3.10 3.11 Notes A
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This document provides homework questions on solving equations and graphing equations. It includes: 1) Solving equations with one variable like 3x + 2 = 5x + 3 and determining the number of solutions. 2) Solving a two variable equation 2x + 3y = 12 and finding points that satisfy the equation. 3) Graphing equations like y = x^2 - 3x + 2 on a graphing calculator and determining features of the graph like where it crosses the x-axis. 4) Applying a transformation like (x,y) -> (x + 5, y) to the points on a graph. 5) Identifying points that do or do not
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(2) The graph of a cubic function is a continuous smooth curve that can have up to two turning points, where the curve changes from increasing to decreasing. These turning points can be local maxima or minima.
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1. Using any two points on the line.
2. Using the x-intercept and y-intercept.
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2. Details about grading which includes assignments, homework, tests, the final exam, and notebook checks.
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3.12 3.13 Notes A
1) The document discusses graphing equations and finding intersection points of graphs. It includes examples of graphing equations like y = x2 - 4x + 3 and finding the intersection points of the graphs y = x and y = 1000 - x.
2) Tony and Sasha's method for graphing y = x2 - 4x + 3 involved choosing x-values of 0 and -1 to plot points and draw a curve between them.
3) The document provides homework problems from pages 263 and 267 involving graphing equations and finding intersection points.
3.12 3.13 Notes B
This document contains a lesson plan on graphing equations. It includes instructions for students to graph equations like y=x^2 - 4x + 3 by plotting points, determine intersection points between two graphs, and identify the region each quadrant represents on a coordinate plane. It provides guidance for students to practice graphing, identifies key ideas to remember, and assigns homework problems for students to complete.
Lecture 07 graphing linear equations
This document provides an overview of graphing linear equations. It defines key terms like solutions, intercepts, and linear models. Examples are given to show how to graph equations by finding intercepts or using a table of points. Horizontal and vertical lines are discussed as special cases of linear equations. The document concludes with an example of using a linear equation to model a real-world situation involving monthly phone costs.
Math Algebra
The document provides an overview of key algebra concepts covered on the ACT exam, including expressions, equations, inequalities, functions, and quadratic equations. It discusses how to simplify and combine like terms in expressions, solve different types of equations (linear, quadratic, absolute value), factor expressions using techniques like greatest common factor and difference of squares, work with fractions and systems of equations. Example problems and step-by-step explanations are provided for many of the concepts. The document is intended as a review of essential algebra skills and strategies for tackling related questions that may appear on the ACT.
4 7 Notes B
This document provides instruction on writing and graphing linear equations in different forms. It begins with a review of equivalent forms of linear equations, including point-slope, slope-intercept, and standard form. Students are then given examples of writing equations of lines given a point and slope, or two points. The document concludes with examples of graphing lines by using the point-slope and slope-intercept forms to easily find points and draw the line. Students are challenged to graph a line given in standard form and explain their process.
5HBC: How to Graph Implicit Relations Intro Packet!
This document discusses five methods for graphing implicit functions on a TI-83 graphing calculator:
1. Using function mode, programming, and Euler's method to graph solutions to a differential equation defined by the implicit function.
2. Using parametric mode and the quadratic formula to solve the implicit function for x as a parametric function of t.
3. Using function mode, solving for x as a function of y, and using DrawInv to graph the inverse relation.
4. Using function mode and the Solve() command to numerically solve the implicit equation for y as a function of x.
5. Using polar mode by rewriting the implicit equation in terms of r and θ and graphing r
4 7 Notes A
Day 2 Opener:
- Review problem solving linear equations from the previous day
- Introduce different forms of writing linear equations: point-slope form, slope-intercept form, standard form
- Students work with a partner to practice identifying properties of lines from their equations
Section 4.7 Jiffy Graphs:
- Explain how to quickly write an equation of a line given certain information using the different forms
- Demonstrate graphing lines using point-slope and slope-intercept form by plotting points
- Ask students to consider how to graph a line in standard form for the exit slip
4 7 Notes A
Day 2 Opener:
- Review problem solving linear equations from the previous day
- Introduce different forms of writing linear equations: point-slope form, slope-intercept form, standard form
- Students work with a partner to practice identifying properties of lines from their equations
Section 4.7 Jiffy Graphs:
- Explain how to quickly write an equation of a line given certain information using the different forms
- Demonstrate graphing lines using point-slope and slope-intercept form by plotting points
- Ask students to consider how to graph a line in standard form for the exit slip
7.1
This document discusses how to solve systems of linear equations by graphing. There are 4 steps: 1) Put both equations in slope-intercept form; 2) Graph both equations on the same coordinate plane; 3) Estimate where the graphs intersect, which is the solution; 4) Check that the solution makes both equations true. It provides examples of solving systems of two linear equations with two variables by graphing them and finding the point where the lines intersect.
3.1 solving systems graphically
The document discusses solving systems of linear equations by graphing. It explains that a system of linear equations contains two or more linear equations using the same variables. To solve a linear system graphically, the two equations must be put into slope-intercept form and graphed on the same coordinate plane. The point where the lines intersect satisfies both equations and is the solution to the system. The solution must be verified by substituting the coordinates into the original equations.
Calculator Strategies
This document provides calculator strategies for the TAKS test, including how to clear memory, use the y= feature to match equations to graphs and tables, ask for specific x-values in a table, graph systems of equations, change between decimals and fractions, navigate the home screen, and use parentheses properly. It demonstrates how to use the calculator to solve problems involving areas, points on a coordinate grid, linear systems, and evaluating expressions for a given value. Graphing and table features are used to verify answer choices for several problems.
April 7, 2014
1) The document outlines a class agenda that includes reviewing quadratics and new solving methods in preparation for a Friday test. It also notes a third quarter exam on Friday covering quadratics, factoring, exponents, and polynomials.
2) The class will begin with warm-up problems reviewing graphing quadratics and finding axes of symmetry. Steps for graphing the equation y + 6x = x^2 + 9 are provided.
3) Additional examples are given for graphing a quadratic function modeling the height of a basketball after being shot, finding the maximum height and time to reach it by using the vertex of the parabola.
Quadratic equations
This document discusses quadratic equations and methods for solving them. It begins by defining quadratic equations as second degree polynomial equations of the form ax^2 + bx + c = 0, where a is not equal to 0. It then presents several methods for finding the roots or solutions of quadratic equations: factoring, completing the square, and using the quadratic formula. Examples are provided to illustrate each method. The document also discusses graphing quadratic functions and key features of parabolas such as vertex, axis of symmetry, and direction of opening.
Xmss ict lesson sec 2 qe wksheet
1. The document provides instructions for students to use a graphing calculator application called Nspire to explore and analyze graphs of quadratic equations.
2. Students are asked to vary the values of a, b, and c in different quadratic equations and record the shape of the graph, location of maximum/minimum points, and equation of the line of symmetry.
3. The summary explains that graphs of quadratic equations with a positive coefficient of x^2 open up and have a maximum point, while those with a negative coefficient of x^2 open down and have a minimum point. The graph is always symmetrical around the line of symmetry passing through the maximum or minimum point.
2010 assessment review_homework_1-4-done
The document provides an assessment review with multiple choice questions about math concepts like algebra, geometry, and coordinate planes. It includes 15 questions testing skills like simplifying expressions, solving equations, factoring polynomials, and graphing lines. The questions are formatted with explanations of steps required to arrive at the answers.
GraphTransformations.pptx
1. This document discusses graph transformations of functions, including translations, stretches, and reflections. It provides rules for how modifications inside and outside the function f(x) will affect the x-values and y-values of the graph.
2. Examples are given of applying transformation rules to specific points on a graph and determining the new coordinates. The document also demonstrates sketching a transformed graph using key points.
3. An exercise section provides multiple choice and short answer questions to test understanding of describing transformations and finding coordinates of transformed points.
Líneas rectas (slide share)
This document defines key concepts about linear equations including slope, graphing lines, finding the slope between two points, and writing equations of lines in various forms (point-slope, slope-intercept). It provides examples of finding the slope and equation of a line given points, graphing lines, and identifying vertical and horizontal lines. The objectives are to define slope as a rate of change, graph a line given a point and slope, find the slope from two points, and write equations of vertical and horizontal lines in different forms. Practice problems are included to reinforce these concepts.
Gr 8O's 2nd Online Session
1. The document provides instructions for solving a pair of simultaneous equations graphically. It includes creating tables of values from the equations, plotting the points on a graph, drawing the lines represented by the equations, and finding the point where the lines intersect, which is the solution to the simultaneous equations.
2. An example problem is worked through step-by-step, with the equations y = x + 1 and x + y = 3. Tables of values are completed and the corresponding points are plotted and lines are drawn on the same graph.
3. Another simultaneous equations problem is then posed for students to solve graphically using the same process: if x + y = 1 and x - y = -5,
Quarter 2 Week 1 Math 8 Danao.pptx
This module provides lessons on linear inequalities in two variables, including:
1) Defining linear equations and inequalities, and differentiating between the two. Linear inequalities divide the plane into two half-planes, while equations represent a single line.
2) Explaining how to read and determine solutions to linear inequalities in two variables by substituting values. Graphs of inequalities show the solution set as the shaded region.
3) Demonstrating how to graph linear inequalities by plotting the boundary line and shading the correct half-plane based on testing a point. Steps are provided to graph inequalities in slope-intercept and standard form.
4) Presenting examples of solving word problems
1.Select the graph of the quadratic function ƒ(x) = 4 – x2. Iden.docx
1.
Select the graph of the quadratic function ƒ(x) = 4 – x2. Identify the vertex and axis of symmetry. Identify the correct graph by noting it in the space below: 1st, 2nd, 3rd, 4th, or 5th.
2.
Select the graph of the quadratic function ƒ(x) = x2 + 3. Identify the vertex and axis of symmetry. Identify which of the graphs listed below is the correct one: 1st, 2nd, 3rd, 4th, or 5th.
3.
Determine the x-intercept(s) of the quadratic function: ƒ(x) = x2 + 4x – 32
(-4,0), (8,0)
(0,0), (7,0)
(4,0), (-8,0)
(0,0), (-7,0)
no x-intercept(s)
4.
Perform the operation and write the result in standard form: (3x2 + 5) – (x2 – 4x + 5)
3x2 + 4x
2x2 + 4x + 5
2x2 - 4x
2x2 + 4x
2x2 + 4x - 5
5.
Multiply or find the special product: (x+4)(x+9)
x2 + 13x
x2 + 4x + 36
x2 + 36
x2 + 13x + 36
x2 + 13x + 9
6.
Evaluate the function
1/8
1/6
1/4
1/7
1/5
7.
The expression 9/5 C+32 where C stands for temperature in degrees Celsius, is used to convert Celsius to Fahrenheit. If the temperature is 45 degrees Celsius, find the temperature in degrees Fahrenheit.
8.
If 3 is subtracted from twice a number, the result is 8 less than the number. Write an equation to solve this problem.
9.
Plot the points and find the slope of the line passing through the pair of points (0,6), (4,0). Identify the correct graph from the ones listed below: 1st, 2nd, 3rd, or 4th.
10.
Graphically estimate the x- and y- intercepts of the graph:
y = x3 - 9x
11.
Find the slope of a line that passes through the given points
(-2,1) (3,4)
3/5
5/3
-7/5
1/2
12.
Determine whether the lines are parallel, perpendicular, both, or neither.
Parallel
Perpendicular
Both
Neither
13.
Mike works for $12 an hour. A total of 15% of his salary is deducted for taxes and insurance. He is trying to save $700 for a new bicycle. Write an equation to help determine how many hours he must work to take home $700 if he saves all of her earnings?
12h - .15 = 700
12h + .15(12h) = 700
h - .15(12h) = 700
12h - .15(12h) = 700
14.
Which of the following would NOT represent a parabola in real life?
The McDonald’s arches
The trajectory of a ball thrown up in the air
The cables on a suspension bridge
A pitched roof
15.
Determine whether the value of x=0 is a solution of the equation.
5x-3 = 3x+5
True
False
16.
Which of the following represents the general formula of a circle?
y = ax2 + bx +c
x2 + y2 = r2
Ax + By = C
y = mx + b
17.
When should you use the quadratic formula?
When a quadratic equation CANNOT be factored easily or at all
When a quadratic equation CAN be factored easily
When a linear equation CANNOT be factored easily or at all
When a linear equation CAN be factored easily
18.
Factor the Trinomial: x2 + 14x + 45
(x-5)(x-9)
(x+5)(x-9)
(x+5)(x+9)
(x-5)(x+9)
19.
Solve the following by extracting the square roots:
X^2-4=0
20.
In a given amount of time, James drove twice as far as Rachel. Alto ...
Similar to 3.10 3.11 Notes A (20)
5HBC: How to Graph Implicit Relations Intro Packet!
5HBC: How to Graph Implicit Relations Intro Packet!
1.Select the graph of the quadratic function ƒ(x) = 4 – x2. Iden.docx
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Unit 1 test 2.9 notes b
1) The document provides 4 math word problems and equations to solve using techniques like backtracking and defining variables.
2) It asks the reader to simplify the expression 2(x+6) - 4(3x - 1) and evaluate it for x = -2.
3) Another problem defines a variable to represent the number of giraffes at Brookfield Zoo and writes an expression relating it to the number of giraffes at Lincoln Park Zoo.
Unit 1 test 2.9 notes a
1) The document provides 4 math word problems and equations to solve using techniques like backtracking and defining variables.
2) It asks the reader to simplify the expression 2(x+6) - 4(3x - 1) and evaluate it for x = -2.
3) Another problem defines a variable to represent the number of giraffes at Brookfield Zoo and writes an expression relating it to the number of giraffes at Lincoln Park Zoo.
1.6 1.7 notes b
This document provides examples and explanations of the midpoint and distance formulas, as well as transformations in the coordinate plane. It includes 4 examples of using the midpoint and distance formulas to find midpoints and lengths of segments. It also defines different types of transformations, such as translations, reflections, and rotations. Students are given homework problems applying these concepts, and examples of identifying transformations and describing them with arrow notation.
2.8 notes a
1. The document discusses solving equations by backtracking. It begins by defining an equation and identifying true and false equations. It then discusses the concept of solutions - the values that make the equation true.
2. Examples are provided to illustrate finding solutions to equations by reversing the steps of number tricks. Backtracking involves working backwards through the operations to determine the starting value.
3. Readers are instructed to solve sample equations by listing the number trick steps and then reversing the steps through backtracking to find the solution values.
Chapter 1 rev notes (a)
This document provides homework questions and a review packet for a chapter. It lists 7 numbered sections that appear to be questions or tasks related to reviewing material from the first chapter. The document aims to help students review and reinforce their understanding of the concepts covered in the initial chapter through completing the homework and review activities.
1.6 1.7 notes b
This document provides examples and explanations of the midpoint and distance formulas, as well as transformations in the coordinate plane. It includes 4 examples of using the midpoint and distance formulas to find midpoints and lengths of segments. It also defines different types of transformations, such as translations, reflections, and rotations. Students are given homework problems applying these concepts, and examples of identifying transformations and describing them with arrow notation.
1.6 1.7 notes a
This document provides examples and explanations of the midpoint and distance formulas, as well as transformations in the coordinate plane. It includes 4 examples of using the midpoint and distance formulas to find midpoints and lengths of line segments. It also discusses the different types of transformations (translations, reflections, rotations, and dilations) and provides examples of identifying each type using arrow notation. The homework assignments are to complete practice problems from sections 1.6 and 1.7 in the textbook, as listed.
1.6 1.7 notes a
This document provides examples and explanations of the midpoint and distance formulas, as well as transformations in the coordinate plane. It includes 4 examples of using the midpoint and distance formulas to find midpoints and lengths of line segments. It also discusses the different types of transformations - translations, reflections, rotations, and dilations - and provides examples of identifying each type of transformation using arrow notation. The homework assignments are to complete practice problems from sections 1.6 and 1.7 in the textbook, as listed.
1.5 1.6 notes b
1. The document provides instructions for a quiz and homework assignments on geometry chapters 1.5 and 1.6.
2. Students are asked to work on their vocabulary packet silently after finishing the quiz. Then the class will review geometry formulas.
3. The homework assignments are to complete problems on page 38 from section 1.5 and page 47 from section 1.6 in the textbook. These cover midpoints, distances, and the midpoint and distance formulas.
1.5 1.6 notes b
1. The document provides instructions for a quiz and homework assignments on geometry chapters 1.5 and 1.6.
2. Students are asked to work on their vocabulary packet silently after finishing the quiz. Then the class will review geometry formulas.
3. The homework assignments are to complete problems on page 38 from section 1.5 and page 47 from section 1.6 in the textbook. These cover midpoints, distances, and the midpoint and distance formulas.
1.5 1.6 notes a
1. The document provides instructions for a quiz and homework assignments on geometry chapters 1.5 and 1.6. Students are asked to work on vocabulary and postulates after the quiz.
2. Examples are given for finding midpoints and distances between points on a coordinate plane using formulas like the midpoint formula, Pythagorean theorem, and distance formula.
3. Homework assignments include problems from the textbook on the topics of formulas in geometry, midpoints, and distance.
2.6 2.7 notes a
1. The document provides instructions and tasks for students to complete mathematical expressions, homework questions, and a lesson on reversible and non-reversible operations.
2. Students are asked to simplify expressions, complete homework problems, and determine whether example operations are reversible by considering if the starting number can be determined.
3. The document demonstrates how to "backtrack" through a multi-step operation to find the original starting number using reversible operations.
2.5 notes b
This document contains notes and instructions for a math lesson that includes:
1) Solving expressions and evaluating them for given values.
2) Completing an in-class activity with partners to review basic arithmetic rules.
3) Practicing the basic rules of arithmetic through examples of simplifying expressions using properties like commutative, associative, and distributive properties.
2.4 notes b
1. Students were asked to put math problems on the board from previous homework. The document then provides examples of expressions and teaches how to simplify them using order of operations and properties like the distributive property. Students are asked to simplify sample expressions involving variables.
2. The document reviews that expressions need to have "like terms" to be simplified, such as terms with the same variables. Students practice simplifying expressions with multiple variables and terms by combining like terms.
3. To conclude, students are instructed to write their name and ID number on raffle tickets and provide just the simplified answer, practicing the skills of defining a variable, writing an expression, simplifying it, and evaluating it.
2.2 notes a
1. When entering class each day, students should say hi, have their homework out, write any questions on the board, and start the opener problem.
2. The document then provides examples of algebra problems involving variables to represent unknown quantities and expressions combining variables, operators, and constants.
3. Students are instructed to complete a set of practice problems from page 94 in their workbook and have their work checked by the teacher.
8.7 notes b
This document contains notes from multiple sections on solving systems of equations by graphing. It includes questions on graphing systems like y = 3x - 4 and y = -2x + 5. It also asks how to solve an absolute value equation like |x+4| = x^2 - 3 and discusses the pros and cons of the graphing method, listing advantages and disadvantages. Finally, it addresses solving an equation like x^3 + 5x = 7x^2 - 5 by graphing and reflects on graphing techniques from prior lessons.
8.7 notes b
This document contains notes from multiple sections on solving systems of equations by graphing. It includes questions about graphing systems like y = 3x - 4 and y = -2x + 5. It also asks how to solve an absolute value equation like |x+4| = x^2 - 3 and discusses the pros and cons of the graphing method, listing advantages and disadvantages.
8.7 notes
The document discusses a 4 step process but provides no details on the actual steps or content of the process. It references numbered sections but provides no information within those sections. Overall, the document does not contain any substantive information that could be summarized due to the lack of details provided within the numbered sections.
8.7 d2 a
This document contains notes from multiple sections on solving systems of equations by graphing. It includes questions on graphing systems like y = 3x - 4 and y = -2x + 5. It also asks how to solve an absolute value equation like |x+4| = x^2 - 3 and discusses the pros and cons of the graphing method, listing advantages and disadvantages.
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3.10 3.11 Notes A
- 1. Homework Questions: 1
- 2. 12/7 and 12/8: 3.10 Getting Started with Equations and Their Graphs Launch: How many solutions do each of the following equations have: a.) 3x +2 = 5x + 3 b.) 3x + 3 = 3x + 3 c.) 3x + 3 = 3x ‐ 3 d.) | x | = 15 2
- 3. Determining the How many solutions does the following equation Solution for 2 have? Variable Equations: 2x + 3y = 12 Find four points that satisfy the equation above. x y What is the easiest way to view ALL of the solutions? 3
- 4. Introduction to Graph the following equation on your TI‐Nspire: Graphing equations: y = x2 ‐ 3x + 2 1.) Press HOME and then 6: NEW DOCUMENTS (Do NOT Save) 2.) Add 2: GRAPHS & GEOMETRY 3.) Enter in the equation after "f1(x) =" and then press ENTER. What shape is the graph? What values of x make y =0 in this equation? Explain. ( Press MENU then WINDOW. Choose 3: Zoom‐In. Move your cursor to the middle of the U and press enter. Then press esc.) Apply the transformation (x,y) → (x +5, y) to each point on the graph of y = x2 ‐3x + 2. Sketch the resulting graph on your graph paper. What is the equation of the new graph? 4
- 5. 3.11 Equations as Point Testers Using Equations as What are two points that do not satisfy the following Point Testers equation? 2x + 3y = 12 Points that pass the test... • • Points that fail the test... • • Graph of An The collection of all points with coordinates that make Equation: the equation true. 5
- 6. Graphing Vertical Draw the graph of the equation y = ‐1. and Horizontal Lines: Draw the graph of a horizontal line that passes through point (3, 7). Also, write an equation for the line. 6
- 7. Homework: Pg. 251 # 6‐10 Pg. 255 #7‐8, 10‐11, 14, & 16‐17 7
- 8. 8
- 9. 9