ 4
Y5 ϭ 2x Ϫ 4
Y3 ϭ 2x Ϫ 1
Y2 ϭ 2x ϩ 1
Y1 ϭ 2x
Ϫ6 6
Figure 40 y = 2x + b
7 Identify the Slope and yIntercept of a Line from Its Equation
When the equation of a line is written in slope–intercept form, it is easy to find the
slope m and yintercept b of the line. For example, suppose that the equation of a
line is
Compare it to .
The slope of this line is and its yintercept is 7.
Now Work P R O B L E M 7 1
2
y = mx + b
cc
y = 2x + 7
y = mx + b
y = 2x + 7
Seeing the Concept
To see the role that the slope m plays,graph the following lines on the same screen.
See Figure 39. What do you conclude about the lines y = mx + 2?
Y5 = 3x + 2
Y4 = 3x + 2
Y3 = x + 2
Y2 = x + 2
Y1 = 2
Finding the Slope and yIntercept
Find the slope m and yintercept b of the equation .Graph the equation.2x + 4y = 8
EXAMPLE 8
Solution To obtain the slope and yintercept, write the equation in slope–intercept form by
solving for y.
The coefficient of x, is the slope, and the yintercept is 2. Graph the line using
the fact that the yintercept is 2 and the slope is Then, starting at the point
go to the right 2 units and then down 1 unit to the point . See Figure 41.
Now Work P R O B L E M 7 7
12, 1210, 22,

1
2
.

1
2
,
y = mx + by = 
1
2
x + 2
4y = 2x + 8
2x + 4y = 8
x
y
–3 3
(0, 2)
(2, 1)
2
1
4
Figure 41
᭹
57.
The xintercept is 4 and the point is on the graph of the equation.
To obtain the yintercept, let in the equation and solve for y.
Let
Divide both sides by 4.
The yintercept is 2 and the point is on the graph of the equation.
Plot the points and and draw the line through the points. See
Figure 42.
Now Work P R O B L E M 9 1
Every line has an equation that is equivalent to an equation written in general
form. For example, a vertical line whose equation is
can be written in the general form
A horizontal line whose equation is
can be written in the general form
A = 0, B = 1, C = b0 # x + 1 # y = b
y = b
A = 1, B = 0, C = a1 # x + 0 # y = a
x = a
10, 2214, 02
10, 22
y = 2
4y = 8
x = 0.2102 + 4y = 8
2x + 4y = 8
x = 0
14, 02
x
y
–3 3
(0, 2)
(4, 0)
4
Figure 42
᭹
26 CHAPTER 1 Graphs
8 Graph Lines Written in General Form Using Intercepts
Refer to Example 8. The form of the equation of the line is called the
general form.
2x + 4y = 8
DEFINITION The equation of a line is in general form* when it is written as
(4)
where A, B, and C are real numbers and A and B are not both 0.
Ax + By = C
If in (4), then and the graph of the equation is a vertical line:
. If in (4), then we can solve the equation for y and write the equation
in slope–intercept form as we did in Example 8.
Another approach to graphing the equation (4) would be to find its intercepts.
Remember, the intercepts of the graph of an equation are the points where the
graph crosses or touches a coordinate axis.
B Z 0x =
C
A
A Z 0B = 0
Graphing an Equation in General Form Using Its Intercepts
Graph the equation by finding its intercepts.2x + 4y = 8
EXAMPLE 9
Solution To obtain the xintercept, let in the equation and solve for x.
Let
Divide both sides by 2.x = 4
2x = 8
y = 0.2x + 4102 = 8
2x + 4y = 8
y = 0
*Some books use the term standard form.
58.
SECTION 1.3 Lines 27
9 Find Equations of Parallel Lines
When two lines (in the plane) do not intersect (that is, they have no points in
common), they are said to be parallel. Look at Figure 43.There we have drawn two
parallel lines and have constructed two right triangles by drawing sides parallel to
the coordinate axes.The right triangles are similar. (Do you see why? Two angles are
equal.) Because the triangles are similar, the ratios of corresponding sides are equal.
THEOREM Criterion for Parallel Lines
Two nonvertical lines are parallel if and only if their slopes are equal and they
have different yintercepts.
The use of the words “if and only if” in the preceding theorem means that
actually two statements are being made, one the converse of the other.
If two nonvertical lines are parallel, then their slopes are equal and they
have different yintercepts.
If two nonvertical lines have equal slopes and they have different yintercepts,
then they are parallel.
Showing That Two Lines Are Parallel
Show that the lines given by the following equations are parallel:
L1: 2x + 3y = 6, L2: 4x + 6y = 0
EXAMPLE 10
y
Rise
Run Run
Rise
x
Figure 43
Lines that are neither vertical nor horizontal have general equations of the form
and
Because the equation of every line can be written in general form, any equation
equivalent to equation (4) is called a linear equation.
B Z 0A Z 0Ax + By = C
Solution To determine whether these lines have equal slopes and different yintercepts, write
each equation in slope–intercept form:
Because these lines have the same slope, but different yintercepts, the lines are
parallel. See Figure 44.

2
3
,
Slope = 
2
3
; yintercept = 2 Slope = 
2
3
; yintercept = 0
y = 
2
3
x + 2 y = 
2
3
x
3y = 2x + 6 6y = 4x
L1: 2x + 3y = 6 L2: 4x + 6y = 0
᭹
y
5
5
L1
L2
Ϫ5 x
Figure 44
Finding a Line That Is Parallel to a Given Line
Find an equation for the line that contains the point and is parallel to the
line .2x + y = 6
12, 32
EXAMPLE 11
Solution Since the two lines are to be parallel, the slope of the line that we seek equals the
slope of the line . Begin by writing the equation of the line
in slope–intercept form.
y = 2x + 6
2x + y = 6
2x + y = 62x + y = 6
59.
Proof Let and denote the slopes of the two lines. There is no loss in gener
ality (that is, neither the angle nor the slopes are affected) if we situate the lines so
that they meet at the origin. See Figure 47.The point is on the line having
slope and the point is on the line having slope (Do you see why
this must be true?)
Suppose that the lines are perpendicular. Then triangle OAB is a right triangle.
As a result of the Pythagorean Theorem, it follows that
(5)
Using the distance formula, the squares of these distances are
3d1A, B242
= 11  122
+ 1m2  m122
= m2
2
 2m1m2 + m1
2
3d1O, B242
= 11  022
+ 1m1  022
= 1 + m1
2
3d1O, A242
= 11  022
+ 1m2  022
= 1 + m2
2
3d1O, A242
+ 3d1O, B242
= 3d1A, B242
m1.B = 11, m12m2,
A = 11, m22
m2m1
x
y
Run = 1
O Rise = m1
Rise = m2
A = (1, m2)
B = (1, m1
)
Slope m2
Slope m1
1
Figure 47
In Problem 128 you are asked to prove the “if” part of the theorem; that is:
If two nonvertical lines have slopes whose product is , then the lines are
perpendicular.
1
28 CHAPTER 1 Graphs
y
6
Ϫ5
6Ϫ6
(2, Ϫ3)
2x ϩ y ϭ 1
2x ϩ y ϭ 6
x
Figure 45
y
90°
x
Figure 46
᭹
10 Find Equations of Perpendicular Lines
When two lines intersect at a right angle (90°), they are said to be perpendicular. See
Figure 46.
The following result gives a condition, in terms of their slopes, for two lines to
be perpendicular.
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