Precalculus, 8th Edition
GRAPHS OF PARENT FUNCTIONS Linear Function Absolute Value Function x, x 0 f x x x, f x mx b y Squar
Author: Ron Larson
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GRAPHS OF PARENT FUNCTIONS Linear Function
Absolute Value Function x, x 0 f x x
x,
f x mx b y
Square Root Function f x x
x < 0
y
y
4
2
f(x) = ⏐x⏐ x
−2
(− mb , 0( (− mb , 0( f(x) = mx + b, m>0
3
1
(0, b)
2
2
1
−1
f(x) = mx + b, m0 x
−1
4
−1
Domain: , Range: , x-intercept: bm, 0 y-intercept: 0, b Increasing when m > 0 Decreasing when m < 0
y
x
x
(0, 0)
−1
f(x) =
1
2
3
4
f(x) = ax 2 , a < 0
(0, 0) − 3 −2
−1
−2
−2
−3
−3
Domain: , Range a > 0: 0, Range a < 0 : , 0 Intercept: 0, 0 Decreasing on , 0 for a > 0 Increasing on 0, for a > 0 Increasing on , 0 for a < 0 Decreasing on 0, for a < 0 Even function y-axis symmetry Relative minimum a > 0, relative maximum a < 0, or vertex: 0, 0
x
1
2
f(x) = x 3
Domain: , Range: , Intercept: 0, 0 Increasing on , Odd function Origin symmetry
3
Rational (Reciprocal) Function f x
1 x
Exponential Function
Logarithmic Function
f x ax, a > 0, a 1
f x loga x, a > 0, a 1
y
y
y
3
f(x) =
2
1 x f(x) = a −x (0, 1)
f(x) = a x
1 x
−1
1
2
f(x) = loga x
1
(1, 0)
3
x
1 x
2
−1
Domain: , 0 傼 0, ) Range: , 0 傼 0, ) No intercepts Decreasing on , 0 and 0, Odd function Origin symmetry Vertical asymptote: y-axis Horizontal asymptote: x-axis
Domain: , Range: 0, Intercept: 0, 1 Increasing on , for f x ax Decreasing on , for f x ax Horizontal asymptote: x-axis Continuous
Domain: 0, Range: , Intercept: 1, 0 Increasing on 0, Vertical asymptote: y-axis Continuous Reflection of graph of f x ax in the line y x
Sine Function f x sin x
Cosine Function f x cos x
Tangent Function f x tan x
y
y
3
y
3
f(x) = sin x
2
2
3
f(x) = cos x
2
1
1 x
−π
f(x) = tan x
π 2
π
2π
x −π
−
π 2
π 2
−2
−2
−3
−3
Domain: , Range: 1, 1 Period: 2 x-intercepts: n, 0 y-intercept: 0, 0 Odd function Origin symmetry
π
2π
Domain: , Range: 1, 1 Period: 2 x-intercepts: n, 0 2 y-intercept: 0, 1 Even function y-axis symmetry
x
π − 2
π 2
3π 2
n 2 Range: , Period: x-intercepts: n, 0 y-intercept: 0, 0 Vertical asymptotes: x n 2 Odd function Origin symmetry Domain: all x
π
Cosecant Function f x csc x
Secant Function f x sec x
f(x) = csc x =
y
1 sin x
y
Cotangent Function f x cot x
f(x) = sec x =
1 cos x
f(x) = cot x =
y
3
3
3
2
2
2
1
1 tan x
1 x
−π
π 2
π
2π
x −π
π − 2
π 2
π
3π 2
2π
x −π
π − 2
π 2
π
2π
−2 −3
Domain: all x n Range: , 1 傼 1, Period: 2 No intercepts Vertical asymptotes: x n Odd function Origin symmetry
Domain: all x
n 2 Range: , 1 傼 1, Period: 2 y-intercept: 0, 1 Vertical asymptotes: x n 2 Even function y-axis symmetry
Domain: all x n Range: , Period: x-intercepts: n, 0 2 Vertical asymptotes: x n Odd function Origin symmetry
Inverse Sine Function f x arcsin x
Inverse Cosine Function f x arccos x
Inverse Tangent Function f x arctan x
y
y
π 2
y
π 2
π
f(x) = arccos x x
−1
−2
1
x
−1
1
f(x) = arcsin x −π 2
Domain: 1, 1 Range: , 2 2 Intercept: 0, 0 Odd function Origin symmetry
2
f(x) = arctan x π − 2
x
−1
1
Domain: 1, 1 Range: 0, y-intercept: 0, 2
Domain: , Range: , 2 2 Intercept: 0, 0 Horizontal asymptotes: y± 2 Odd function Origin symmetry
Precalculus Eighth Edition
Ron Larson The Pennsylvania State University The Behrend College With the assistance of
David C. Falvo The Pennsylvania State University The Behrend College
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Precalculus, Eighth Edition Ron Larson Publisher: Charlie VanWagner Acquiring Sponsoring Editor: Gary Whalen Development Editor: Stacy Green Assistant Editor: Cynthia Ashton Editorial Assistant: Guanglei Zhang
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Contents A Word from the Author (Preface) vii
chapter 1
Functions and Their Graphs
1
Rectangular Coordinates 2 Graphs of Equations 13 Linear Equations in Two Variables 24 Functions 39 Analyzing Graphs of Functions 54 A Library of Parent Functions 66 Transformations of Functions 73 Combinations of Functions: Composite Functions 83 Inverse Functions 92 Mathematical Modeling and Variation Chapter Summary Review Exercises Chapter Test Proofs in Mathematics Problem Solving
chapter 2
Polynomial and Rational Functions
Quadratic Functions and Models Polynomial Functions of Higher Degree Polynomial and Synthetic Division Complex Numbers Zeros of Polynomial Functions Rational Functions Nonlinear Inequalities Chapter Summary Review Exercises Chapter Test Proofs in Mathematics Problem Solving
chapter 3
Exponential and Logarithmic Functions
Exponential Functions and Their Graphs Logarithmic Functions and Their Graphs Properties of Logarithms Exponential and Logarithmic Equations
iii
iv
Contents
Exponential and Logarithmic Models Chapter Summary Review Exercises Chapter Test Cumulative Test for Chapters 1–3 Proofs in Mathematics Problem Solving
chapter 4
Trigonometry
Radian and Degree Measure Trigonometric Functions: The Unit Circle Right Triangle Trigonometry Trigonometric Functions of Any Angle Graphs of Sine and Cosine Functions Graphs of Other Trigonometric Functions Inverse Trigonometric Functions Applications and Models Chapter Summary Review Exercises Chapter Test Proofs in Mathematics Problem Solving
chapter 5
Analytic Trigonometry
Using Fundamental Identities Verifying Trigonometric Identities Solving Trigonometric Equations Sum and Difference Formulas Multiple-Angle and Product-to-Sum Formulas Chapter Summary Review Exercises Chapter Test Proofs in Mathematics Problem Solving
chapter 6
Additional Topics in Trigonometry
Law of Sines Law of Cosines Vectors in the Plane Vectors and Dot Products Trigonometric Form of a Complex Number Chapter Summary Review Exercises Chapter Test Cumulative Test for Chapters 4–6 Proofs in Mathematics Problem Solving
Contents
chapter 7
Systems of Equations and Inequalities
Linear and Nonlinear Systems of Equations Two-Variable Linear Systems Multivariable Linear Systems Partial Fractions Systems of Inequalities Linear Programming Chapter Summary Review Exercises Chapter Test Proofs in Mathematics Problem Solving
chapter 8
Matrices and Determinants
Matrices and Systems of Equations Operations with Matrices The Inverse of a Square Matrix The Determinant of a Square Matrix Applications of Matrices and Determinants Chapter Summary Review Exercises Chapter Test Proofs in Mathematics Problem Solving
chapter 9
Sequences, Series, and Probability
Sequences and Series Arithmetic Sequences and Partial Sums Geometric Sequences and Series Mathematical Induction The Binomial Theorem Counting Principles Probability Chapter Summary Review Exercises Chapter Test Cumulative Test for Chapters 7–9 Proofs in Mathematics Problem Solving
v
vi
Contents
chapter 10
Topics in Analytic Geometry
Lines Introduction to Conics: Parabolas Ellipses Hyperbolas Rotation of Conics Parametric Equations Polar Coordinates Graphs of Polar Equations Polar Equations of Conics Chapter Summary Review Exercises Chapter Test Proofs in Mathematics Problem Solving
Appendix A Review of Fundamental Concepts of Algebra A.1 A.2 A.3 A.4 A.5 A.6 A.7
Real Numbers and Their Properties A1 Exponents and Radicals A14 Polynomials and Factoring A27 Rational Expressions A39 Solving Equations A49 Linear Inequalities in One Variable A63 Errors and the Algebra of Calculus A73
Answers to Odd-Numbered Exercises and Tests Index
A
Index of Applications (web) Appendix B Concepts in Statistics (web) B.1 B.2 B.3
Representing Data Measures of Central Tendency and Dispersion Least Squares Regression
A81
A1
A Word from the Author Welcome to the Eighth Edition of Precalculus! We are proud to offer you a new and revised version of our textbook. With each edition, we have listened to you, our users, and have incorporated many of your suggestions for improvement.
8th
4th
7th
3rd
6th
2nd
5th
1st
In the Eighth Edition, we continue to offer instructors and students a text that is pedagogically sound, mathematically precise, and still comprehensible. There are many changes in the mathematics, art, and design; the more significant changes are noted here. • New Chapter Openers Each Chapter Opener has three parts, In Mathematics, In Real Life, and In Careers. In Mathematics describes an important mathematical topic taught in the chapter. In Real Life tells students where they will encounter this topic in real-life situations. In Careers relates application exercises to a variety of careers. • New Study Tips and Warning/Cautions Insightful information is given to students in two new features. The Study Tip provides students with useful information or suggestions for learning the topic. The Warning/Caution points out common mathematical errors made by students. • New Algebra Helps Algebra Help directs students to sections of the textbook where they can review algebra skills needed to master the current topic. • New Side-by-Side Examples Throughout the text, we present solutions to many examples from multiple perspectives—algebraically, graphically, and numerically. The side-by-side format of this pedagogical feature helps students to see that a problem can be solved in more than one way and to see that different methods yield the same result. The side-by-side format also addresses many different learning styles.
vii
viii
A Word from the Author
• New Capstone Exercises Capstones are conceptual problems that synthesize key topics and provide students with a better understanding of each section’s concepts. Capstone exercises are excellent for classroom discussion or test prep, and teachers may find value in integrating these problems into their reviews of the section. • New Chapter Summaries The Chapter Summary now includes an explanation and/or example of each objective taught in the chapter. • Revised Exercise Sets The exercise sets have been carefully and extensively examined to ensure they are rigorous and cover all topics suggested by our users. Many new skill-building and challenging exercises have been added. For the past several years, we’ve maintained an independent website— gwd.es—that provides free solutions to all odd-numbered exercises in the text. Thousands of students using our textbooks have visited the site for practice and help with their homework. For the Eighth Edition, we were able to use information from gwd.es, including which solutions students accessed most often, to help guide the revision of the exercises. I hope you enjoy the Eighth Edition of Precalculus. As always, I welcome comments and suggestions for continued improvements.
Acknowledgments I would like to thank the many people who have helped me prepare the text and the supplements package. Their encouragement, criticisms, and suggestions have been invaluable. Thank you to all of the instructors who took the time to review the changes in this edition and to provide suggestions for improving it. Without your help, this book would not be possible.
Reviewers Chad Pierson, University of Minnesota-Duluth; Sally Shao, Cleveland State University; Ed Stumpf, Central Carolina Community College; Fuzhen Zhang, Nova Southeastern University; Dennis Shepherd, University of Colorado, Denver; Rhonda Kilgo, Jacksonville State University; C. Altay Özgener, Manatee Community College Bradenton; William Forrest, Baton Rouge Community College; Tracy Cook, University of Tennessee Knoxville; Charles Hale, California State Poly University Pomona; Samuel Evers, University of Alabama; Seongchun Kwon, University of Toledo; Dr. Arun K. Agarwal, Grambling State University; Hyounkyun Oh, Savannah State University; Michael J. McConnell, Clarion University; Martha Chalhoub, Collin County Community College; Angela Lee Everett, Chattanooga State Tech Community College; Heather Van Dyke, Walla Walla Community College; Gregory Buthusiem, Burlington County Community College; Ward Shaffer, College of Coastal Georgia; Carmen Thomas, Chatham University; Emily J. Keaton My thanks to David Falvo, The Behrend College, The Pennsylvania State University, for his contributions to this project. My thanks also to Robert Hostetler, The Behrend College, The Pennsylvania State University, and Bruce Edwards, University of Florida, for their significant contributions to previous editions of this text. I would also like to thank the staff at Larson Texts, Inc. who assisted with proofreading the manuscript, preparing and proofreading the art package, and checking and typesetting the supplements. On a personal level, I am grateful to my spouse, Deanna Gilbert Larson, for her love, patience, and support. Also, a special thanks goes to R. Scott O’Neil. If you have suggestions for improving this text, please feel free to write to me. Over the past two decades I have received many useful comments from both instructors and students, and I value these comments very highly.
Ron Larson
ix
Supplements
Supplements for the Instructor Annotated Instructor’s Edition This AIE is the complete student text plus point-ofuse annotations for the instructor, including extra projects, classroom activities, teaching strategies, and additional examples. Answers to even-numbered text exercises, Vocabulary Checks, and Explorations are also provided. Complete Solutions Manual This manual contains solutions to all exercises from the text, including Chapter Review Exercises and Chapter Tests. Instructor’s Companion Website of instructor resources.
This free companion website contains an abundance
PowerLecture™ with ExamView® The CD-ROM provides the instructor with dynamic media tools for teaching college algebra. PowerPoint® lecture slides and art slides of the figures from the text, together with electronic files for the test bank and a link to the Solution Builder, are available. The algorithmic ExamView allows you to create, deliver, and customize tests (both print and online) in minutes with this easy-to-use assessment system. Enhance how your students interact with you, your lecture, and each other. Solutions Builder This is an electronic version of the complete solutions manual available via the PowerLecture and Instructor’s Companion Website. It provides instructors with an efficient method for creating solution sets to homework or exams that can then be printed or posted. Online AIE to the Note Taking Guide in the innovative Note Taking Guide.
x
This AIE includes the answers to all problems
Supplements
xi
Supplements for the Student Student Companion Website student resources.
This free companion website contains an abundance of
Instructional DVDs Keyed to the text by section, these DVDs provide comprehensive coverage of the course—along with additional explanations of concepts, sample problems, and applications—to help students review essential topics. Student Study and Solutions Manual This guide offers step-by-step solutions for all odd-numbered text exercises, Chapter and Cumulative Tests, and Practice Tests with solutions. Premium eBook The Premium eBook offers an interactive version of the textbook with search features, highlighting and note-making tools, and direct links to videos or tutorials that elaborate on the text discussions. Enhanced WebAssign Enhanced WebAssign is designed for you to do your homework online. This proven and reliable system uses pedagogy and content found in Larson’s text, and then enhances it to help you learn Precalculus more effectively. Automatically graded homework allows you to focus on your learning and get interactive study assistance outside of class. Note Taking Guide This is an innovative study aid, in the form of a notebook organizer, that helps students develop a section-by-section summary of key concepts.
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1
Functions and Their Graphs
Rectangular Coordinates
Graphs of Equations
Linear Equations in Two Variables
Functions
Analyzing Graphs of Functions
A Library of Parent Functions
Transformations of Functions
Inverse Functions
Combinations of Functions: Composite Functions
Mathematical Modeling and Variation
In Mathematics Functions show how one variable is related to another variable.
Functions are used to estimate values, simulate processes, and discover relationships. For instance, you can model the enrollment rate of children in preschool and estimate the year in which the rate will reach a certain number. Such an estimate can be used to plan measures for meeting future needs, such as hiring additional teachers and buying more books. (See Exercise , page )
Jose Luis Pelaez/Getty Images
In Real Life
IN CAREERS There are many careers that use functions. Several are listed below. • Financial analyst Exercise 95, page 51
• Tax preparer Example 3, page
• Biologist Exercise 73, page 91
• Oceanographer Exercise 83, page
1
2
Chapter 1
Functions and Their Graphs
RECTANGULAR COORDINATES What you should learn
The Cartesian Plane
• Plot points in the Cartesian plane. • Use the Distance Formula to find the distance between two points. • Use the Midpoint Formula to find the midpoint of a line segment. • Use a coordinate plane to model and solve real-life problems.
Just as you can represent real numbers by points on a real number line, you can represent ordered pairs of real numbers by points in a plane called the rectangular coordinate system, or the Cartesian plane, named after the French mathematician René Descartes (–). The Cartesian plane is formed by using two real number lines intersecting at right angles, as shown in Figure The horizontal real number line is usually called the x-axis, and the vertical real number line is usually called the y-axis. The point of intersection of these two axes is the origin, and the two axes divide the plane into four parts called quadrants.
Why you should learn it The Cartesian plane can be used to represent relationships between two variables. For instance, in Exercise 70 on page 11, a graph represents the minimum wage in the United States from to
y-axis
Quadrant II
3 2 1
Origin −3
−2
−1
Quadrant I
Directed distance x
(Vertical number line) x-axis
−1 −2
Quadrant III
−3
FIGURE
y-axis
1
2
(x, y)
3
(Horizontal number line)
Directed y distance
Quadrant IV
FIGURE
x-axis
© Ariel Skelly/Corbis
Each point in the plane corresponds to an ordered pair (x, y) of real numbers x and y, called coordinates of the point. The x-coordinate represents the directed distance from the y-axis to the point, and the y-coordinate represents the directed distance from the x-axis to the point, as shown in Figure Directed distance from y-axis
4
(3, 4)
3
Example 1
(−1, 2)
−4 −3
−1
−1 −2
(−2, −3) FIGURE
−4
Directed distance from x-axis
The notation x, y denotes both a point in the plane and an open interval on the real number line. The context will tell you which meaning is intended.
y
1
x, y
(0, 0) 1
(3, 0) 2
3
4
x
Plotting Points in the Cartesian Plane
Plot the points 1, 2, 3, 4, 0, 0, 3, 0, and 2, 3.
Solution To plot the point 1, 2, imagine a vertical line through 1 on the x-axis and a horizontal line through 2 on the y-axis. The intersection of these two lines is the point 1, 2. The other four points can be plotted in a similar way, as shown in Figure Now try Exercise 7.
Section
Rectangular Coordinates
3
The beauty of a rectangular coordinate system is that it allows you to see relationships between two variables. It would be difficult to overestimate the importance of Descartes’s introduction of coordinates in the plane. Today, his ideas are in common use in virtually every scientific and business-related field.
Example 2 Subscribers, N
From through , the numbers N (in millions) of subscribers to a cellular telecommunication service in the United States are shown in the table, where t represents the year. Sketch a scatter plot of the data. (Source: CTIA-The Wireless Association)
Solution To sketch a scatter plot of the data shown in the table, you simply represent each pair of values by an ordered pair t, N and plot the resulting points, as shown in Figure For instance, the first pair of values is represented by the ordered pair , . Note that the break in the t-axis indicates that the numbers between 0 and have been omitted.
N
Number of subscribers (in millions)
Year, t
Sketching a Scatter Plot
Subscribers to a Cellular Telecommunication Service
50 t
Year FIGURE
Now try Exercise In Example 2, you could have let t 1 represent the year In that case, the horizontal axis would not have been broken, and the tick marks would have been labeled 1 through 14 (instead of through ).
T E C H N O LO G Y The scatter plot in Example 2 is only one way to represent the data graphically. You could also represent the data using a bar graph or a line graph. If you have access to a graphing utility, try using it to represent graphically the data given in Example 2.
4
Chapter 1
Functions and Their Graphs
The Pythagorean Theorem and the Distance Formula a2 + b2 = c2
The following famous theorem is used extensively throughout this course.
c
a
Pythagorean Theorem For a right triangle with hypotenuse of length c and sides of lengths a and b, you have a 2 b2 c 2, as shown in Figure (The converse is also true. That is, if a 2 b2 c 2, then the triangle is a right triangle.) b
FIGURE
Suppose you want to determine the distance d between two points x1, y1 and x2, y2 in the plane. With these two points, a right triangle can be formed, as shown in Figure The length of the vertical side of the triangle is y2 y1 , and the length of the horizontal side is x2 x1 . By the Pythagorean Theorem, you can write
y
y
(x1, y1 )
1
y 2 − y1
2
d x2 x1 2 y2 y1 2 x2 x12 y2 y12. y
2
This result is the Distance Formula.
(x1, y2 ) (x2, y2 ) x1
x2
x
x 2 − x1 FIGURE
d 2 x2 x1 2 y2 y1
d
The Distance Formula The distance d between the points x1, y1 and x2, y2 in the plane is d x2 x12 y2 y12.
Example 3
Finding a Distance
Find the distance between the points 2, 1 and 3, 4.
Algebraic Solution Let x1, y1 2, 1 and x2, y2 3, 4. Then apply the Distance Formula. d x2 x12 y2 y12 3 22 4 12
Distance Formula Substitute for x1, y1, x2, and y2.
5 2 32
Simplify.
34
Simplify.
Use a calculator.
Graphical Solution Use centimeter graph paper to plot the points A2, 1 and B3, 4. Carefully sketch the line segment from A to B. Then use a centimeter ruler to measure the length of the segment.
cm 1 2 3 4 5
Distance checks.
✓
7
34 34
6
So, the distance between the points is about units. You can use the Pythagorean Theorem to check that the distance is correct. ? d 2 32 52 Pythagorean Theorem 2 ? Substitute for d. 34 32 52
FIGURE
The line segment measures about centimeters, as shown in Figure So, the distance between the points is about units. Now try Exercise
Section
y
Example 4
Rectangular Coordinates
5
Verifying a Right Triangle
(5, 7)
7
Show that the points 2, 1, 4, 0, and 5, 7 are vertices of a right triangle.
6 5
Solution d1 = 45
4
The three points are plotted in Figure Using the Distance Formula, you can find the lengths of the three sides as follows.
d3 = 50
3 2 1
d2 4 2 2 0 1 2 4 1 5
(4, 0) 1 FIGURE
d1 5 2 2 7 1 2 9 36 45
d2 = 5
(2, 1) 2
3
4
5
x 6
7
d3 5 4 2 7 0 2 1 49 50 Because
d12 d22 45 5 50 d32 you can conclude by the Pythagorean Theorem that the triangle must be a right triangle. Now try Exercise
You can review the techniques for evaluating a radical in Appendix A
The Midpoint Formula To find the midpoint of the line segment that joins two points in a coordinate plane, you can simply find the average values of the respective coordinates of the two endpoints using the Midpoint Formula.
The Midpoint Formula The midpoint of the line segment joining the points x1, y1 and x 2, y 2 is given by the Midpoint Formula Midpoint
x1 x 2 y1 y2 , . 2 2
For a proof of the Midpoint Formula, see Proofs in Mathematics on page
Example 5
Finding a Line Segment’s Midpoint
Find the midpoint of the line segment joining the points 5, 3 and 9, 3.
Solution Let x1, y1 5, 3 and x 2, y 2 9, 3.
y
6
(9, 3) (2, 0) −6
x
−3
(−5, −3)
3 −3 −6
FIGURE
Midpoint
6
9
x1 x2 y1 y2
2 , 2 5 9 3 3 , 2 2
Midpoint
3
2, 0
Midpoint Formula
Substitute for x1, y1, x2, and y2. Simplify.
The midpoint of the line segment is 2, 0, as shown in Figure Now try Exercise 47(c).
6
Chapter 1
Functions and Their Graphs
Applications Example 6
Finding the Length of a Pass
A football quarterback throws a pass from the yard line, 40 yards from the sideline. The pass is caught by a wide receiver on the 5-yard line, 20 yards from the same sideline, as shown in Figure How long is the pass?
Solution You can find the length of the pass by finding the distance between the points 40, 28 and 20, 5.
Football Pass
Distance (in yards)
35
d x2 x12 y2 y12
(40, 28)
30 25
40 20 28 5 2
20 15 10
(20, 5)
5
Distance Formula
2
Substitute for x1, y1, x2, and y2.
Simplify.
Simplify.
30
Use a calculator.
5 10 15 20 25 30 35 40
So, the pass is about 30 yards long.
Distance (in yards) FIGURE
Now try Exercise
In Example 6, the scale along the goal line does not normally appear on a football field. However, when you use coordinate geometry to solve real-life problems, you are free to place the coordinate system in any way that is convenient for the solution of the problem.
Example 7
Estimating Annual Revenue
Barnes & Noble had annual sales of approximately $ billion in , and $ billion in Without knowing any additional information, what would you estimate the sales to have been? (Source: Barnes & Noble, Inc.)
Solution
Sales (in billions of dollars)
y
One solution to the problem is to assume that sales followed a linear pattern. With this assumption, you can estimate the sales by finding the midpoint of the line segment connecting the points , and , .
Barnes & Noble Sales
(, )
x1 x2 y1 y2 , 2 2
, 2 2
(, ) Midpoint
(, )
Year
, x
FIGURE
Midpoint
Midpoint Formula
Substitute for x1, x2, y1 and y2. Simplify.
So, you would estimate the sales to have been about $ billion, as shown in Figure (The actual sales were about $ billion.) Now try Exercise
Section
Example 8
7
Rectangular Coordinates
Translating Points in the Plane
The triangle in Figure has vertices at the points 1, 2, 1, 4, and 2, 3. Shift the triangle three units to the right and two units upward and find the vertices of the shifted triangle, as shown in Figure y
y
5
5 4
4
(2, 3)
Paul Morrell
(−1, 2)
3 2 1
Much of computer graphics, including this computer-generated goldfish tessellation, consists of transformations of points in a coordinate plane. One type of transformation, a translation, is illustrated in Example 8. Other types include reflections, rotations, and stretches.
x
−2 −1
1
2
3
4
5
6
7
−2
1
2
3
5
6
7
−2
−3
−3
(1, −4)
−4 FIGURE
x
−2 −1
−4
FIGURE
Solution To shift the vertices three units to the right, add 3 to each of the x-coordinates. To shift the vertices two units upward, add 2 to each of the y-coordinates. Original Point 1, 2
Translated Point 1 3, 2 2 2, 4
1, 4
1 3, 4 2 4, 2
2, 3
2 3, 3 2 5, 5 Now try Exercise
The figures provided with Example 8 were not really essential to the solution. Nevertheless, it is strongly recommended that you develop the habit of including sketches with your solutions—even if they are not required.
CLASSROOM DISCUSSION Extending the Example Example 8 shows how to translate points in a coordinate plane. Write a short paragraph describing how each of the following transformed points is related to the original point. Original Point x, y
Transformed Point ⴚx, y
x, y
x, ⴚy
x, y
ⴚx, ⴚy
8
Chapter 1
Functions and Their Graphs
EXERCISES
See gwd.es for worked-out solutions to odd-numbered exercises.
VOCABULARY 1. Match each term with its definition. (a) x-axis (i) point of intersection of vertical axis and horizontal axis (b) y-axis (ii) directed distance from the x-axis (c) origin (iii) directed distance from the y-axis (d) quadrants (iv) four regions of the coordinate plane (e) x-coordinate (v) horizontal real number line (f) y-coordinate (vi) vertical real number line In Exercises 2– 4, fill in the blanks. 2. An ordered pair of real numbers can be represented in a plane called the rectangular coordinate system or the ________ plane. 3. The ________ ________ is a result derived from the Pythagorean Theorem. 4. Finding the average values of the representative coordinates of the two endpoints of a line segment in a coordinate plane is also known as using the ________ ________.
SKILLS AND APPLICATIONS In Exercises 5 and 6, approximate the coordinates of the points. y
5.
A
6
D
y
6. C
4
2
D
2
−6 −4 −2 −2 B −4
4
x 2
4
−6
−4
−2
C
x 2
B −2 A
−4
In Exercises 7–10, plot the points in the Cartesian plane. 7. 8. 9.
4, 2, 3, 6, 0, 5, 1, 4 0, 0, 3, 1, 2, 4, 1, 1 3, 8, , 1, 5, 6, 2, 1, 13 , 34, 3, 3, 4, 43, 32
In Exercises 11–14, find the coordinates of the point. The point is located three units to the left of the y-axis and four units above the x-axis. The point is located eight units below the x-axis and four units to the right of the y-axis. The point is located five units below the x-axis and the coordinates of the point are equal. The point is on the x-axis and 12 units to the left of the y-axis.
In Exercises 15–24, determine the quadrant(s) in which x, y is located so that the condition(s) is (are) satisfied.
x > 0 and y < 0 x 4 and y > 0 y < 5 x < 0 and y > 0 xy > 0
x < 0 and y < 0 x > 2 and y 3 x > 4 x > 0 and y < 0 xy < 0
In Exercises 25 and 26, sketch a scatter plot of the data shown in the table. NUMBER OF STORES The table shows the number y of Wal-Mart stores for each year x from through (Source: Wal-Mart Stores, Inc.) Year, x
Number of stores, y
Section
METEOROLOGY The table shows the lowest temperature on record y (in degrees Fahrenheit) in Duluth, Minnesota for each month x, where x 1 represents January. (Source: NOAA) Month, x
Temperature, y
1 2 3 4 5 6 7 8 9 10 11 12
39 39 29 5 17 27 35 32 22 8 23 34
In Exercises 43–46, show that the points form the vertices of the indicated polygon.
Right triangle: 4, 0, 2, 1, 1, 5 Right triangle: 1, 3), 3, 5, 5, 1 Isosceles triangle: 1, 3, 3, 2, 2, 4 Isosceles triangle: 2, 3, 4, 9, 2, 7
1, 4, 8, 4 3, 4, 3, 6 8, 5, 0, 20 1, 3, 3, 2 23, 3, 1, 54
1, 1, 9, 7 4, 10, 4, 5 1, 2, 5, 4 12, 1, 52, 43 , , ,
In Exercises 39– 42, (a) find the length of each side of the right triangle, and (b) show that these lengths satisfy the Pythagorean Theorem. y
4
(13, 5) (1, 0)
4
(4, 2)
x 4
x 1
2
3
4
8
(13, 0)
5
y
30 20 10
(12, 18)
Distance (in yards) 8
(0, 2)
(50, 42)
10 20 30 40 50 60
3 2
1, 12, 6, 0 7, 4, 2, 8 2, 10, 10, 2 13, 13 , 16, 12 , , ,
40
(4, 5)
5
1
50
y
FLYING DISTANCE An airplane flies from Naples, Italy in a straight line to Rome, Italy, which is kilometers north and kilometers west of Naples. How far does the plane fly? SPORTS A soccer player passes the ball from a point that is 18 yards from the endline and 12 yards from the sideline. The pass is received by a teammate who is 42 yards from the same endline and 50 yards from the same sideline, as shown in the figure. How long is the pass? Distance (in yards)
6, 3, 6, 5 3, 1, 2, 1 2, 6, 3, 6 1, 4, 5, 1 12, 43 , 2, 1 , , , , , ,
SALES In Exercises 59 and 60, use the Midpoint Formula to estimate the sales of Big Lots, Inc. and Dollar Tree Stores, Inc. in , given the sales in and Assume that the sales followed a linear pattern. (Source: Big Lots, Inc.; Dollar Tree Stores, Inc.) Big Lots
y
(1, 5)
6
4
(9, 4)
Year
Sales (in millions)
$ $
4 2
(9, 1)
2
(5, −2)
x
(−1, 1)
6
9
In Exercises 47–56, (a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points.
In Exercises 27–38, find the distance between the points.
Rectangular Coordinates
x
8 −2
(1, −2)
6
Dollar Tree Year
Sales (in millions)
$ $
In Exercises 61–64, the polygon is shifted to a new position in the plane. Find the coordinates of the vertices of the polygon in its new position. y
(−3, 6) 7 (−1, 3) 5 6 units
3 units
4
(−1, −1)
x
−4 −2
2
(−2, − 4)
(−3, 0) (−5, 3)
2 units (2, −3)
x 1
3
Original coordinates of vertices: 7, 2,2, 2, 2, 4, 7, 4 Shift: eight units upward, four units to the right Original coordinates of vertices: 5, 8, 3, 6, 7, 6, 5, 2 Shift: 6 units downward, 10 units to the left RETAIL PRICE In Exercises 65 and 66, use the graph, which shows the average retail prices of 1 gallon of whole milk from to (Source: U.S. Bureau of Labor Statistics) Average price (in dollars per gallon)
Year FIGURE FOR
y
5 units
Cost of second TV spot (in thousands of dollars)
Functions and Their Graphs
67
(a) Estimate the percent increase in the average cost of a second spot from Super Bowl XXXIV in to Super Bowl XXXVIII in (b) Estimate the percent increase in the average cost of a second spot from Super Bowl XXXIV in to Super Bowl XLII in ADVERTISING The graph shows the average costs of a second television spot (in thousands of dollars) during the Academy Awards from to (Source: Nielson Monitor-Plus) Cost of second TV spot (in thousands of dollars)
Chapter 1
Year
Year
Approximate the highest price of a gallon of whole milk shown in the graph. When did this occur? Approximate the percent change in the price of milk from the price in to the highest price shown in the graph. ADVERTISING The graph shows the average costs of a second television spot (in thousands of dollars) during the Super Bowl from to (Source: Nielson Media and TNS Media Intelligence)
(a) Estimate the percent increase in the average cost of a second spot in to the cost in (b) Estimate the percent increase in the average cost of a second spot in to the cost in MUSIC The graph shows the numbers of performers who were elected to the Rock and Roll Hall of Fame from through Describe any trends in the data. From these trends, predict the number of performers elected in (Source: gwd.es) 10
Number elected
10
8 6 4 2
Year
Section
Minimum wage (in dollars)
LABOR FORCE Use the graph below, which shows the minimum wage in the United States (in dollars) from to (Source: U.S. Department of Labor)
Year, x
Pieces of mail, y
8 7 6 5 4 3 2 1
Year
(a) Which decade shows the greatest increase in minimum wage? (b) Approximate the percent increases in the minimum wage from to and from to (c) Use the percent increase from to to predict the minimum wage in (d) Do you believe that your prediction in part (c) is reasonable? Explain. SALES The Coca-Cola Company had sales of $19, million in and $28, million in Use the Midpoint Formula to estimate the sales in Assume that the sales followed a linear pattern. (Source: The Coca-Cola Company) DATA ANALYSIS: EXAM SCORES The table shows the mathematics entrance test scores x and the final examination scores y in an algebra course for a sample of 10 students. x
22
29
35
40
44
48
53
58
65
76
y
53
74
57
66
79
90
76
93
83
99
(a) Sketch a scatter plot of the data. (b) Find the entrance test score of any student with a final exam score in the 80s. (c) Does a higher entrance test score imply a higher final exam score? Explain. DATA ANALYSIS: MAIL The table shows the number y of pieces of mail handled (in billions) by the U.S. Postal Service for each year x from through (Source: U.S. Postal Service)
Rectangular Coordinates
TABLE FOR
11
73
(a) Sketch a scatter plot of the data. (b) Approximate the year in which there was the greatest decrease in the number of pieces of mail handled. (c) Why do you think the number of pieces of mail handled decreased? DATA ANALYSIS: ATHLETICS The table shows the numbers of men’s M and women’s W college basketball teams for each year x from through (Source: National Collegiate Athletic Association) Year, x
Men’s teams, M
Women’s teams, W
(a) Sketch scatter plots of these two sets of data on the same set of coordinate axes.
12
Chapter 1
Functions and Their Graphs
(b) Find the year in which the numbers of men’s and women’s teams were nearly equal. (c) Find the year in which the difference between the numbers of men’s and women’s teams was the greatest. What was this difference?
EXPLORATION A line segment has x1, y1 as one endpoint and xm, ym as its midpoint. Find the other endpoint x2, y2 of the line segment in terms of x1, y1, xm, and ym. Use the result of Exercise 75 to find the coordinates of the endpoint of a line segment if the coordinates of the other endpoint and midpoint are, respectively, (a) 1, 2, 4, 1 and (b) 5, 11, 2, 4. Use the Midpoint Formula three times to find the three points that divide the line segment joining x1, y1 and x2, y2 into four parts. Use the result of Exercise 77 to find the points that divide the line segment joining the given points into four equal parts. (a) 1, 2, 4, 1 (b) 2, 3, 0, 0 MAKE A CONJECTURE Plot the points 2, 1, 3, 5, and 7, 3 on a rectangular coordinate system. Then change the sign of the x-coordinate of each point and plot the three new points on the same rectangular coordinate system. Make a conjecture about the location of a point when each of the following occurs. (a) The sign of the x-coordinate is changed. (b) The sign of the y-coordinate is changed. (c) The signs of both the x- and y-coordinates are changed. COLLINEAR POINTS Three or more points are collinear if they all lie on the same line. Use the steps below to determine if the set of points A2, 3, B2, 6, C6, 3 and the set of points A8, 3, B5, 2, C2, 1 are collinear. (a) For each set of points, use the Distance Formula to find the distances from A to B, from B to C, and from A to C. What relationship exists among these distances for each set of points? (b) Plot each set of points in the Cartesian plane. Do all the points of either set appear to lie on the same line? (c) Compare your conclusions from part (a) with the conclusions you made from the graphs in part (b). Make a general statement about how to use the Distance Formula to determine collinearity.
TRUE OR FALSE? In Exercises 81 and 82, determine whether the statement is true or false. Justify your answer. In order to divide a line segment into 16 equal parts, you would have to use the Midpoint Formula 16 times. The points 8, 4, 2, 11, and 5, 1 represent the vertices of an isosceles triangle. THINK ABOUT IT When plotting points on the rectangular coordinate system, is it true that the scales on the x- and y-axes must be the same? Explain. CAPSTONE Use the plot of the point x0 , y0 in the figure. Match the transformation of the point with the correct plot. Explain your reasoning. [The plots are labeled (i), (ii), (iii), and (iv).] y
(x0 , y0 ) x
(i)
y
y
(ii)
x
(iii)
y
x
y
(iv)
x
(a) x0, y0 (c) x0, 12 y0
x
(b) 2x0, y0 (d) x0, y0
PROOF Prove that the diagonals of the parallelogram in the figure intersect at their midpoints. y
(b , c)
(a + b , c)
(0, 0)
(a, 0)
x
Section
Graphs of Equations
13
GRAPHS OF EQUATIONS What you should learn • Sketch graphs of equations. • Find x- and y-intercepts of graphs of equations. • Use symmetry to sketch graphs of equations. • Find equations of and sketch graphs of circles. • Use graphs of equations in solving real-life problems.
Why you should learn it The graph of an equation can help you see relationships between real-life quantities. For example, in Exercise 87 on page 23, a graph can be used to estimate the life expectancies of children who are born in
The Graph of an Equation In Section , you used a coordinate system to represent graphically the relationship between two quantities. There, the graphical picture consisted of a collection of points in a coordinate plane. Frequently, a relationship between two quantities is expressed as an equation in two variables. For instance, y 7 3x is an equation in x and y. An ordered pair a, b is a solution or solution point of an equation in x and y if the equation is true when a is substituted for x and b is substituted for y. For instance, 1, 4 is a solution of y 7 3x because 4 7 31 is a true statement. In this section you will review some basic procedures for sketching the graph of an equation in two variables. The graph of an equation is the set of all points that are solutions of the equation.
Example 1
Determining Solution Points
Determine whether (a) 2, 13 and (b) 1, 3 lie on the graph of y 10x 7.
Solution a.
y 10x 7 ? 13 102 7 13 13
Write original equation. Substitute 2 for x and 13 for y.
2, 13 is a solution.
✓
The point 2, 13 does lie on the graph of y 10x 7 because it is a solution point of the equation. b.
y 10x 7 ? 3 101 7 3 17
Write original equation. Substitute 1 for x and 3 for y.
1, 3 is not a solution.
© John Griffin/The Image Works
The point 1, 3 does not lie on the graph of y 10x 7 because it is not a solution point of the equation. Now try Exercise 7. The basic technique used for sketching the graph of an equation is the point-plotting method.
Sketching the Graph of an Equation by Point Plotting 1. If possible, rewrite the equation so that one of the variables is isolated on one side of the equation. When evaluating an expression or an equation, remember to follow the Basic Rules of Algebra. To review these rules, see Appendix A
2. Make a table of values showing several solution points. 3. Plot these points on a rectangular coordinate system. 4. Connect the points with a smooth curve or line.
14
Chapter 1
Functions and Their Graphs
Example 2
Sketching the Graph of an Equation
Sketch the graph of y 7 3x.
Solution Because the equation is already solved for y, construct a table of values that consists of several solution points of the equation. For instance, when x 1, y 7 31 10 which implies that 1, 10 is a solution point of the graph. x
y 7 3x
x, y
1
10
1, 10
0
7
0, 7
1
4
1, 4
2
1
2, 1
3
2
3, 2
4
5
4, 5
From the table, it follows that
1, 10, 0, 7, 1, 4, 2, 1, 3, 2, and 4, 5 are solution points of the equation. After plotting these points, you can see that they appear to lie on a line, as shown in Figure The graph of the equation is the line that passes through the six plotted points. y
(− 1, 10) 8 6 4
(0, 7) (1, 4)
2
(2, 1) x
−4 −2 −2 −4 −6 FIGURE
Now try Exercise
2
4
6
8 10
(3, −2)
(4, − 5)
Section
Example 3
15
Graphs of Equations
Sketching the Graph of an Equation
Sketch the graph of y x 2 2.
Solution Because the equation is already solved for y, begin by constructing a table of values. 2
1
0
1
2
3
2
1
2
1
2
7
2, 2
1, 1
0, 2
1, 1
2, 2
3, 7
x y x2 2 One of your goals in this course is to learn to classify the basic shape of a graph from its equation. For instance, you will learn that the linear equation in Example 2 has the form
x, y
Next, plot the points given in the table, as shown in Figure Finally, connect the points with a smooth curve, as shown in Figure y
y
y mx b and its graph is a line. Similarly, the quadratic equation in Example 3 has the form y
ax 2
bx c
and its graph is a parabola.
(3, 7)
(3, 7) 6
6
4
4
2
2
y = x2 − 2
(−2, 2) −4
x
−2
(−1, −1)
FIGURE
(−2, 2)
(2, 2) 2
(1, −1) (0, −2)
−4
4
−2
(−1, −1)
FIGURE
(2, 2) x 2
(1, −1) (0, −2)
4
Now try Exercise The point-plotting method demonstrated in Examples 2 and 3 is easy to use, but it has some shortcomings. With too few solution points, you can misrepresent the graph of an equation. For instance, if only the four points
2, 2, 1, 1, 1, 1, and 2, 2 in Figure were plotted, any one of the three graphs in Figure would be reasonable. y
y
4
4
4
2
2
2
x
−2
FIGURE
y
2
−2
x 2
−2
x 2
16
Chapter 1
Functions and Their Graphs
y
T E C H N O LO G Y To graph an equation involving x and y on a graphing utility, use the following procedure. 1. Rewrite the equation so that y is isolated on the left side.
x
2. Enter the equation into the graphing utility. No x-intercepts; one y-intercept
3. Determine a viewing window that shows all important features of the graph.
y
4. Graph the equation.
Intercepts of a Graph It is often easy to determine the solution points that have zero as either the x-coordinate or the y-coordinate. These points are called intercepts because they are the points at which the graph intersects or touches the x- or y-axis. It is possible for a graph to have no intercepts, one intercept, or several intercepts, as shown in Figure Note that an x-intercept can be written as the ordered pair x, 0 and a y-intercept can be written as the ordered pair 0, y. Some texts denote the x-intercept as the x-coordinate of the point a, 0 [and the y-intercept as the y-coordinate of the point 0, b] rather than the point itself. Unless it is necessary to make a distinction, we will use the term intercept to mean either the point or the coordinate.
x
Three x-intercepts; one y-intercept y
x
Finding Intercepts
One x-intercept; two y-intercepts
1. To find x-intercepts, let y be zero and solve the equation for x.
y
2. To find y-intercepts, let x be zero and solve the equation for y.
Finding x- and y-Intercepts
Example 4
Find the x- and y-intercepts of the graph of y x3 4x.
x
Solution
No intercepts FIGURE
Let y 0. Then 0 x3 4x xx2 4 y
has solutions x 0 and x ± 2.
y = x 3 − 4x 4 (0, 0)
(−2, 0)
Let x 0. Then
(2, 0) x
−4
4 −2 −4
FIGURE
x-intercepts: 0, 0, 2, 0, 2, 0
y 03 40 has one solution, y 0. y-intercept: 0, 0
See Figure
Now try Exercise
Section
Graphs of Equations
17
Symmetry Graphs of equations can have symmetry with respect to one of the coordinate axes or with respect to the origin. Symmetry with respect to the x-axis means that if the Cartesian plane were folded along the x-axis, the portion of the graph above the x-axis would coincide with the portion below the x-axis. Symmetry with respect to the y-axis or the origin can be described in a similar manner, as shown in Figure y
y
y
(x, y) (x, y)
(−x, y)
(x, y) x
x x
(x, −y) (−x, −y)
x-axis symmetry FIGURE
y-axis symmetry
Origin symmetry
Knowing the symmetry of a graph before attempting to sketch it is helpful, because then you need only half as many solution points to sketch the graph. There are three basic types of symmetry, described as follows.
Graphical Tests for Symmetry 1. A graph is symmetric with respect to the x-axis if, whenever x, y is on the graph, x, y is also on the graph. 2. A graph is symmetric with respect to the y-axis if, whenever x, y is on the graph, x, y is also on the graph. 3. A graph is symmetric with respect to the origin if, whenever x, y is on the graph, x, y is also on the graph.
y
7 6 5 4 3 2 1
(− 3, 7)
(− 2, 2)
(3, 7)
(2, 2) x
− 4 −3 − 2
(−1, −1) −3
FIGURE
You can conclude that the graph of y x 2 2 is symmetric with respect to the y-axis because the point x, y is also on the graph of y x2 2. (See the table below and Figure )
2 3 4 5
x
3
2
1
1
2
3
y
7
2
1
1
2
7
3, 7
2, 2
1, 1
1, 1
2, 2
3, 7
x, y
(1, −1)
y = x2 − 2
y-axis symmetry
Algebraic Tests for Symmetry 1. The graph of an equation is symmetric with respect to the x-axis if replacing y with y yields an equivalent equation. 2. The graph of an equation is symmetric with respect to the y-axis if replacing x with x yields an equivalent equation. 3. The graph of an equation is symmetric with respect to the origin if replacing x with x and y with y yields an equivalent equation.
18
Chapter 1
Functions and Their Graphs
Example 5
Testing for Symmetry
Test y 2x3 for symmetry with respect to both axes and the origin.
y 2
Solution
(1, 2)
y 2x3
x-axis:
y = 2x 3 1
Write original equation.
y 2x
3
x −2
−1
1
y 2x3
y-axis:
2
−2
y 2x
Replace x with x.
y 2x3
Simplify. Result is not an equivalent equation.
y 2x
Write original equation.
3
Origin: FIGURE
Write original equation. 3
−1
(−1, −2)
Replace y with y. Result is not an equivalent equation.
y 2x3
Replace y with y and x with x.
y 2x
Simplify.
3
y
y 2x3
x − y2 = 1
2
(5, 2) 1
Of the three tests for symmetry, the only one that is satisfied is the test for origin symmetry (see Figure ).
(2, 1) (1, 0)
Now try Exercise
x 2
3
4
Equivalent equation
5
−1
Example 6
−2
Using Symmetry as a Sketching Aid
Use symmetry to sketch the graph of x y 2 1.
FIGURE
Solution Of the three tests for symmetry, the only one that is satisfied is the test for x-axis symmetry because x y2 1 is equivalent to x y2 1. So, the graph is symmetric with respect to the x-axis. Using symmetry, you only need to find the solution points above the x-axis and then reflect them to obtain the graph, as shown in Figure
Now try Exercise
In Example 7, x 1 is an absolute value expression. You can review the techniques for evaluating an absolute value expression in Appendix A
Example 7
Sketching the Graph of an Equation
Sketch the graph of y x 1 .
Solution This equation fails all three tests for symmetry and consequently its graph is not symmetric with respect to either axis or to the origin. The absolute value sign indicates that y is always nonnegative. Create a table of values and plot the points, as shown in Figure From the table, you can see that x 0 when y 1. So, the y-intercept is 0, 1. Similarly, y 0 when x 1. So, the x-intercept is 1, 0.
y 6 5
y = ⏐x − 1⏐
(−2, 3) 4 3
(4, 3) (3, 2) (2, 1)
(−1, 2) 2 (0, 1) −3 −2 −1
x x
(1, 0) 2
3
4
5
y x1
x, y
2
1
0
1
2
3
4
3
2
1
0
1
2
3
2, 3
1, 2
0, 1
1, 0
2, 1
3, 2
4, 3
−2 FIGURE
Now try Exercise
Section
y
Graphs of Equations
19
Throughout this course, you will learn to recognize several types of graphs from their equations. For instance, you will learn to recognize that the graph of a seconddegree equation of the form y ax 2 bx c Center: (h, k)
is a parabola (see Example 3). The graph of a circle is also easy to recognize.
Circles
Radius: r Point on circle: (x, y)
Consider the circle shown in Figure A point x, y is on the circle if and only if its distance from the center h, k is r. By the Distance Formula, x
FIGURE
x h2 y k2 r.
By squaring each side of this equation, you obtain the standard form of the equation of a circle.
Standard Form of the Equation of a Circle The point x, y lies on the circle of radius r and center (h, k) if and only if
x h 2 y k 2 r 2.
WARNING / CAUTION Be careful when you are finding h and k from the standard equation of a circle. For instance, to find the correct h and k from the equation of the circle in Example 8, rewrite the quantities x 12 and y 22 using subtraction.
From this result, you can see that the standard form of the equation of a circle with its center at the origin, h, k 0, 0, is simply x 2 y 2 r 2.
Example 8
Circle with center at origin
Finding the Equation of a Circle
The point 3, 4 lies on a circle whose center is at 1, 2, as shown in Figure Write the standard form of the equation of this circle.
x 12 x 12, y 22 y 22
Solution
So, h 1 and k 2.
The radius of the circle is the distance between 1, 2 and 3, 4. r x h2 y k2
y
6
(3, 4) 4
(−1, 2) −6
FIGURE
x
−2
2
4
Distance Formula
3 1 4 2
Substitute for x, y, h, and k.
42 22
Simplify.
16 4
Simplify.
20
Radius
2
2
Using h, k 1, 2 and r 20, the equation of the circle is
x h2 y k2 r 2
Equation of circle
−2
x 1 2 y 22 20
−4
x 1 y 2
2
2
2
Now try Exercise
Substitute for h, k, and r. Standard form
20
Chapter 1
Functions and Their Graphs
Application In this course, you will learn that there are many ways to approach a problem. Three common approaches are illustrated in Example 9. You should develop the habit of using at least two approaches to solve every problem. This helps build your intuition and helps you check that your answers are reasonable.
A Numerical Approach: Construct and use a table. A Graphical Approach: Draw and use a graph. An Algebraic Approach: Use the rules of algebra.
Example 9
Recommended Weight
The median recommended weight y (in pounds) for men of medium frame who are 25 to 59 years old can be approximated by the mathematical model y x 2 x ,
62 x 76
where x is the man’s height (in inches). (Source: Metropolitan Life Insurance Company) a. Construct a table of values that shows the median recommended weights for men with heights of 62, 64, 66, 68, 70, 72, 74, and 76 inches. b. Use the table of values to sketch a graph of the model. Then use the graph to estimate graphically the median recommended weight for a man whose height is 71 inches. c. Use the model to confirm algebraically the estimate you found in part (b).
Solution Weight, y
62 64 66 68 70 72 74 76
a. You can use a calculator to complete the table, as shown at the left. b. The table of values can be used to sketch the graph of the equation, as shown in Figure From the graph, you can estimate that a height of 71 inches corresponds to a weight of about pounds. y
Recommended Weight
Weight (in pounds)
Height, x
x 62 64 66 68 70 72 74 76
Height (in inches) FIGURE
c. To confirm algebraically the estimate found in part (b), you can substitute 71 for x in the model. y (71)2 (71) So, the graphical estimate of pounds is fairly good. Now try Exercise
Section
EXERCISES
21
Graphs of Equations
See gwd.es for worked-out solutions to odd-numbered exercises.
VOCABULARY: Fill in the blanks. 1. An ordered pair a, b is a ________ of an equation in x and y if the equation is true when a is substituted for 2. 3. 4. 5. 6.
x, and b is substituted for y. The set of all solution points of an equation is the ________ of the equation. The points at which a graph intersects or touches an axis are called the ________ of the graph. A graph is symmetric with respect to the ________ if, whenever x, y is on the graph, x, y is also on the graph. The equation x h2 y k2 r 2 is the standard form of the equation of a ________ with center ________ and radius ________. When you construct and use a table to solve a problem, you are using a ________ approach.
SKILLS AND APPLICATIONS In Exercises 7–14, determine whether each point lies on the graph of the equation. 7. 8. 9.
Equation y x 4 y 5 x y x 2 3x 2 y4 x2 y x1 2 2x y 3 0 x2 y2 20 y 13x3 2x 2
(a) (a) (a) (a) (a) (a) (a) (a)
Points 0, 2 (b) 1, 2 (b) 2, 0 (b) 1, 5 (b) 2, 3 (b) 1, 2 (b) 3, 2 (b) 2, (b)
y 5 x 2
5, 3 5, 0 2, 8 6, 0 1, 0 1, 1 4, 2 3, 9
In Exercises 19–22, graphically estimate the x- and y-intercepts of the graph. Verify your results algebraically. y x 32
y 20
10 8 6 4 2
0
1
2
−4 −2
5 2
8 4
2
0
1
4 3
3
y
5 4 3 2
2
x 1
y2 4 x y 3 1 x −1
1 2
4 5
x
y
−4 −3 −2 −1
x, y
1
−3
1
In Exercises 23–32, find the x- and y-intercepts of the graph of the equation.
y x 2 3x
x, y
−1
2 4 6 8
y x 2
y 34 x 1
y
y 16 4x 2
y
x, y
x
2
x
y
x
1
x, y
y 2x 5 1
0
y
In Exercises 15–18, complete the table. Use the resulting solution points to sketch the graph of the equation.
x
1
2
x
0
1
2
3
y 5x 6 y x 4 y 3x 7 y 2x3 4x 2 y2 6 x
y 8 3x y 2x 1 y x 10 y x 4 25 y2 x 1
22
Chapter 1
Functions and Their Graphs
In Exercises 33– 40, use the algebraic tests to check for symmetry with respect to both axes and the origin. x 2 y 0 y x 3 x y 2 x 1 xy 2 10 0
x y 2 0 y x 4 x 2 3 y
1 x2 1
xy 4
y
y
y 6 x x y 2 x
In Exercises 69–76, write the standard form of the equation of the circle with the given characteristics.
In Exercises 41– 44, assume that the graph has the indicated type of symmetry. Sketch the complete graph of the equation. To print an enlarged copy of the graph, go to the website gwd.es
y x x 6 y x 3
4
Center: 0, 0; Radius: 4 Center: 0, 0; Radius: 5 Center: 2, 1; Radius: 4 Center: 7, 4; Radius: 7 Center: 1, 2; Solution point: 0, 0 Center: 3, 2; Solution point: 1, 1 Endpoints of a diameter: 0, 0, 6, 8 Endpoints of a diameter: 4, 1, 4, 1
4 2
2 x
−4
2
x
4
2
−2
4
6
8
−4
y-axis symmetry
x-axis symmetry
y
−4
−2
y
4
4
2
2 x 2
−4
4
−2 −4
−2
x 2
4
−2 −4
Origin symmetry
y-axis symmetry
In Exercises 45–56, identify any intercepts and test for symmetry. Then sketch the graph of the equation.
y 3x 1 y x 2 2x y x3 3 y x 3 y x6 x y2 1
y 2x 3 y x 2 2x y x3 1 y 1 x y1 x x y2 5
In Exercises 57–68, use a graphing utility to graph the equation. Use a standard setting. Approximate any intercepts. 1 y 3 2x y x 2 4x 3 2x y x1 3 x 2 y
The symbol
2 y 3x 1 y x 2 x 2 4 y 2 x 1 3 x 1 y
In Exercises 77– 82, find the center and radius of the circle, and sketch its graph. x 2 y 2 25 x 12 y 32 9 x 12 2 y 12 2 94 x 22 y 32
x 2 y 2 16 x 2 y 1 2 1
DEPRECIATION A hospital purchases a new magnetic resonance imaging (MRI) machine for $, The depreciated value y (reduced value) after t years is given by y , 40,t, 0 t 8. Sketch the graph of the equation. CONSUMERISM You purchase an all-terrain vehicle (ATV) for $ The depreciated value y after t years is given by y t, 0 t 6. Sketch the graph of the equation. GEOMETRY A regulation NFL playing field (including the end zones) of length x and width y has a perimeter 2 of or 3 yards. (a) Draw a rectangle that gives a visual representation of the problem. Use the specified variables to label the sides of the rectangle. (b) Show that the width of the rectangle is y x and its area is A x x . 3 3
(c) Use a graphing utility to graph the area equation. Be sure to adjust your window settings. (d) From the graph in part (c), estimate the dimensions of the rectangle that yield a maximum area. (e) Use your school’s library, the Internet, or some other reference source to find the actual dimensions and area of a regulation NFL playing field and compare your findings with the results of part (d).
indicates an exercise or a part of an exercise in which you are instructed to use a graphing utility.
Section
GEOMETRY A soccer playing field of length x and width y has a perimeter of meters. (a) Draw a rectangle that gives a visual representation of the problem. Use the specified variables to label the sides of the rectangle. (b) Show that the width of the rectangle is y x and its area is A x x. (c) Use a graphing utility to graph the area equation. Be sure to adjust your window settings. (d) From the graph in part (c), estimate the dimensions of the rectangle that yield a maximum area. (e) Use your school’s library, the Internet, or some other reference source to find the actual dimensions and area of a regulation Major League Soccer field and compare your findings with the results of part (d). POPULATION STATISTICS The table shows the life expectancies of a child (at birth) in the United States for selected years from to (Source: U.S. National Center for Health Statistics) Year
Life Expectancy, y
A model for the life expectancy during this period is y t 2 t , 20 t where y represents the life expectancy and t is the time in years, with t 20 corresponding to (a) Use a graphing utility to graph the data from the table and the model in the same viewing window. How well does the model fit the data? Explain. (b) Determine the life expectancy in both graphically and algebraically. (c) Use the graph to determine the year when life expectancy was approximately Verify your answer algebraically. (d) One projection for the life expectancy of a child born in is How does this compare with the projection given by the model?
Graphs of Equations
23
(e) Do you think this model can be used to predict the life expectancy of a child 50 years from now? Explain. ELECTRONICS The resistance y (in ohms) of feet of solid copper wire at 68 degrees Fahrenheit can be approximated by the model y
10, , 5 x x2
where x is the diameter of the wire in mils ( inch). (Source: American Wire Gage) (a) Complete the table. x
5
10
20
30
40
50
y x
60
70
80
90
y (b) Use the table of values in part (a) to sketch a graph of the model. Then use your graph to estimate the resistance when x (c) Use the model to confirm algebraically the estimate you found in part (b). (d) What can you conclude in general about the relationship between the diameter of the copper wire and the resistance?
EXPLORATION THINK ABOUT IT Find a and b if the graph of y ax 2 bx 3 is symmetric with respect to (a) the y-axis and (b) the origin. (There are many correct answers.) CAPSTONE Match the the given characteristic. (i) y 3x3 3x (ii) (iii) y 3x 3 (iv) 2 (v) y 3x 3 (vi) (a) (b) (c) (d) (e) (f)
equation or equations with y x 32 3 x y y x 3
Symmetric with respect to the y-axis Three x-intercepts Symmetric with respect to the x-axis 2, 1 is a point on the graph Symmetric with respect to the origin Graph passes through the origin
24
Chapter 1
Functions and Their Graphs
LINEAR EQUATIONS IN TWO VARIABLES What you should learn • Use slope to graph linear equations in two variables. • Find the slope of a line given two points on the line. • Write linear equations in two variables. • Use slope to identify parallel and perpendicular lines. • Use slope and linear equations in two variables to model and solve real-life problems.
Why you should learn it Linear equations in two variables can be used to model and solve real-life problems. For instance, in Exercise on page 36, you will use a linear equation to model student enrollment at the Pennsylvania State University.
Using Slope The simplest mathematical model for relating two variables is the linear equation in two variables y mx b. The equation is called linear because its graph is a line. (In mathematics, the term line means straight line.) By letting x 0, you obtain y m0 b
Substitute 0 for x.
b. So, the line crosses the y-axis at y b, as shown in Figure In other words, the y-intercept is 0, b. The steepness or slope of the line is m. y mx b Slope
y-Intercept
The slope of a nonvertical line is the number of units the line rises (or falls) vertically for each unit of horizontal change from left to right, as shown in Figure and Figure y
y
y-intercept
1 unit
y = mx + b
m units, m0
(0, b)
y-intercept
1 unit
y = mx + b
Courtesy of Pennsylvania State University
x
Positive slope, line rises. FIGURE
x
Negative slope, line falls.
FIGURE
A linear equation that is written in the form y mx b is said to be written in slope-intercept form.
The Slope-Intercept Form of the Equation of a Line The graph of the equation y mx b is a line whose slope is m and whose y-intercept is 0, b.
Section
y
Once you have determined the slope and the y-intercept of a line, it is a relatively simple matter to sketch its graph. In the next example, note that none of the lines is vertical. A vertical line has an equation of the form
(3, 5)
5
25
Linear Equations in Two Variables
4
x a.
x=3
Vertical line
The equation of a vertical line cannot be written in the form y mx b because the slope of a vertical line is undefined, as indicated in Figure
3 2
(3, 1)
1
Example 1
Graphing a Linear Equation
x 1 FIGURE
2
4
5
Sketch the graph of each linear equation.
Slope is undefined.
a. y 2x 1 b. y 2 c. x y 2
Solution a. Because b 1, the y-intercept is 0, 1. Moreover, because the slope is m 2, the line rises two units for each unit the line moves to the right, as shown in Figure b. By writing this equation in the form y 0x 2, you can see that the y-intercept is 0, 2 and the slope is zero. A zero slope implies that the line is horizontal—that is, it doesn’t rise or fall, as shown in Figure c. By writing this equation in slope-intercept form xy2
Write original equation.
y x 2
Subtract x from each side.
y 1x 2
Write in slope-intercept form.
you can see that the y-intercept is 0, 2. Moreover, because the slope is m 1, the line falls one unit for each unit the line moves to the right, as shown in Figure y
y
5
y = 2x + 1
4 3
y
5
5
4
4
y=2
3
3
m=2
2
(0, 2)
2
m=0
(0, 2) x
1
m = −1
1
1
(0, 1)
y = −x + 2
2
3
4
5
When m is positive, the line rises. FIGURE
x
x 1
2
3
4
5
When m is 0, the line is horizontal. FIGURE
Now try Exercise
1
2
3
4
5
When m is negative, the line falls. FIGURE
26
Chapter 1
Functions and Their Graphs
Finding the Slope of a Line y
y2 y1
Given an equation of a line, you can find its slope by writing the equation in slopeintercept form. If you are not given an equation, you can still find the slope of a line. For instance, suppose you want to find the slope of the line passing through the points x1, y1 and x2, y2 , as shown in Figure As you move from left to right along this line, a change of y2 y1 units in the vertical direction corresponds to a change of x2 x1 units in the horizontal direction.
(x 2, y 2 ) y2 − y1
(x 1, y 1) x 2 − x1 x1
FIGURE
x2
y2 y1 the change in y rise
x
and x2 x1 the change in x run The ratio of y2 y1 to x2 x1 represents the slope of the line that passes through the points x1, y1 and x2, y2 . Slope
change in y change in x
rise run
y2 y1 x2 x1
The Slope of a Line Passing Through Two Points The slope m of the nonvertical line through x1, y1 and x2, y2 is m
y2 y1 x2 x1
where x1 x2.
When this formula is used for slope, the order of subtraction is important. Given two points on a line, you are free to label either one of them as x1, y1 and the other as x2, y2 . However, once you have done this, you must form the numerator and denominator using the same order of subtraction. m
y2 y1 x2 x1
Correct
m
y1 y2 x1 x2
Correct
m
y2 y1 x1 x2
Incorrect
For instance, the slope of the line passing through the points 3, 4 and 5, 7 can be calculated as m
74 3 53 2
or, reversing the subtraction order in both the numerator and denominator, as m
4 7 3 3 . 3 5 2 2
Section
Example 2
Linear Equations in Two Variables
27
Finding the Slope of a Line Through Two Points
Find the slope of the line passing through each pair of points. a. 2, 0 and 3, 1
b. 1, 2 and 2, 2
c. 0, 4 and 1, 1
d. 3, 4 and 3, 1
Solution a. Letting x1, y1 2, 0 and x2, y2 3, 1, you obtain a slope of To find the slopes in Example 2, you must be able to evaluate rational expressions. You can review the techniques for evaluating rational expressions in Appendix A
m
y2 y1 10 1 . x2 x1 3 2 5
See Figure
b. The slope of the line passing through 1, 2 and 2, 2 is m
22 0 0. 2 1 3
See Figure
c. The slope of the line passing through 0, 4 and 1, 1 is m
1 4 5 5. 10 1
See Figure
d. The slope of the line passing through 3, 4 and 3, 1 is m
1 4 3 . 33 0
See Figure
Because division by 0 is undefined, the slope is undefined and the line is vertical. y
y
4
In Figures to , note the relationships between slope and the orientation of the line. a. Positive slope: line rises from left to right b. Zero slope: line is horizontal c. Negative slope: line falls from left to right d. Undefined slope: line is vertical
4
3
m=
2
(3, 1) (−2, 0) −2 −1
FIGURE
(−1, 2)
1 x
1
−1
2
3
−2 −1
FIGURE
(0, 4)
3
m = −5
2
2
−1
2
3
(3, 4) Slope is undefined. (3, 1)
1
1 x
2
(1, − 1)
−1
FIGURE
x
1
4
3
−1
(2, 2)
1
y
y
4
m=0
3
1 5
3
4
Now try Exercise
−1
x
1
−1
FIGURE
2
4
28
Chapter 1
Functions and Their Graphs
Writing Linear Equations in Two Variables If x1, y1 is a point on a line of slope m and x, y is any other point on the line, then y y1 m. x x1 This equation, involving the variables x and y, can be rewritten in the form y y1 mx x1 which is the point-slope form of the equation of a line.
Point-Slope Form of the Equation of a Line The equation of the line with slope m passing through the point x1, y1 is y y1 mx x1.
The point-slope form is most useful for finding the equation of a line. You should remember this form.
Example 3 y
y = 3x − 5
Find the slope-intercept form of the equation of the line that has a slope of 3 and passes through the point 1, 2.
1 −2
x
−1
1
3
−1 −2 −3
3
4
Solution Use the point-slope form with m 3 and x1, y1 1, 2. y y1 mx x1
1 (1, −2)
−4 −5 FIGURE
Using the Point-Slope Form
y 2 3x 1 y 2 3x 3 y 3x 5
-
-
-