Essentials of Precalculus

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ESSENTIALS OF PRECALCULUS

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ESSENTIALS OF PRECALCULUS

Richard N. Aufmann Richard D. Nation Palomar College

Houghton Mifflin Company Boston New York

Publisher: Jack Shira Senior Sponsoring Editor: Lynn Cox Associate Editor: Jennifer King Assistant Editor: Melissa Parkin Senior Project Editor: Tamela Ambush Editorial Assistant: Sage Anderson Manufacturing Manager: Karen Banks Senior Marketing Manager: Danielle Potvin Marketing Coordinator: Nicole Mollica

Cover photograph: PunchStock

PHOTO CREDITS: Chapter 1: p. 1 Charles O’Rear / CORBIS; p. 8 The Granger Collection; p. 77 CORBIS. Chapter 2: p. 115 Sonda Dawes / The Image Works, Inc.; p. 116 David Young-Wolff / PhotoEdit, Inc.; p. 136 Syndicated Features Limited / The Image Works, Inc.; p. 157 The Granger Collection; p. 167 Bettmann/CORBIS; p. 169 Bettman / CORBIS; p. 190 Richard T. Nowitz / CORBIS; Chapter 3: p. 201 Chris McLaughlin / CORBIS; p. 221 Bettman / CORBIS; p. 227 Charles O’Rear / CORBIS; p. 227 David James / Getty Images; p. 231 Bettman / CORBIS. Chapter 4: p. 304 Courtesy of NASA and STScI; p. 318 Reuters / CORBIS. Chapter 5: p. 357 Massimo Listri / CORBIS; p. 373 Art; p. 373 Tony Craddock / Getty Images; p. 402 Courtesy of Richard Nation. Chapter 6: p. 461 Stephen Johnson / Getty Images; p. 461 Stephen Johnson / Getty Images.

Copyright © 2006 Houghton Mifflin Company. All rights reserved. No part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system without the prior written permission of Houghton Mifflin Company unless such copying is expressly permitted by federal copyright law. Address inquiries to College Permissions, Houghton Mifflin Company, 222 Berkeley Street, Boston, MA 02116-3764. Printed in the U.S.A. Library of Congress Control Number: 2003110130 ISBNs: Student's Edition: 0-618-44702-4 Instructor's Annotated Edition: 0-618-44703-2 1 2 3 4 5 6 7 8 9-VH-09 08 07 06 05

CONTENTS Preface

1

ix

FUNCTIONS 1.1 1.2 1.3 1.4 1.5 1.6 1.7

AND

GRAPHS

1

N S E A N Q O E S IN T U A D IL Equations and Inequalities 3 A Two-Dimensional Coordinate System and Graphs 16 Introduction to Functions 30 Linear Functions 51 Quadratic Functions 67 Properties of Graphs 81 The Algebra of Functions 96

EXPLORING CONCEPTS WITH TECHNOLOGY: Graphing Piecewise Functions with a Graphing Calculator 108 Chapter 1 Summary 109 Chapter 1 True/False Exercises 111 Chapter 1 Review Exercises 111 Chapter 1 Test 114

2

P O LY N O M I A L 2.1 2.2 2.3 2.4 2.5 2.6

AND

R AT I O N A L F U N C T I O N S

115

Complex Numbers 117 The Remainder Theorem and the Factor Theorem 127 Polynomial Functions of Higher Degree 138 Zeros of Polynomial Functions 154 The Fundamental Theorem of Algebra 168 Graphs of Rational Functions and Their Applications 177

EXPLORING CONCEPTS WITH TECHNOLOGY: Finding Zeros of a Polynomial Using Mathematica 193 Chapter 2 Summary 194 Chapter 2 True/False Exercises 195 Chapter 2 Review Exercises 196 Chapter 2 Test 198 Cumulative Review Exercises 199

v

vi

Contents

3

E XPONENTIAL 3.1 3.2 3.3 3.4 3.5 3.6

AND

L OGARITHMIC F UNCTIONS

Inverse Functions 203 Exponential Functions and Their Applications 215 Logarithmic Functions and Their Applications 230 Logarithms and Logarithmic Scales 243 Exponential and Logarithmic Equations 256 Exponential Growth and Decay 268

EXPLORING CONCEPTS WITH TECHNOLOGY: Using a Semilog Graph to Model Exponential Decay 281 Chapter 3 Summary 282 Chapter 3 True/False Exercises 284 Chapter 3 Review Exercises 284 Chapter 3 Test 286 Cumulative Review Exercises 287

4

T RIGONOMETRIC F UNCTIONS 4.1 4.2 4.3 4.4 4.5

289

Angles and Arcs 291 Trigonometric Functions of Real Numbers 307 Graphs of the Sine and Cosine Functions 322 Graphs of the Other Trigonometric Functions 331 Graphing Techniques 340

EXPLORING CONCEPTS WITH TECHNOLOGY: Sinusoidal Families Chapter 4 Summary 351 Chapter 4 True/False Exercises 352 Chapter 4 Review Exercises 352 Chapter 4 Test 354 Cumulative Review Exercises 354

5

A P P L I C AT I O N S O F T R I G O N O M E T RY T R I G O N O M E T R I C I D E N T I T I E S 357 5.1 5.2 5.3 5.4

Trigonometric Functions of Angles 359 Verification of Trigonometric Identities 375 More on Trigonometric Identities 388 Inverse Trigonometric Functions 402

AND

350

201

Contents

5.5 5.6 5.7

Trigonometric Equations 414 The Law of Sines and the Law of Cosines 425 Vectors 438

EXPLORING CONCEPTS WITH TECHNOLOGY: Approximate an Inverse Trigonometric Function with Polynomials 453 Chapter 5 Summary 454 Chapter 5 True/False Exercises 455 Chapter 5 Review Exercises 456 Chapter 5 Test 458 Cumulative Review Exercises 459

6

ADDITIONAL 6.1 6.2 6.3 6.4 6.5

TOPICS IN

M AT H E M AT I C S

461

Conic Sections 463 Polar Coordinates 484 Parametric Equations 499 Sequences, Series, and Summation Notation 508 The Binomial Theorem 525

EXPLORING CONCEPTS WITH TECHNOLOGY: Using a Graphing Calculator to Find the nth Roots of z 531 Chapter 6 Summary 532 Chapter 6 True/False Exercises 533 Chapter 6 Review Exercises 534 Chapter 6 Test 535 Cumulative Review Exercises 536

ALGEBRA REVIEW APPENDIX A.1 A.2 A.3

Integer and Rational Number Exponents 537 Polynomials 550 Factoring 556

SOLUTIONS TO THE TRY EXERCISES ANSWERS TO SELECTED EXERCISES INDEX

I1

S1 A1

vii

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PREFACE Essentials of Precalculus provides students with material that focuses on selected key concepts of precalculus and how those concepts can be applied to a variety of problems. To help students master these concepts, we have tried to maintain a balance among theory, application, modeling, and drill. Carefully developed mathematics is complemented by applications that are both contemporary and representative of a wide range of disciplines. Many application exercises are accompanied by a diagram that helps the student visualize the mathematics of the application. Technology is introduced naturally to support and advance better understanding of a concept. The optional Integrating Technology boxes and graphing calculator exercises are designed to promote an appreciation of both the power and the limitations of technology.

Features Interactive Presentation Essentials of Precalculus is written in a style that encourages the student to interact with the textbook. At various places throughout the text, we pose a question to the student about the material being presented. This question encourages the reader to pause and think about the current discussion and to answer the question. To ensure that the student does not miss important information, the answer to the question is provided as a footnote on the same page. Each section contains a variety of worked examples. Each example is given a title so that the student can see at a glance the type of problem being illustrated. Most examples are accompanied by annotations that assist the student in moving from step to step. Following the worked example is a suggested exercise for the student to work. The complete solution to that exercise can be found in an appendix in the text. This feature allows students to self-assess their progress and to get immediate feedback by means of not just an answer, but a complete solution. Focus on Problem Solving Each chapter begins with a Focus on Problem Solving that demonstrates various strategies that are used by successful problem solvers. At the completion of the Focus on Problem Solving, the student is directed to an exercise in the text that can be solved using the problem-solving strategy that was just discussed. Mathematics and Technology Technology is introduced in the text to illustrate or enhance a concept. We attempt to foster the idea that technology, combined with analytical thinking, can lead to deeper understanding of a concept. The optional Integrating Technology boxes and graphing calculator exercises are designed to develop an awareness of technology’s capabilities and limitations. Topics for Discussion are found at the end of each section of the text. These topics can form the basis for a group discussion or serve as writing assignments.

ix

x

Preface Extensive Exercise Sets The exercise sets of Essentials of Precalculus are carefully developed to provide the student with a variety of exercises. The exercises range from drill and practice to interesting challenges and were chosen to illustrate the many facets of the topics discussed in the text. Each exercise set emphasizes concept building, skill building and maintenance, and, as appropriate, applications. Projects are included at the end of each exercise set and are designed to encourage students to research and write about mathematics and its applications. These projects encourage critical thinking beyond the scope of the regular problem sets. Responses to the projects are given in the Instructor’s Solutions Manual. Prepare for Next Section Exercises are found at the end of each exercise set, except for the last section of a chapter. These exercises concentrate on topics from previous sections of the text that are particularly relevant to the next section of the text. By completing these exercises, the student reviews some of the concepts and skills necessary for success in the next section. Chapter Review Exercises allow the student to review concepts and skills presented in the chapter. Answers to all chapter review exercises are included in the student answer section. If a student incorrectly answers an exercise, there is a section reference next to each answer that directs the student to the section from which that exercise was taken. Using this reference, the student can review the concepts that are required to correctly solve the exercise. Chapter Tests provide students with an opportunity to self-assess their understanding of the concepts presented in the chapter. The answers to all exercises in the chapter tests are included in the student answer section. As with the answers to the chapter review exercises, there is a section reference next to each answer that directs the student to the section from which that exercise was taken. Cumulative Review Exercises end every chapter after Chapter 1. These exercises allow students to refresh their knowledge of previously studied skills and concepts and help them maintain skills that promote success in precalculus. Answers to all cumulative review exercises are included in the student answer section. As with the answers to the chapter review exercises, there is a section reference next to each answer that directs the student to the section from which that exercise was taken.

xi

Preface

C H A P T E R O P E N E R F E AT U R E S Page 1

CHAPTER OPENER

C H A P T E R

Each chapter begins with a Chapter Opener that illustrates a specific application of a concept from the chapter. There is a reference to a particular exercise within the chapter that asks the student to solve a problem related to the chapter opener topic.

1 1.1 1.2

FUNCTIONS AND GRA

PHS

1.3 1.4 1.5 1.6 1.7

VIDEO & DVD

EQUATIONS AND INEQUALITIES A TWO-DIMENSION AL COORDINATE SYSTE M AND GRAPHS INTRODUCTION TO FUNCTIONS LINEAR FUNCTIONS QUADRATIC FUNCT IONS PROPERTIES OF GRAPH S THE ALGEBRA OF FUNCTIONS

Functions as Models The Golden Gate Bridg e spans the Golden Gate Strait, which is the entra to the San Francisco Bay nce from the Pacific Ocean . Designed by Joseph the Golden Gate Bridg Strass, e is a suspension bridg e. A quadratic function, one topics of this chapter, of the can be used to model a cable of this bridge. Exercise 74 on page See 79. Strass had many skept ics who did not believ e the bridge could be Nonetheless, the bridg built. e opened on May 27, 1937, a little over 4 years after construction began. When it was completed, Strass composed the following poem.

The icons SSG at the bottom of the page let students know of additional resources available on CD, video/DVD, in the Student Study Guide, and online at math.college.hmco.com/students.

The Mighty Task Is Done At last the mighty task is done; Resplendent in the weste rn sun; The Bridge looms moun tain high. On its broad decks in rightful pride, The world in swift parad e shall ride Throughout all time to be.

Page 116

VIDEO & DVD SSG

Narrow the Search Find and Use Clues to

game you . At the beginning of the classic whodunit game find and use The game of Clue is a ered! It is your job to Boddy has been murd in which the murroom are informed that Mr. the and on, murderer, the weap ver? clues to determine the rd room with the revol it Miss Scarlet in the billia the rope? der was committed. Was the conservatory with commit the murder in nine rooms in and ons, Or did Professor Plum weap er six possible murd solutions to There are six suspects,  6  9  324 possible There are a total of 6 Mr. Boddy’s mansion. each game.

Launched midst a thous and hopes and fears, Damned by a thousand hostile sneers, Yet ne’er its course was stayed. But ask of those who met the foe, Who stood alone when faith was low, Ask them the price they paid. High overhead its lights shall gleam, Far, far below 74. GOLDEN GATE BRIDGE The suspension cables of the main life’s restless stream , Unceasingly shallspan flow. .of . . the Golden Gate Bridge are in the shape of a

parabola. If a coordinate system is drawn as shown, find the quadratic function that models a suspension cable for the main span of the bridge.

(−2100, 500)

(2100, 500)

500 400 300 200 100

−2000

(10, 6)

−1000

1000

2000

x

Page 79

funczeros of a polynomial often need to find the d than In this chapter you will can be more complicate of a polynomial function er is a possible zero. numb tion. Finding the zeros lex comp or After all, any real several thesolving a game of Clue. ion are found by using of a polynomial funct to find Quite often the zeros e theorem can be used h. In most cases no singl the results of sevining orems to narrow the searc comb by but omial function, In Expolyn . zeros given a the of find zeros to the gh clues to find often able to gather enou eral theorems, we are ems from this chapter will apply several theor ercise 52, page 165, you l function. the zeros of a polynomia

FOCUS ON PROBLEM SOLVING

A Focus on Problem Solving follows the Chapter Opener. This feature highlights and demonstrates a problem-solving strategy that may be used to successfully solve some of the problems presented in the chapter.

xii

Preface

AUFMANN INTERACTIVE METHOD (AIM) Page 208

INTERACTIVE PRESENTATION

EXAMPLE 3

Essentials of Precalculus is written in a style that encourages the student to interact with the textbook.

Find the inverse of fx  3x  8.

Find the Inverse of a Function

Solution fx  3x  8 y  3x  8

• Replace f(x) by y.

x  3y  8

• Interchange x and y.

x  8  3y

EXAMPLES

• Solve for y.

x8 y 3

Each section contains a variety of worked examples. Examples are titled so that the student can see at a glance the type of problem being illustrated, often accompanied by annotations that assist the student in moving from step to step; and offers the final answer in color so that it is readily identifiable.

8 1 x   f 1x 3 3

• Replace y by f 1(x).

The inverse function is given by f 1x 

1 8 x . 3 3

TRY EXERCISE 28, PAGE 213



TRY EXERCISES 28.

Following every example is a suggested Try Exercise from that section’s exercise set for the student to work. The exercises are color coded by number in the exercise set and the complete solution to that exercise can be found in an appendix to the text.

Thus f and g are inverses. f x  4x  8 y  4x  8 x  4y  8

• Replace f (x) by y. • Interchange x and y. • Solve for y.

x  8  4y

1 f 1x  4 x  2

27. f x  2x  4

 28. f x  4x  8 29. f x  3x  7

1 x  8  y 4 1 y x2 4

x

log bb x   x

In Exercises 27 to 44, find f 1(x). State any restrictions on the domain of f 1(x).

Page 213 –1 • Replace y by f (x).

Page S10

2

blogb  x

and

onal ents to establish the following additi We can use the properties of expon rithmic properties.

loga-

Properties of Logarithms

take note Pay close attention to these properties. Note that logbMN  logb M  logb N

In the following properties, b, M, Product property

b  1. and N are positive real numbers log bMN  log b M  log b N

Quotient property Power property

and logb

logb M M  logb N N

Also,

log b

M  log b M  log b N N

QUESTION/ANSWER

p log bM   p log b M

implies log b M  log b N M  log b N implies M  N

Logarithm-of-each-side property

MN

One-to-one property

log b

log M  N  logb M  logb N b

In fact, the expression logb M  N cannot be expanded at all.

5  ln 10  ln 50? QUES TION Is it true that ln exare often used to rewrite logarithmic The above properties of logarithms pressions in an equivalent form.

ANSW ER

Yes. By the product property, ln 5

 ln 10  ln5  10.

Page 243

In every section, we pose at least one Question to the student about the material being presented. This question encourages the reader to pause and think about the current discussion and to answer the question. To make sure that the student does not miss important information, the Answer to the question is provided as a footnote on the same page.

xiii

Preface

R E A L D ATA A N D A P P L I C AT I O N S 318

Chapter 4

APPLICATIONS

Trigonometric Functions Because  t , cos t is negative. Therefore, 2 cos t   1

One way to motivate an interest in mathematics is through applications. Applications require the student to use problem-solving strategies, along with the skills covered in a section, to solve practical problems. This careful integration of applications generates student awareness of the value of algebra as a real-life tool. Applications are taken from many disciplines including agriculture, business, chemistry, construction, Earth science, education, economics, manufacturing, nutrition, real estate, and sociology.

Thus

tan t  

sin t

1  sin2 t

•

 sin2 t.

 t  2

TRY EXERCISE 80, PAGE 320

AN APPLICATION INVO LVIN G A T RIGO NOM ETR IC FUNCTION EXAMPLE 6

Determine a Height as a Function of Time

The Millennium Whee l, in London, is the world ’s largest Ferris wheel. has a diameter of 450 It feet. When the Millennium Wheel is in uniform motion, it completes one revolution every 30 minu tes. The height h, in feet above the Thames River , of a person riding on the Millennium Whee estimated by l can be

 

ht  255  225 cos  t 15 tes since the person starte d the ride. a. How high is the perso n at the start of the ride t  0? b. How high is the perso n after 18.0 minutes? where t is the time in minu

Solution a.

 

h0  255  225 cos  0 15  255  225  30 At the start of the ride, the person is 30 feet abov e the Thames.

The Millennium Wheel , on the banks of the Thames River, London.





 b. h18.0  255  225 cos  18.0 15

255  182  437 After 18.0 minutes, the person is about 437 feet above the Thames.

TRY EXERCISE 84, PAGE 320

TOPICS

FOR D ISCU SSIO N Is Wt a number? Expla in. 2. Explain how to find the exact value of cos 13 . 6 3. Is fx  cos3 x an even function or an odd function? Explain how your decision. you made 4. Explain how to make use of a unit circle to show that sint  sin t. 1.

Page 62 62

Functions and Graphs

Chapter 1 s

(100, 4)

File size (in megabytes)

4

2

57.

(25, 1)

1

40

60

80

100 t

Time (in seconds)

below OLOG Y The table AUTOMOTIVE TECHN my values for selected shows the EPA fuel econo e: www. 2003 model year. (Sourc two-seater cars for the fueleconomy.gov.) Cars s for Selected Two-Seater EPA Fuel Economy Value Highway mpg City mpg Car 29 20 Audi, TT Roadster 21 13 BMW, Z8 16 11 r Spide 360 i, Ferrar 13 9 Lamborghini, L-174 22 15 Lotus, Esprit V8 17 11 Maserati, Spider GT the Audi, find the Lamborghini and a. Using the data for per galpredicts highway miles a linear function that slope per gallon. Round the lon in terms of city miles . redth to the nearest hund

55.

per galpredict the highway miles b. Using your model, ncy is , whose city fuel efficie lon for a Porsche Boxer st whole number. neare the to Round . 18 miles per gallon conamount of revolving CONSUMER CREDIT The tment credit cards and depar sumer credit (such as is given in the table 2003 to 1997 years store cards) for the of the org, Board of Governor’s below. (Source: www.nber. .) Federal Reserve System Year 1997 1998 1999 2000 2001 2002 2003

Page 318

tenth. mer credit in what year will consu b. Using this model, ? first exceed $850 billion

3

20

 56.

linear model 1997 and 2003, find a a. Using the data for credit nt of revolving consumer that predicts the amou to the nearest slope the d Roun t. (in billions) for year

billions of $) Consumer Credit (in 531.0 562.5 598.0 667.4

ding to the Bureau of Labor LABOR MARKET Accor hwere 38,000 desktop publis Statistics (BLS), there BLS in the year 2000. The States d ing jobs in the Unite hing jobs publis op deskt be 63,000 projects that there will in 2010. op pubfind the number of deskt a. Using the BLS data, function of the year. lishing jobs as a linear number of in what year will the b. Using your model, first exceed 60,000? desktop publishing jobs

from a kiln and of pottery is removed temper58. POTTERY A piece lled environment. The allowed to cool in a contro y after it is repotter the of nheit) ature (in degrees Fahre minutes) is for various times (in moved from the kiln . shown in the table below Time, min

Temperature, F

15

2200

20

2150

30

2050

60

1750

l for the temperature a. Find a linear mode after t minutes.

of the pottery

in of the slope of this line Explain the meaning em. the context of the probl at the continues to decrease c. Assuming temperature pottery the temperature of the same rate, what will be

REAL DATA

b.

in 3 hours?

of board-feet For a log, the number 59. LUMBER INDUSTRY ds on the died from the log depen (bf) that can be obtain its length. The table below and log the of s, inche in ameter, er that can be oblumb of -feet board of shows the number is 32 feet long. tained from a log that bf Diameter, inches 180 16 240 18

701.3

20

300

712.0

22

360

725.0

 

Real data examples and exercises, identified by , ask students to analyze and create mathematical models from actual situations. Students are often required to work with tables, graphs, and charts drawn from a variety of disciplines.

xiv

Preface

TECHNOLOGY 410

INTEGRATING TECHNOLOGY

The Integrating Technology feature contains optional discussions that can be used to further explore a concept using technology. Some introduce technology as an alternative way to solve certain problems and others provide suggestions for using a calculator to solve certain problems and applications. Additionally, optional graphing calculator examples and exercises (identified by ) are presented throughout the text.

Chapter 5

Applications of Trigonome try and Trigonometric Ident ities  cos 1cos cos   sin sin   cos 1 1  x 2 x

 cos 1 0   2

y

−2

x

2 −

• Addition identity for cosine

TRY EXERCISE 64, PAGE 412

π 2

y = sin−1 x

 x 1  x 2 

π 2

y = sin−1 (x − 2)

FIGURE 5.36

GRAPHS OF INVERSE TRIGONOMETRIC FUNC TION S The inverse trigonome tric functions can be graphed by using the stretching, shrinking, procedures of and translation that were discussed earlier in the instance, the graph of text. For y  sin1x  2 is a horiz ontal shift 2 units to the the graph of y  sin1 right of x, as shown in Figure 5.36. EXAMPLE 7

Graph: y  cos1 x 

y

Graph an Inverse Func tion 1

Solution π y = cos−1 x

y=

cos−1

x+1

π 2

−1

Recall that the graph of y  fx  c is a vertic al translation of the graph Because c  1, a posit of f. ive number, the graph of y  cos1 x  1 is the of y  cos1 x shifted graph 1 unit up. See Figure 5.37. TRY EXERCISE 68, PAGE 412

1

x

FIGURE 5.37

INTEGRATING TECHNOLOGY 2π

−3

3

−2π

y  3 sin1 0.5x

FIGURE 5.38

Page 281

281

Technology Exploring Concepts with

EXPLORING CONCEPTS WITH TECHNOLOGY TABLE 3.13 90

700

100

500

110

350

120

250

130

190

140

150

Page 410 ial Decay h to Model Exponent Using a Semilog Grap V of SAE 40 h shows the viscosity

120

V

V 700 600 500

Viscosity

1n 700

Natural logarithm of viscosit

y

150

400 300 200 100 90

100

110

120

130

140

Temperature, in °F

150

T

1n 500 1n 300

1n 100 90

100

110

120

130

140

150

T

Temperature, in °F

FIGURE 3.47

FIGURE 3.46

TABLE 3.14 A

t

 The range of y  sin1 x is   y   . Thus the range of 2 2 y  3 sin1 0.5x is  3 3 y . This is also consistent 2 with the graph. 2 Verifying some of the properties of y  sin1 x serves as a check that have correctly entered you the equation for the graph .

Table 3.13, whic below, Consider the data in of these data is shown eratures T. The graph in Figure 3.46 apmotor oil at various temp gh the points. The graph throu s passe that curve along with a model. of an exponential decay of an expears to have the shape Figure 3.46 is the graph r, whether the graph in One way to determine paper. On this graph pape data on semilog graph the plot c scale. to is ithmi ion ponential funct al axis uses a logar ins the same, but the vertic this time the verbut 3.47, e the horizontal axis rema Figur in are graphed again ht line. The data in Table 3.13 is approximately a straig logarithm axis. This graph tical axis is a natural

V

T

When you use a graph ing utility to draw the graph of an inverse trigonometric function, use the properties of these functions to verify the correctness of your graph . For instance, the graph of y  3 sin1 0.5x is shown in Figure 5.38. The domain of y  sin1 x is 1  x  1. There the domain of y  3 sin1 fore, 0.5x is 1  0.5x  1 or, multiplying the inequality by 2, 2  x  2. This is consistent with the graph in Figure 5.38.

1

91.77

4

70.92

8

50.30

15

27.57

20

17.95

30

7.60

st ten-thousandth, is Figure 3.47, to the neare The slope of the line in ln 500  ln 120 0.0285 m 100  150 have V replaced by ln V, we point-slope formula with Using this slope and the 85T  150 ln V  ln 120  0.02 (1) ln V 0.0285T  9.062 og coordinate grid. tion of the line on a semil equa the is (1) tion Equa for V. Now solve Equation (1) 0.0285T9.062 e ln V  e 0.0285T e 9.062 Ve (2) 0.0285T V 8621e m shown in ngular coordinate syste recta the in data the l of Equation (2) is a mode Figure 3.46.

1.

tics of iodine-131. A 100mine the decay characteris s A chemist wishes to deter y period. Table 3.14 show observed over a 30-da is e-131 iodin of le mg samp ining after t days. rams) of iodine-131 rema the amount A (in millig r pape log Semi paper. (Note: pairs t, A on semilog paper a. Graph the ordered are based on semilog ties. Our calculations comes in different varie axis.) al ithm scale on the vertic that has a natural logar

EXPLORING CONCEPTS WITH TECHNOLOGY

A special end-of-chapter feature, Exploring Concepts with Technology, extends ideas introduced in the text by using technology (graphing calculator, CAS, etc.) to investigate extended applications or mathematical topics. These explorations can serve as group projects, class discussions, or extra-credit assignments.

xv

Preface

STUDENT PEDAGOGY TOPIC LIST

At the beginning of each section is a list of the major topics covered in the section.

Page 463 6.1 SECTION

KEY TERMS AND CONCEPTS

6.1

CONIC SECTIONS

PARABOLAS ELLIPSES HYPERBOLAS APPLICATIONS

Key terms, in bold, emphasize important terms. Key concepts are presented in blue boxes in order to highlight these important concepts and to provide for easy reference.

Axis

These margin notes contain interesting sidelights about mathematics, its history, or its application.

463

The graph of a para bola, a circle, an ellipse, or a hyperbo intersection of a plane and a cone la can be formed . Hence these figu by the tions. See Figure res are referred to 6.1. as conic sec-

M AT H M AT TERS Appollonius (262 –200 B.C.) wrote an eight-volume treat ise entitled On Conic Sections in whic h he derived the formulas for all the conic sections. He was the first to use the words parabola, ellips e, and hyperbola.

MATH MATTERS

Conic Sections

Axis

Axis

E Hyperbola

Ellipse Circle C

take note If the intersection of a plane and a cone is a point, a line, or two intersecting lines, then the intersection is calle d a degenerate conic section.

TAKE NOTE

These margin notes alert students to a point requiring special attention or are used to amplify the concept under discussion.

Parabola

FIGURE 6.1

Cones intersecte d by

planes

A plane perpend icular to the axis (plane C). The plan of the cone intersec e E, tilted so that ts the cone in a it is not perpend circle the cone in an ellip icular to the axis se. When the plan , intersects e is parallel to a cone, the plane inte line on the surface rsects the cone in of the a parabola. Whe portions of the cone n the plane intersec , a hyperbola is form ts both ed.

PARABOLAS Axis of symmetry

REVIEW NOTES

Besides the geom etric description of a conic section be defined as a set just given, a coni of points. This met c section can the curve to dete hod uses some spec rmine which poin ified conditions ts in a coordina about graph. For exam te system are poin ple, a parabola can ts of the be defined by the following set of poin ts.

y

Def inition of a Par abola

Focus

A directs the student to the place in the text where the student can review a concept that was previously discussed.

x Directrix Vertex

FIGURE 6.2

To review AXIS OF SYMMETRY, see p.

A parabola is the set of points in the plane that are equ line (the directrix idistant from a fixe ) and a fixed poin d t (the focus) not on the directrix. The line that pass es through the focu called the axis of s and is perpend symmetry of the icular to the dire parabola. The mid ctrix is tween the focus point of the line and directrix on segment bethe axis of sym parabola, as show metry is the vert n in Figure 6.2. ex of the

67.

and a element of the domain determined.

Page 55 EXAMPLE 4

h f(x)  b Domain of f for whic Find the Value in the

Find the value x in the

4 for which fx  5. domain of fx  3x 

Algebraic Solution fx  3x  4 5  3x  4 9  3x 3x

solve for x. • Replace f(x) by 5 and

Visualize the Solution By graphing y  5 and see that fx  3x  4, we can fx  5 when x  3. y

domain of f is paired This means that 3 in the ed When x  3, fx  5. g this is that the order f. Another way of statin with 5 in the range of of f. pair 3, 5 is an element

y=5

8 4

−8

−4

(3, 5) 4

−4

8

x

f(x) = 3x − 4

−8

61 TRY EXERCISE 42, PAGE

n, the r functions in this sectio ip ly concerned with linea a powerful relationsh Although we are main functions. It illustrates aph of y  fx th rem applies to all f th i

VISUALIZE THE SOLUTION

For appropriate examples within the text, we have provided both an algebraic solution and a graphical representation of the solution. This approach creates a link between the algebraic and visual components of a solution.

Preface

EXERCISES

ION TOPICS FOR DISCUSS

2.

TOPICS FOR DISCUSSION

3. 4.

These special exercises provide questions related to key concepts in the section. Instructors can use these to initiate class discussions or to ask students to write about concepts presented.

5.

152 t is a complex number? Wha Chapnum ter 2ber?Polyn omial and Rational Funct What is an imaginary ions numbers? related to the complex approreal ximatnum es thebers U.S. marriage rate for How are the the years 1900 t  0 to 1999 t  99. ber? L Is zero a complex num U.S. Marriage Rate, 20 999ber? num a complex1900–1 of e ugat conj 16 15 What is the has real ys 2 alwa 0  12  bx  2 w that the equation x 12 b. 8 Explain how you kno value of the real number the of ss rdle rega 4 s number solution 9 Milligrams of pseudo ephedr hydrochloride in the bloodst ine ream

1.

6

00

The exercise sets in Essentials of Precalculus were carefully developed to provide a wide variety of exercises. The exercises range from drill and practice to interesting challenges. They were chosen to illustrate the many facets of topics discussed in the text. Each exercise set emphasizes skill building, skill maintenance, and, as appropriate, applications. Icons identify appropriate writing , group , data analysis ,and graphing calculator

exercises.

54.

a. Use a graph of P to determine the absolute minimum gazelle population (roun ded to the nearest single gazelle) that is attained during this time period .

TS CON NEC TIN G CON CEP verify the identity. In Exercises 91 to 95, cos x 1  sin x  cos x  91. x1 1  sin x  cos x sin 92.

 sec x 1  tan x  sec x  1 tan x 1  tan x  sec x

x sin y cos z  cos x cos y cos z  sin 93. cosx  y  z  x sin y sin z sin x cos y sin z  cos cos h  1 sin h  sin x sinx  h  sin x  cos x h h 94. h sin h 1  h cos x  sin x h cosx  h  cos  cos x h 95. h on a fish when The drag (resistance) 96. MODEL RESISTANCE when it is to three times the drag it is swimming is two in a sawfor this, some fish swim gliding. To compensate figure. The n in the accompanying tooth pattern, as show

N 5.3 PRE PAR E FOR SEC TIO 2 . [5.2] sin   to rewrite sin 97. Use the identity for  to rewrite cos 2 . [5.2]  

cos for ty 98. Use the identi tan 2 . [5.2] tan   to rewrite 99. Use the identity for

PRO JEC TS r’s se that you are a teache GRADING A QUIZ Suppo a assist the teacher of assistant. You are to n quiz. Each grading a four-questio strigonometry class by find a trigonometric expre to nt stude the asks question r has preapplication. The teache sion that models a given the next answers are shown in pared an answer key. These ssions as answers the expre column. A student gives probn. Determine for which colum right far the in shown response. t correc a given has lems the student

57. by

BEAM DEFLECTION The deflection D, in feet, of an 8-foot beam that is center loaded is given

Dx  0.00254x 3  3  8x 2, 0 x  4 where x is the distance, in feet, from one end of the beam.

D x 8 ft

a. Determine the deflec tion of the beam when x  3 feet. Round to the nearest hund redth of an inch. b. At what point does the beam achieve its maximum deflection? What is the maximum deflection? Round to the nearest hundredth of an inch. c. What is the deflection

at x  5 feet?

387

ds when of energy the fish expen ratio of the amount g down at angle  and then glidin swimming upward at ntally is expends swimming horizo it y energ the to

angle given by  k sin  sin ER  k sin   depends on that 2  k  3, and k such value a is k where drag exabout the amount of make we the assumptions 2,  10 , and  k for E R Find perienced by the fish.   20 .

α β

ty. Hint: Find a  2 sin is not an identi does not 101. Verify that sin 2

left side of the equation value of for which the equal the right side. [5.1] 1

ty. [5.1]  cos is not an identi 102. Verify that cos 2 2

sin

for  60 ,  90 , and and 100. Compare tan 2 1  cos

 120 . [5.1]

1.

b. The absolute maxim um of P is attained at the endpoint, where t  12. What is this absolute maxim um (rounded to the neare st single squirrel)?

55.

Page 387 tric Identities Verification of Trigonome

SQUIRREL POPULATION The population P of squirrels in a wilderness area is given by Pt  0.6t 4  13.7t 3  104.5t 2  243.8t  360, where 0  t  12 years. a. What is the absolu te minimum number of squirrels (rounded to the neare st single squirrel) attain ed on the interval 0  t  12?

b. Use a graph of P to determine the absolute maximum gazelle population (roun ded to the nearest single gazelle) that is attained during this time period . MEDICATION LEVEL Pseud oephedrine hydrochloride is an allergy medic ation. The function Lt  0.03t 4  0.4t 3  7.3t 2  23.1t where 0  t  5, mode ls the level of pseud oephedrine hydrochloride, in millig rams, in the bloodstream patient t hours after of a 30 milligrams of the medication have been taken. a. Use a graphing utility and the function Lt to determine the maximum level of pseudoephe drine hydrochloride in the patien t’s bloodstream. Roun d your result to the nearest 0.01 milligram.

5.2

56.

b. the relative minim um marriage rate, round ed to the nearest 0.1, during the period from 1950 to 1970. GAZELLE POPULATION A herd of 204 African gazelles is introduced into a wild animal park. The population of the gazelles, Pt, after t years is given by Pt  0.7t 3  18.7t 2  69.5t  204, where 0 t  18.

t

b. At what time t, to the nearest minute, is this maximum level of pseudoephedrine hydrochloride reached?

20 40 60 80 99 Year (00 represents 1900)

Use Mt and a graph ing utility to estimate a. during what year the U.S. marriage rate reached its maximum for the period from 1900 to 1999.

L(t) = 0.03t4 + 0.4t3 − 7.3t2 + 23.1t

1 2 3 4 5 Time (in hours)

3 0

EXERCISES

, web

Page 124

equatio discriminant is zero, the the left. . See the Take Note at solution is a double root

Numb er of marria ges per thousand population

xvi

Page 152

Included in each exercise set are Connecting Concepts exercises. These exercises extend some of the concepts discussed in the section and require students to connect ideas studied earlier with new concepts. EXERCISES TO PREPARE FOR THE NEXT SECTION

Every section’s exercise set (except for the last section of a chapter) contains exercises that allow students to practice the previously-learned skills they will need to be successful in the next section. Next to each question, in brackets, is a reference to the section of the text that contains the concepts related to the question for students to easily review. All answers are provided in the Answer Appendix. PROJECTS

Answer Key 1. csc x sec x 2 2. cos x 3. cos x cot x 4. csc x cot x

Student’s Response 1. cot x  tan x 2. 1  sin x1  sin x 3. csc x  sec x 3 4. sin xcot x  cot x

Projects are provided at the end of each exercise set. They are designed to encourage students to do research and write about what they have learned. These Projects generally emphasize critical thinking skills and can be used as collaborative learning exercises or as extra-credit assignments.

xvii

Preface

END OF CHAPTER Page 454 Chapter 5

454

CHAPTER SUMMARY

At the end of each chapter there is a Chapter Summary that provides a concise section-by-section review of the chapter topics.

CHAPTER

5

metric Identities Applications of Trigonometry and Trigono

S U M M A RY • The cofunction identities are

Angles 5.1 Trigonometric Functions of

adj hyp

sec  

hyp adj

opp tan   adj

cot  

adj opp

cos  

TRUE/FALSE EXERCISES

Following each chapter summary are true/false exercises. These exercises are intended to help students understand concepts and can be used to initiate class discussions.

x r

sec  

r , x0 x

y tan   , x  0 x

cot  

x , y0 y

t i Id

tities

CHAPTER

5

CHAPTER

12.

5

 

1 13. tan 67 2

functions of an the six trigonometric 1. Find the values of 3 on the on with the point P1, angle in standard positi . terminal side of the angle of 2. Find the exact value

3.

c. cot225 

d. cos

2 3

nearest of the following to the Find the value of each ten-thousandth. b. cot 4.22 a. cos 123 c. sec 612

d. tan

5

2 5

TEST

    sin 2 . cos 3 ct value of tan 6 1. Find the exa top of a tree is point A to the a line through of elevation from from A and on 2. The angle B, 5.24 meters ation is 37.4 . nt elev poi of At . le 42.2 and A, the ang tree the of e the bas ht of the tree. Find the heig 2 2 x. 2 x sec x  sec identity 1  sin 13. Verify the identity 14. Verify the 1  2 tan x 1  sec x  tan x sec x  tan x 1  cos x .  cot x  sin x identity csc x 15. Verify the

sin x x  tan cos y y

3 cos   2 ,

2 , in Quadrant I, and 18. Given sin  2 .

2 sin  in Quadrant IV, find expression as a single 19 to 22 write the given

i

and cos 3 Quadrant III,

  5 , in 17. Given sin  . II, find sin 

nt dra Qua in  identity 8. Verify the

tan

2  2 ,



1

sin cos exact value of 11. Find the

x f x   3 . x2

4. Determine whether f x  x  sin x is an even function or an odd function. 5. Find the inverse of f x



5x . x1

6. Writex  25 in logari thmic form. 7. Evaluate:log 1000





1  cos

14. sin 2  2 sin for all

15. sin    sin

 sin  16. If tan  tan , then

 . 17. cos 1cos x  x

18. If 1  x  1, then coscos 1 x  x. 9. The Law of Cosin es can be used to solve any triangle given two sides and an angle.

Page 455



12 . 13

Page 458

12. Evaluate:sin1 1 2 13. Use interval notati on to state the domain of f x  cos 1 x. 14. Use interval notati on to state the range of f x  tan1 x.

  

15. Evaluate tan sin1 12 13

CHAPTER REVIEW EXERCISES

Review exercises are found at the end of each chapter. These exercises are selected to help the student integrate all of the topics presented in the chapter.

The Chapter Test exercises are designed to simulate a possible test of the material in the chapter.

C U M U L AT I V E R E V I E W EXERCISES

3. Find the vertical asym ptote for the graph of



CHAPTER TEST

cos   csc .   sin  2

15 cos 75 . ct value of sin 9. Find the exa 1

3 sin x  cos x in the form  2 10. Write y  2 radians. measured in

, where is y  k sinx 

2. Explain how to use the graph of y  f x to produce the graph of y  f x.

• The half-angle identities are

14. sin 112.5

sin 195 . exact value of 16. Find the

1. Explain how to use the graph of y  f x to produce the graph of y  f x  1  2.

2 tan

2 1  tan

3 1 cos    2 , , in Quadrant IV, and 17. Given sin   2 .

2 tan  in Quadrant III, find

of 4.5 for a dis; v   1, 3 4. A car climbs a hill that has a constant angle altitude? 77. u  3, 7 is the car’s increase in tance of 1.14 miles. What 5; u  2, 1 78. v  8, the angle of eleh d w of 8 55 feet when

CHAPTER

2  2 cos  1

s of the given find the exact value In Exercises 15 to 18, functions. 1 1 cos   2 ,  in

in Quadrant I, and 15. Given sin  2 , .  

cos find IV, Quadrant 1 cos    2 ,

3 , in Quadrant II, and 16. Given sin  2 .  

sin find  in Quadrant III,

  

a. sec 150

3 b. tan  4

sin 2  2 sin cos

2 2 cos 2  cos  sin

2  1  2 sin

TRUE/FALSE EXERC ISES

13. sin1 x  csc x1

S REVIEW EXERCISE

• The double-angle identities are

tions are

11. The angle  measu red in degrees is in stand ard position with the terminal side in the second quadr ant. The reference angle of is 180°  .

equal.

measure, replace 90° where  is in degrees. If  is in radian  . with 2

tan 2 

In Exercises 1 to 12, answer true or false. If the statement is false, give a reason or an example to show that the statement is false.

Page 456

csc90    sec 

sec90    csc 

es 5.3 More on Trigonometric Identiti

origin, on the terminal side • Let Px, y be a point, except the . The six trigonometric of an angle  in standard position functions of  are r y csc   , y  0 sin   y r cos  

cos90    sin  cot90    tan 

sin90    cos  tan90    cot 

triangle. The six trigono• Let  be an acute angle of a right metric functions of  are given by hyp opp csc   sin   opp hyp

.

16. Solve: 2 cos 2 x  sin x1

0, for 0  x 2 17. Find the magnitude and direction angle for the vector 3, 4. Round the angle to the nearest tenth of a degree. 18. Find the smallest positive angle betwe en the vectors v  2 3 d 

Page 459

CUMULATIVE REVIEW EXERCISES

Cumulative Review Exercises, which appear at the end of each chapter (except Chapter 1), help students maintain skills learned in previous chapters. The answers to all Chapter Review Exercises, all Chapter Test Exercises, and all Cumulative Review Exercises are given in the Answer Section. Along with the answer, there is a reference to the section that pertains to each exercise.

xviii

Preface

I N S T R U C T O R ’ S A N N O TAT E D E D I T I O N Page 206

The Instructor’s Annotated Edition includes:

206

Chapter 3

Exponential and Logarithmi c Functions

COMPOSITION OF A F UNC TION AND I TS INVERSE Observe the effect, as show n below, of taking the composition of functions inverses of one another. that are

MARGIN NOTES

There is an Alternative Example for every numbered Example that can be used as another in-class example or as an in-class exercise for students to try.

fx  2x



f gx  2 1 x 2 f gx  x

gx  • Replace x by g(x).

1 x 2

g fx  1 2x 2

• Replace x by f(x).

g fx  x

This property of the comp osition of inverse funct ions always holds true. taking the compositio n of inverse functions, When the inverse function effect of the original funct reverses the ion. For the two funct ions above, f doubles and g halves a numb a number, er. If you double a numb er and then take one-h result, you are back to alf of the the original number.

Instructor Notes give suggestions for teaching concepts, warnings about common student errors, or historical notes. take note

Composition of Inver se Functions Property

If we think of a function as a machine, then the Compo sition of Inverse Functions Proper ty can be represented as shown below. Take any input x for f. Use the output of f as the input for f 1. The result is the original input, x.

DIGITAL ART AND TABLES

Next to many of the graphs and tables in the text, there is a that indicates that a Microsoft PowerPoint® slide of that figure is available. These slides (along with PowerPoint Viewer) are available on the ClassPrep CD and also can be downloaded from our website at math.college.hmco.com/instructors. These slides also can be printed as transparency masters.

If f is a one-to-one funct ion, then f 1 is the inver se function of f if and only if  f  f 1x  f  f 1x x for all x in the domain of f 1 and  f 1  f x  f 1 fx

x

for all x in the domain

of f.

x

f(x)

Use the Composition of Inverse Functions Property

EXAMPLE 2

f function x f −1 function

Use composition of funct ions to show that f 1x  3x  6 is the inverse function of fx  1 x  2. 3 Solution We must show that f  1 f x  x and f 1 fx  x. fx 

1 x2 3

f  f x  1 3x  6 2 3 1

f  f 1x  x

180

Chapter 2

T

f 1x  3x  6





f 1 fx  3 1 x  2 6 3 f 1 fx  x

E

Polynomial and Rational Functions Vertical asymptotes of the graph of a rational function can be found by using the following theorem.

Theorem on Vertical Asymptotes

Page 371 5.1

371

of Angles Trigonometric Functions

If the real number a is a zero of the denominator Qx, then the graph of Fx  PxQx, where Px and Qx have no common factors, has the vertical asymptote x  a.

Page 180

EXERCISE SET 5.1

A 12-foot ladder FROM SLANT HEIGHT 52 with 19. VERTICAL HEIGHT and makes an angle of is resting against a wall reach t to which the ladder will heigh the Find d. the groun on the wall.

six trigonometf ind the values of the In Exercises 1 to 10, the given sides. the right triangle with ric functions of for 2. 1.

distance AB across A MARSH Find the 20. DISTANCE ACROSS accompanying figure. the marsh shown in the

7 12 θ

θ

A

3 5

4.

3. 7

9

3

θ

4

52°

θ

C

B

31 m

 6.

5.

8 5

5

θ

θ

2

8.

7. 3

n of a piece LIGHT For best illuminatio mends 21. PLACEMENT OF A list for an art gallery recom of art, a lighting specia piece of art light be 6 feet from the that a ceiling-mounted How ssion of the light be 38 . depre of angle the and that that the recthe light be placed so far from a wall should e that the Notic met? are list specia ommendations of the inches from the wall. art extends outward 4

10 5

θ

θ

2

4 in.

6

θ

θ

3

In Exercises 11 to 18, 11. sin 45  cos 45 13

6 ft

10.

9.

0.8

1

each expression. find the exact value of 12. csc 45  sec 45

in 30 cos 60  tan 45

of elevation OWER The angle OF THE EIFFEL T Eiffel Tower s from the base of the from a point 116 meter the approximate Find 68.9 . is tower to the top of the

 22. HEIGHT

A Suggested Assignment is provided for each section.

38°

Answers for all exercises are provided.

Preface

xix

Instructor Resources Essentials of Precalculus has a complete set of support materials for the instructor. Instructor’s Annotated Edition This edition contains a replica of the student text with additional resources for the instructor. These include: Instructor Notes, Alternative Example notes, PowerPoint icons, Suggested Assignments, and answers to all exercises. Instructor’s Solutions Manual The Instructor’s Solutions Manual contains worked-out solutions for all exercises in the text. Instructor’s Resource Manual with Testing This resource includes six ready-touse printed Chapter Tests per chapter, and a Printed Test Bank providing a printout of one example of each of the algorithmic items on the HM Testing CD-ROM program. HM ClassPrep w/ HM Testing CD-ROM HM ClassPrep contains a multitude of text-specific resources for instructors to use to enhance the classroom experience. These resources can be easily accessed by chapter or resource type and also can link you to the text’s website. HM Testing is our computerized test generator and contains a database of algorithmic test items, as well as providing online testing and gradebook functions. Instructor Text-specific Website The resources available on the ClassPrep CD are also available on the instructor website at math.college.hmco.com/instructors. Appropriate items are password protected. Instructors also have access to the student part of the text’s website.

Student Resources Student Study Guide The Student Study Guide contains complete solutions to all odd-numbered exercises in the text, as well as study tips and a practice test for each chapter. Math Study Skills Workbook by Paul D. Nolting This workbook is designed to reinforce skills and minimize frustration for students in any math class, lab, or study skills course. It offers a wealth of study tips and sound advice on note taking, time management, and reducing math anxiety. In addition, numerous opportunities for self assessment enable students to track their own progress. HM Eduspace® Online Learning Environment Eduspace is a text-specific, webbased learning environment that combines an algorithmic tutorial program with homework capabilities. Specific content is available 24 hours a day to help you further understand your textbook. HM mathSpace® Tutorial CD-ROM This tutorial CD-ROM allows students to practice skills and review concepts as many times as necessary by providing algorithmically-generated exercises and step-by-step solutions for practice. SMARTHINKING™ Live, Online Tutoring Houghton Mifflin has partnered with SMARTHINKING to provide an easy-to-use and effective online tutorial service. Whiteboard Simulations and Practice Area promote real-time visual interaction.

xx

Preface Three levels of service are offered. Text-specific Tutoring provides real-time, one-on-one instruction with a specially qualified ‘e-structor.’ Questions Any Time allows students to submit questions to the tutor outside the scheduled hours and receive a reply within 24 hours. Independent Study Resources connect students with around-the-clock access to additional educational services, including interactive websites, diagnostic tests, and Frequently Asked Questions posed to SMARTHINKING e-structors. Houghton Mifflin Instructional Videos and DVDs Text-specific videos and DVDs, hosted by Dana Mosely, cover all sections of the text and provide a valuable resource for further instruction and review. Student Text-specific Website Online student resources can be found at this text’s website at math.college.hmco.com/students.

Acknowledgments The authors would like to thank the people who have provided many valuable suggestions during the development of this text. Ioannis K. Argyros, Cameron University, OK Timothy D. Beaver, Isothermal Community College, NC Norma Bisulca, The University of Maine–Augusta, ME C. Allen Brown, Wabash Valley College, IL Larry Buess, Trevecca Nazarene University, TN Dr. Warren J. Burch, Brevard Community College, FL Alice Burstein, Middlesex Community College, CT Sharon Rose Butler, Pikes Peak Community College, CO Michael Button, The Master’s College, CA Harold Carda, South Dakota School of Mines and Technology, SD Charles Cheney, Spring Hill College, AL Oiyin Pauline Chow, Harrisburg Area Community College, PA Dr. Greg Clements, Midland Lutheran College, NE Jacqueline Coomes, Eastern Washington University, WA Anna Cox, Kellogg Community College, MI Ellen Cunningham, Saint Mary-of-the-Woods College, IN Rohan Dalpatadu, University of Nevada, NV Jerry Davis, Johnson State College, VT Lucy S. DeComo, Walsh University, OH Blair T. Dietrich, Georgia College and State University, GA Sharon Dunn, Southside Virginia Community College, VA Heather Van Dyke, Manhattan Christian College, NY Noelle Eckley, Lassen Community College, CA David Ellis, San Francisco State University, CA Theodore S. Erickson, Wheeling Jesuit University, WV Dr. Hamidullah Farhat, Hampton University, VA Cathy Ferrer, Valencia Community College, FL Dr. William P. Fox, Francis Marion University, SC Tarsh Freeman, Bevill State Community College–Brewer Campus, AL

Preface Allen G. Fuller, Gordon College, GA John D. Gieringer, Alvernia College, PA Earl Gladue, Roger Williams University, RI John T. Gordon, Southern Polytechnic State University, GA Cornell Grant, DeKalb Technical College, GA Allen Hamlin, Palm Beach Community College, FL Ronald E. Harrell, Allegheny College, PA Gregory P. Henderson, Hillsborough Community College–Plant City Campus, FL Dr. Shahryar Heydari, Piedmont College, GA Kaat Higham, Bergen Community College, NJ Dr. Philip Holladay, Geneva College, PA Matthew Hudock, St. Philip’s College, TX John Jacobs, MassBay Community College, MA Dr. Jay M. Jahangiri, Kent State University –Geauga Campus, OH David W. Jessee, Jr., Manatee Community College, FL John M. Johnson, George Fox University, OR Glen A. Just, The Franciscan University, IA Regina Keller, Suffolk County Community College, NY Dr. Gary Kimball, Southeastern College, FL Ellen Knapp, Philadelphia University, PA Dr. F. Kostanyan, DeVry University, NY Carlos de la Lama, San Diego City College, CA Mike Lavinder, Wenatchee Valley College, WA Dr. Shinemin Lin, Savannah State University, GA Domingo Litong, Houston Community College, TX Nicholas Loudin, Alderson-Broaddus College, WV Rich Maresh, Viterbo University, WI Dr. Chris Masters, Doane College, NE Frank Mattero, Quinsigamond Community College, MA Richard McCall, St. Louis College of Pharmacy, MO Robbie McKelvy, Cossatot Community College, AR Dr. Helen Medley Robert Messer, Albion College, MI Dr. Beverly K. Michael, University of Pittsburgh, PA Dr. Cheryl Chute Miller, SUNY Potsdam, NY Deborah Mirdamadi, Pennsylvania State University– Mont Alto, PA Gregory A. Mitchell, Penn Valley Community College, MO Charles D. Mooney, Reinhardt College, GA Dr. John C. Nardo, Oglethorpe University, GA Al Niemier, Holy Cross College, IN Steve Nimmo, Morningside College, IA Neal Ninteman, George Fox University, OR Earl Packard, Kutztown University of Pennsylvania, PA Ron Palcic, Johnson County Community College, KS Leslie A. Palmer, Mercyhurst College, PA Vadim Ponomarenko, Trinity University, TX Stephen J. Ramirez, Crafton Hills College, CA Jane Roads, Moberly Area Community College, MO Pascal Roubides, Georgia Institute of Technology, GA Carol Roush, Community College of Baltimore County–Catonsville Campus, MD

xxi

xxii

Preface Fred Safier, City College of San Francisco, CA Dr. Davis A. Santos, Community College of Philadelphia, PA Kurt Scholz, University of St. Thomas, MN Dr. Bill Schwendner, University of Connecticut–Stamford, CT Lauri Semarne Melody Shipley, North Central Missouri College, MO Caroline Shook, Bellevue Community College, WA Jelinda Spotorno, Clayton College and State University, GA Daryl Stephens, East Tennessee State University, TN David Tanenbaum, Harford Community College, MD Dr. Stephen J. Tillman, Wilkes University, PA William K. Tomhave, Concordia College, MN Dr. Rajah P. Varatharajah, North Carolina A&T State University, NC Dr. Paul Vaz, Arizona State University, AZ Dr. August Waltmann, Wartburg College, IA Connie Wappes, Fond du Lac Tribal and Community College, MN Dr. Peter R. Weidner, Edinboro University of Pennsylvania, PA LeAnn Werner, South Dakota State University, SD Denise Widup, University of Wisconsin –Parkside, WI Robert R. Young, Brevard Community College, FL

C H A P T E R

1 1.1 1.2

FUNCTIONS AND GRAPHS

1.3 1.4 1.5 1.6 1.7

EQUATIONS AND INEQUALITIES A TWO-DIMENSIONAL COORDINATE SYSTEM AND GRAPHS INTRODUCTION TO FUNCTIONS LINEAR FUNCTIONS QUADRATIC FUNCTIONS PROPERTIES OF GRAPHS THE ALGEBRA OF FUNCTIONS

Functions as Models The Golden Gate Bridge spans the Golden Gate Strait, which is the entrance to the San Francisco Bay from the Pacific Ocean. Designed by Joseph Strass, the Golden Gate Bridge is a suspension bridge. A quadratic function, one of the topics of this chapter, can be used to model a cable of this bridge. See Exercise 74 on page 79. Strass had many skeptics who did not believe the bridge could be built. Nonetheless, the bridge opened on May 27, 1937, a little over 4 years after construction began. When it was completed, Strass composed the following poem. The Mighty Task Is Done At last the mighty task is done; Resplendent in the western sun; The Bridge looms mountain high. On its broad decks in rightful pride, The world in swift parade shall ride Throughout all time to be. Launched midst a thousand hopes and fears, Damned by a thousand hostile sneers, Yet ne’er its course was stayed. But ask of those who met the foe, Who stood alone when faith was low, Ask them the price they paid. High overhead its lights shall gleam, Far, far below life’s restless stream, Unceasingly shall flow. . . .

VIDEO & DVD SSG

Difference Tables When devising a plan to solve a problem, it may be helpful to organize information in a table. One particular type of table that can be used to discern some patterns is called a difference table. For instance, suppose that we want to determine the number of square tiles in the tenth figure of a pattern whose first four figures are

We begin by creating a difference table by listing the number of tiles in each figure and the differences between the numbers of tiles on the next line. Tiles

2

5

Differences

8

3

11

3

3

From the difference table, note that each succeeding figure has three more tiles. Therefore, we can find the number of tiles in the tenth figure by extending the difference table. Tiles Differences

2

5 3

8 3

11 3

14 3

17 3

20 3

23 3

26 3

29 3

There are 29 tiles in the tenth figure. Sometimes the first differences are not constant as they were in the preceding example. In this case we find the second differences. For instance, consider the pattern at the right. The difference table is shown below. Tiles

1

5

First differences

11

4

19

6

Second differences

8

2

2

In this case the first differences are not constant, but the second differences are constant. With this information we can determine the number of tiles in succeeding figures. Tiles First differences

1

5 4

Second differences 2

11 6

19 8

2

29 10

2

41 12

2

55 14

2

71 16

2

89 18

2

109 20

2

In this case there are 109 tiles in the tenth figure. If second differences are not constant, try third differences.1 If third differences are not constant, try fourth differences, and so on. 1 Not all lists of numbers will end with a difference row of constants. For instance, consider 1, 1, 2, 3, 5, 8, 13, 21, 34, . . . .

1.1

1.1

SECTION

M AT H M AT T E R S Archimedes (c. 287–212 B.C.) was the first to calculate  with any degree of precision. He was able to show that 10 1   3 71 7

from which we get the approximation 3

1

.The use of the symbol  7

for this quantity was introduced by Leonhard Euler (1707–1783) in 1739, approximately 2000 years after Archimedes. A

B

− 5 − 4 − 3 −2 −1 0

C 1

2

3

4

5

3

4

5

4

5

FIGURE 1.1

− 5 − 4 − 3 −2 −1 0

1

2

The open interval (−2, 4)

−5 − 4 −3 −2 −1 0

1

2

3

The closed interval [1, 5]

− 5 − 4 − 3 −2 −1 0

1

2

3

1

2

3

4

4

The half-open interval (−5 , −2]

FIGURE 1.2

THE REAL NUMBERS The real numbers are used extensively in mathematics. The set of real numbers is quite comprehensive and contains several unique sets of numbers. The integers are the set of numbers . . . , 4, 3, 2, 1, 0, 1, 2, 3, 4, . . . Recall that the brace symbols,  , are used to identify a set. The positive integers are called natural numbers. a The rational numbers are the set of numbers of the form , where a and b b 3 5 are integers and b  0. Thus the rational numbers include  and . Because 4 2 a each integer can be expressed in the form with denominator b  1, the inteb gers are included in the set of rational numbers. Every rational number can be written as either a terminating or a repeating decimal. A number written in decimal form that does not repeat or terminate is called an irrational number. Some examples of irrational numbers are 0.141141114 . . . , 2, and . These numbers cannot be expressed as quotients of integers. The set of real numbers is the union of the sets of rational and irrational numbers. A real number can be represented geometrically on a coordinate axis called a real number line. Each point on this line is associated with a real number called the coordinate of the point. Conversely, each real number can be associated with 7 a point on a real number line. In Figure 1.1, the coordinate of A is  , the coor2 dinate of B is 0, and the coordinate of C is 2. Given any two real numbers a and b, we say that a is less than b, denoted by a  b, if a  b is a negative number. Similarly, we say that a is greater than b, denoted by a  b, if a  b is a positive number. When a equals b, a  b is zero. The symbols  and  are called inequality symbols. Two other inequality symbols,  (less than or equal to) and  (greater than or equal to), are also used. The inequality symbols can be used to designate sets of real numbers. If a  b, the interval notation a, b is used to indicate the set of real numbers between a and b. This set of numbers also can be described using set-builder notation: a, b  x  a  x  b

5

The half-open interval [−4, 0)

−5 − 4 − 3 −2 −1 0

3

EQUATIONS AND INEQUALITIES

THE REAL NUMBERS ABSOLUTE VALUE AND DISTANCE LINEAR AND QUADRATIC EQUATIONS LINEAR AND QUADRATIC INEQUALITIES ABSOLUTE VALUE INEQUALITIES

3

Equations and Inequalities

5

When reading a set written in set-builder notation, we read x as “the set of x such that.” The expression that follows the vertical bar designates the elements in the set. The set a, b is called an open interval. The graph of the open interval consists of all the points on the real number line between a and b, not including a and b. A closed interval, denoted by a, b, consists of all points between a and b, including a and b. We can also discuss half-open intervals. An example of each type of interval is shown in Figure 1.2.

4

Chapter 1

Functions and Graphs 2, 4  x  2  x  4

An open interval

The interval notation 1, 

1, 5  x  1  x  5

A closed interval

represents all real numbers greater

4, 0  x  4  x  0

A half-open interval

than or equal to 1. The interval

5, 2  x  5  x  2 A half-open interval

take note

notation  , 4 represents all real numbers less than 4.

ABSOLUTE VALUE AND DISTANCE The absolute value of a real number is a measure of the distance from zero to the point associated with the number on a real number line. Therefore, the absolute value of a real number is always positive or zero. We now give a more formal definition of absolute value.

Absolute Value For a real number a, the absolute value of a, denoted by a, is a 

A

− 5 −4 − 3 −2 −1 0

1

2

a a

if a  0 if a  0

The distance d between the points A and B with coordinates 3 and 2, respectively, on a real number line is the absolute value of the difference between the coordinates. See Figure 1.3.

B

5 units



3

4

FIGURE 1.3

d  2  3  5

5

Because the absolute value is used, we could also write d  3  2  5 In general, we define the distance between any two points A and B on a real number line as the absolute value of the difference between the coordinates of the points.

Distance Between Two Points on a Coordinate Line Let a and b be the coordinates of the points A and B, respectively, on a real number line. Then the distance between A and B, denoted dA, B, is 5 4

dA, B  a  b

A

3 2 1 9

0 −1

d(A, B) =|4 − (−5)|= 9

This formula applies to any real number line. It can be used to find the distance between two points on a vertical real number line, as shown in Figure 1.4.

−2 −3

LINEAR AND QUADRATIC EQUATIONS

−4 −5

B

FIGURE 1.4

An equation is a statement about the equality of two expressions. Examples of equations follow. 725

x 2  4x  5

3x  2  2x  1  3

1.1

Equations and Inequalities

5

The values of the variable that make an equation a true statement are the roots or solutions of the equation. To solve an equation means to find the solutions of the equation. The number 2 is said to satisfy the equation 2x  1  5, because substituting 2 for x produces 22  1  5, which is a true statement.

Definition of a Linear Equation A linear equation in the single variable x is an equation of the form ax  b  0, where a  0. To solve a linear equation in one variable, isolate the variable on one side of the equals sign. EXAMPLE 1

Solve a Linear Equation

3x  7  5x  2

Solve: 3x  5  2 Solution 3x  5  2 3x  5  5  2  5

• Add 5 to each side of the equation.

3x  7 3x 7  3 3 x

• Divide each side of the equation by 3.

7 3

The solution is

7 . 3

TRY EXERCISE 6, PAGE 13

Definition of a Quadratic Equation A quadratic equation in x is an equation that can be written in the standard quadratic form ax2  bx  c  0 where a, b, and c are real numbers and a  0. M AT H M AT T E R S The term quadratic is derived from the Latin word quadrare, which means “to make square.” Because the area of a square that measures x units on each side is x 2, we refer to equations that can be written in the form ax 2  bx  c  0 as equations that are quadratic in x.

Several methods can be used to solve a quadratic equation. For instance, if you can factor ax2  bx  c into linear factors, then ax2  bx  c  0 can be solved by applying the following property.

The Zero Product Principle If A and B are algebraic expressions such that AB  0, then A  0 or B  0.

6

Chapter 1

Functions and Graphs The zero product principle states that if the product of two factors is zero, then at least one of the factors must be zero. In Example 2, the zero product principle is used to solve a quadratic equation.

Solve by Factoring

EXAMPLE 2

Solve by factoring: 2x2  5x  12 Solution

To review FACTORING, see the Review Appendix, p. 556.

2x2  5x  12 2x2  5x  12  0 x  42x  3  0 x40 2x  3  0 x4 2x  3 3 x 2

• Write in standard quadratic form. • Factor. • Set each factor equal to zero. • Solve each linear equation.

A check shows that 4 and 

3 are both solutions of 2x2  5x  12. 2

TRY EXERCISE 22, PAGE 13 To review RADICAL EXPRESSIONS, see the Review Appendix, p. 543.

The solutions of x2  25 can be found by taking the square root of each side of the equation. x2  25 x2  25 x  5 x  5 x  5 or

• Recall that x2  x.

x  5

We will refer to the preceding method of solving a quadratic equation as the square root procedure.

The Square Root Procedure If x2  c, then x  c or x   c, which can also be written as x  c. This procedure can be used to solve x  12  49. x  12  49 x  1  49 x  1 7 x  8 or

• Apply the Square Root Procedure. • Simplify.

6

The solutions are 8 and 6. Note that in each of the following perfect-square trinomials, the coefficient of x2 is 1, and the constant term is the square of half the coefficient of the x term.

1.1

Equations and Inequalities

x  52  x2  10x  25, x  32  x2  6x  9,



 1  10 2

2

1  6 2

2

7

 25 9

Adding to a binomial of the form x2  bx the constant term that makes the binomial a perfect-square trinomial is called completing the square. For example, to complete the square of x2  8x, add

 1 8 2

2

 16

to produce the perfect-square trinomial x2  8x  16. Completing the square is a powerful procedure because it can be used to solve any quadratic equation. Completing the square by adding the square of half the coefficient of the x term requires that the coefficient of the x2 term be 1. If the coefficient of the x2 term is not 1, then first multiply each term on each side of the equation by the reciprocal of the coefficient of x2 to produce a coefficient of 1 for the x2 term.

Solve by Completing the Square

EXAMPLE 3

Solve 2x2  8x  1  0 by completing the square. Solution 2x2  8x  1  0 2x2  8x  1 1 1 2x2  8x  1 2 2

• Isolate the constant term. • Multiply both sides of the equation by the reciprocal of the coefficient of x2.

1 2 1 x2  4x  4   4 2 9 x  22  2 9 x2

2 x2  4x 

To review SIMPLIFYING RADICAL EXPRESSIONS, see the Review Appendix, p. 543.



3 2 2 4 3 2 x 2 x  2

The solutions are x 

• Complete the square. • Factor and simplify. • Apply the square root procedure. • Solve for x. • Simplify.

4  3 2 4  3 2 . and x  2 2

TRY EXERCISE 28, PAGE 13

8

Chapter 1

Functions and Graphs Completing the square on ax2  bx  c  0 a  0 produces a formula for x in terms of the coefficients a, b, and c. The formula is known as the quadratic formula, and it can be used to solve any quadratic equation.

M AT H M AT T E R S

The Quadratic Formula If ax2  bx  c  0, a  0, then x

b b2  4ac 2a

Proof We assume a is a positive real number. If a were a negative real number, then we could multiply each side of the equation by 1 to make it positive. ax2  bx  c  0 a  0 ax2  bx  c b c x2  x   a a

Evariste Galois (1811–1832)

The quadratic formula provides the solutions to the general quadratic equation ax 2  bx  c  0 and formulas have been developed to solve the general cubic ax 3  bx 2  cx  d  0 and the general quartic

x2 

2

b 2a

2

ax4  bx 3  cx 2  dx  e  0

x

However, the French mathematician Evariste Galois, shown above, was able to prove that there are no formulas that can be used to solve “by radicals” general equations of degree 5 or larger. Shortly after completion of his remarkable proof, Galois was shot in a duel. It has been reported that as Galois lay dying, he asked his brother, Alfred, to “Take care of my work. Make it known. Important.” When Alfred broke into tears, Evariste said, “Don’t cry Alfred. I need all my courage to die at twenty.” (Source: Whom the Gods Love, by Leopold Infeld,The National Council of Teachers of Mathematics, 1978, p. 299.)

 





b b x a 2a b x 2a

x

x

2

b 2 c  2a a 2 b c  2 4a a 



b2 4a c   4a2 4a a

b 

2a

b2  4ac 4a2

b b2  4ac 

2a 2a b b2  4ac x

2a 2a 2 b b  4ac x 2a

• Given. • Isolate the constant term. • Multiply each term on each side 1 of the equation by . a • Complete the square. • Factor the left side. Simplify the powers on the right side. • Use a common denominator to simplify the right side. • Apply the square root procedure. • Because a > 0, 4a2  2a. • Add 

b to each side. 2a

◆

As a general rule, you should first try to solve quadratic equations by factoring. If the factoring process proves difficult, then solve by using the quadratic formula. EXAMPLE 4

Solve by Using the Quadratic Formula

Use the quadratic formula to solve x2  3x  5. Solution The standard form of x2  3x  5 is x2  3x  5  0. Substituting a  1, b  3, and c  5 in the quadratic formula produces

1.1

x  The solutions are x 

Equations and Inequalities

9

3 32  415 21 3 29 2

3  29 3  29 and x  . 2 2

TRY EXERCISE 30, PAGE 13 QUESTION Can the quadratic formula be used to solve any quadratic equa-

tion ax2  bx  c  0 with real coefficients and a  0?

LINEAR AND QUADRATIC INEQUALITIES A statement that contains the symbol , , , or  is called an inequality. An inequality expresses the relative order of two mathematical expressions. The solution set of an inequality is the set of real numbers each of which, when substituted for the variable, results in a true inequality. The inequality x  4 is true for any 17 value of x greater than 4. For instance, 5, 21, and are all solutions of x  4. 3 The solution set of the inequality can be written in set-builder notation as x  x  4 or in interval notation as 4, . Equivalent inequalities have the same solution set. We solve an inequality by producing simpler but equivalent inequalities until the solutions are found. To produce these simpler but equivalent inequalities, we apply the following properties.

Properties of Inequalities Let a, b, and c be real numbers. 1.

Addition Property Adding the same real number to each side of an inequality preserves the direction of the inequality symbol. a  b and a  c  b  c are equivalent inequalities.

2.

Multiplication Properties a. Multiplying each side of an inequality by the same positive real number preserves the direction of the inequality symbol. If c  0, then a  b and ac  bc are equivalent inequalities. b.

Multiplying each side of an inequality by the same negative real number changes the direction of the inequality symbol. If c  0, then a  b and ac  bc are equivalent inequalities.

ANSWER

Yes. However, it is sometimes easier to find the solutions by factoring, by the square root procedure, or by completing the square.

10

Chapter 1

Functions and Graphs Note the difference between Property 2a and Property 2b. Property 2a states that an equivalent inequality is produced when each side of a given inequality is multiplied by the same positive real number and that the direction of the inequality symbol is not changed. By contrast, Property 2b states that when each side of a given inequality is multiplied by a negative real number, we must reverse the direction of the inequality symbol to produce an equivalent inequality. A linear inequality in one variable is one that can be written in the form ax  b  0 or ax  b  0, where a  0. The inequality symbols  and  can also be used. EXAMPLE 5

4x  12  6x  2.

Solve a Linear Inequality

Solve: 2x  3  4x  10. Write the solution set in interval notation. Solution

take note Solutions of inequalities are often stated using set-builder notation or interval notation. For instance, the real numbers that are solutions of the inequality in Example 5 can be written in setbuilder notation as x  x  2 or in interval notation as 2, .

2x  3  4x  10 2x  6  4x  10 2x  4 x  2

• Use the distributive property. • Subtract 4x and 6 from each side of the inequality. • Divide each side by 2 and reverse the inequality symbol.

Thus the original inequality is true for all real numbers greater than 2. The solution set is 2, . TRY EXERCISE 50, PAGE 13

A quadratic inequality in one variable is one that can be written in the form ax2  bx  c  0 or ax2  bx  c  0 , where a  0. The symbols  and  can also be used. Quadratic inequalities can be solved by algebraic means. However, it is often easier to use a graphical method to solve these inequalities. The graphical method is used in the example that follows. To solve x2  x  6  0, factor the trinomial. x2  x  6  0 x  3x  2  0 On a number line, draw vertical lines at the numbers that make each factor equal to zero. See Figure 1.5. x30 x3

−5 −4 −3 −2 −1

x20 x  2

0

1

2

3

4

5

FIGURE 1.5

For each factor, place plus signs above the number line for those regions where the factor is positive and negative signs where the factor is negative, as in Figure 1.6.

1.1 (x − 3)

−

−

−

(x + 2)

−

−

−

−

−

−

−

+

+

+

+

−5 −4 −3 −2 −1

0

1

2

0

0

3

11

Equations and Inequalities +

+

+

+

4

5

• x  3 is positive for x  3. • x  2 is positive for x  2.

FIGURE 1.6 − 5 −4 −3 −2 −1

0

1

2

3

4

5

FIGURE 1.7

Because x2  x  6  0, the solution set will be the regions where one factor is positive and the other factor is negative. This occurs when 2  x  3 or 2, 3. The graph of the solution set of the inequality x2  x  6  0 is shown in Figure 1.7.

Solve a Quadratic Inequality

EXAMPLE 6

Solve and graph the solution set of 2x2  x  3  0. Solution 2x2  x  3  0 2x  3x  1  0 2x  3  0 x10 3 x x  1 2

take note We use the set union symbol  to

(2x − 3)

−

−

−

−

(x + 1)

−

−

−

−

−

− 0 +

+

+

+

0

+

+

+

+

+

+

−5 −4 −3 −2 −1

0

1

2

3

4

5

join intervals that contain solutions to inequalities, as in

− 5 − 4 −3 −2 −1

0

1

2

3

4

5

Example 6.

The solution set is , 1 

 

3 , . 2

TRY EXERCISE 58, PAGE 13

ABSOLUTE VALUE INEQUALITIES − 5 − 4 − 3 −2 −1

0

1

2

3

4

5

2

3

4

5

FIGURE 1.8 − 5 − 4 −3 −2 −1

0

1

FIGURE 1.9

The solution set of the absolute value inequality x  1  3 is the set of all real numbers whose distance from 1 is less than 3. Therefore, the solution set consists of all numbers between 2 and 4. See Figure 1.8. In interval notation, the solution set is 2, 4. The solution set of the absolute value inequality x  1  3 is the set of all real numbers whose distance from 1 is greater than 3. Therefore, the solution set consists of all real numbers less than 2 or greater than 4. See Figure 1.9. In interval notation, the solution set is , 2  4, . The following properties are used to solve absolute value inequalities.

Properties of Absolute Value Inequalities For any variable expression E and any nonnegative real number k, E  k

if and only if

k  E  k

E  k

if and only if

E  k

or

Ek

12

Chapter 1

Functions and Graphs

Solve an Absolute Value Inequality

EXAMPLE 7

Solve: 2  3x  7 Solution 2  3x  7 implies 7  2  3x  7. Solve this compound inequality. 7  2  3x  7 9  3x  5 3

5 − 3 −4 −3

x



• Subtract 2 from each of the three parts of the inequality.

5 3

• Multiply each part of the inequality by 

1 and 3

reverse the inequality symbols.

−2 −1

0

1

2

3



4

In interval notation, the solution set is given by 

FIGURE 1.10

5 , 3 . See Figure 1.10. 3

TRY EXERCISE 70, PAGE 13

EXAMPLE 8

Solve an Absolute Value Inequality

Solve: 4x  3  5 Solution 4x  3  5 implies 4x  3  5 or 4x  3  5. Solving each of these inequalities produces

take note Some inequalities have a solution

4x  3  5 4x  2

set that consists of all real numbers. For example, x  9  0 is true for all values of x. Because

x

an absolute value is always

nonnegative, the equation is always

1 2

Therefore, the solution set is  , 

true.

or

4x  3  5 4x  8 x2

1 2



 2, . See Figure 1.11.

TRY EXERCISE 72, PAGE 13

TOPICS FOR DISCUSSION

− −4 −3

−2 −1

1 2 0

1

2

3

4

1.

Discuss the similarities and differences among natural numbers, integers, rational numbers, and real numbers.

2.

Discuss the differences among an equation, an inequality, and an expression.

3.

Is it possible for an equation to have no solution? If not, explain why. If so, give an example of an equation with no solution.

4.

Is the statement x  x ever true? Explain why or why not.

5.

How do quadratic equations in one variable differ from linear equations in one variable? Explain how the method used to solve an equation depends on whether it is a linear or a quadratic equation.

FIGURE 1.11

1.1

Equations and Inequalities

13

EXERCISE SET 1.1 In Exercises 1 to 18, solve and check each equation. 11. 2x  10  40

12. 3y  20  2

13. 5x  2  2x  10

14. 4x  11  7x  20

15. 2x  3  5  4x  5

 6. 65s  11  122s  5  0 1 2 3 x  4 2 3

18.

1 x 5 4 2

19.

1 2 x5 x3 3 2

10.

1 19 1 x7 x 2 4 2

13.

12. 0.04x  0.2  0.07

36.

1 2 2 x  5x   0 3 2

37. 2x2  3x  2  0

38. 2x2  5x  3  0

39. x2  3x  5

40. x2  7x  1

41. 2x  3  11

42. 3x  5  16

43. x  4  3x  16

44. 5x  6  2x  1

45. 6x  1  19

46. 5x  2  37

47. 3x  2  5x  7

48. 4x  5  2x  15

49. 43x  5  2x  4

3 3 n  5  n  11  0 5 4

14. 

1 2 3 x  x10 2 4

In Exercises 41 to 50, use the properties of inequalities to solve each inequality.Write answers using interval notation.

17.

11. 0.2x  0.4  3.6

35.

2 5  p  11  2p  5  0 7 5

15. 3x  5x  1  3x  4x  2 16. 5x  4x  4  x  35x  4 17. 0.08x  0.124000  x  432

 50. 3x  7  52x  8

In Exercises 51 to 58, solve each quadratic inequality. Use interval notation to write each solution set. 51. x 2  7x  0

52. x 2  5x  0

53. x 2  7x  10  0

54. x 2  5x  6  0

55. x 2  3x  28

56. x 2  x  30

57. 6x 2  4  5x

 58. 12x 2  8x  15

18. 0.075y  0.0610,000  y  727.50 In Exercises 59 to 76, use interval notation to express the solution set of each inequality. In Exercises 19 to 26, solve each quadratic equation by factoring and applying the zero product property.

59. x  4

60. x  2

19. x2  2x  15  0

61. x  1  9

62. x  3  10

63. x  3  30

64. x  4  2

21. 8y 2  189y  72  0

20. y2  3y  10  0

 22. 12w 2  41w  24  0

23. 3x 2  7x  0

24. 5x 2  8x

65. 2x  1  4

66. 2x  9  7

25. x  52  9  0

26. 3x  42  16  0

67. x  3  5

68. x  10  2

In Exercises 27 to 40, solve by completing the square or by using the quadratic formula.

69. 3x  10  14

 70. 2x  5  1

71. 4  5x  24

 72. 3  2x  5

27. x2  2x  15  0

 28. x2  5x  24  0

73. x  5  0

74. x  7  0

29. x2  x  1  0

 30. x 2  x  2  0

75. x  4  0

76. 2x  7  0

31. 2x2  4x  1  0

32. 2x2  4x  1  0

33. 3x2  5x  3  0

34. 3x2  5x  4  0

77. GEOMETRY The perimeter of a rectangle is 27 centimeters, and its area is 35 square centimeters. Find the length and the width of the rectangle.

14

Chapter 1

Functions and Graphs

78. GEOMETRY The perimeter of a rectangle is 34 feet and its area is 60 square feet. Find the length and the width of the rectangle. 79. RECTANGULAR ENCLOSURE A gardener wishes to use 600 feet of fencing to enclose a rectangular region and subdivide the region into two smaller rectangles. The total enclosed area is 15,000 square feet. Find the dimensions of the enclosed region.

w

l

80. RECTANGULAR ENCLOSURE A farmer wishes to use 400 yards of fencing to enclose a rectangular region and subdivide the region into three smaller rectangles. If the total enclosed area is 4800 square yards, find the dimensions of the enclosed region.

w

82. PERSONAL FINANCE You can rent a car for the day from company A for $29.00 plus $0.12 a mile. Company B charges $22.00 plus $0.21 a mile. Find the number of miles m (to the nearest mile) per day for which it is cheaper to rent from company A. 83. PERSONAL FINANCE A sales clerk has a choice between two payment plans. Plan A pays $100.00 a week plus $8.00 a sale. Plan B pays $250.00 a week plus $3.50 a sale. How many sales per week must be made for plan A to yield the greater paycheck? 84. PERSONAL FINANCE A video store offers two rental plans. The yearly membership fee and the daily charge per video are shown below. How many videos can be rented per year if the No-fee plan is to be the less expensive of the plans?

THE VIDEO STORE Rental Plan

Yearly Fee

Daily Charge per Video

Low-rate

$15.00

$1.49

No-fee

None

$1.99

l

81. PERSONAL FINANCE A bank offers two checking account plans. The monthly fee and charge per check for each plan are shown below. Under what conditions is it less expensive to use the LowCharge plan? Account Plan

Monthly Fee

Charge per Check

LowCharge

$5.00

$.01

FeeSaver

$1.00

$.08

85. AVERAGE TEMPERATURES The average daily minimum-tomaximum temperatures for the city of Palm Springs during the month of September are 68F to 104F. What is the corresponding temperature range measured on the Celsius temperature scale?

CONNECTING CONCEPTS 86. A GOLDEN RECTANGLE The ancient Greeks defined a rectangle as a “golden rectangle” if its length l and its width w satisfied the equation l w  w lw a. Solve this formula for w. b. If the length of a golden rectangle is 101 feet, determine its width. Round to the nearest hundredth.

87. SUM OF NATURAL NUMBERS The sum S of the first n natural numbers 1, 2, 3, . . . , n is given by the formula S

n n  1 2

How many consecutive natural numbers starting with 1 produce a sum of 253?

1.1 88. NUMBER OF DIAGONALS The number of diagonals D of a polygon with n sides is given by the formula D

n n  3 2

Equations and Inequalities

15

92. HEIGHT OF A PROJECTILE A ball is thrown directly upward from a height of 32 feet above the ground with an initial velocity of 80 feet per second. Find the time interval during which the ball will be more than 96 feet above the ground. (Hint: See Exercise 91.)

a. Determine the number of sides of a polygon with 464 diagonals. b. Can a polygon have 12 diagonals? Explain. 80 ft/sec

89. REVENUE The monthly revenue R for a product is given by R  420x  2x 2, where x is the price in dollars of each unit produced. Find the interval in terms of x for which the monthly revenue is greater than zero.

32 ft

90. Write an absolute value inequality to represent all real numbers within a. 8 units of 3 b. k units of j (assume k  0) 91. HEIGHT OF A PROJECTILE The equation

93. GEOMETRY The length of the side of a square has been measured accurately to within 0.01 foot. This measured length is 4.25 feet. a. Write an absolute value inequality that describes the relationship between the actual length of each side of the square s and its measured length.

s  16t 2  v0 t  s0 gives the height s, in feet above ground level, of an object t seconds after the object is thrown directly upward from a height s0 feet above the ground with an initial velocity of v0 feet per second. A ball is thrown directly upward from ground level with an initial velocity of 64 feet per second. Find the time interval during which the ball has a height of more than 48 feet.

b. Solve the absolute value inequality you found in part a. for s.

PREPARE FOR SECTION 1.2 94. Evaluate

x1  x2 when x1  4 and x2  7. 2

95. Simplify 50. [A.1] 96. Is y  3x  2 a true equation when y  5 and x  1? [1.1]

97. If y  x2  3x  2, find x when y  0. [1.1] 98. Evaluate x  y when x  3 and y  1. [1.1] 99. Evaluate a2  b2 when a  3 and b  4. [A.1]

PROJECTS 1.

TEACHING MATHEMATICS Prepare a lesson that you could use to explain to someone how to solve linear and quadratic equations. Be sure to include an explanation of the differences between these two types of equations and the different methods that are used to solve them.

2.

CUBIC EQUATIONS Write an essay on the development of the solution of the cubic equation. An excellent source of information is the chapter “Cardano and the Solution of the Cubic” in Journey Through Genius by William Dunham (New York: Wiley, 1990). Another excellent source is A History of Mathematics: An Introduction by Victor J. Katz (New York: Harper Collins, 1993).

16

Chapter 1

SECTION

Functions and Graphs

1.2

CARTESIAN COORDINATE SYSTEMS THE DISTANCE AND MIDPOINT FORMULAS GRAPH OF AN EQUATION INTERCEPTS CIRCLES,THEIR EQUATIONS, AND THEIR GRAPHS

take note Abscissa comes from the same root word as scissors. An open pair of scissors looks like an x.

A TWO-DIMENSIONAL COORDINATE SYSTEM AND GRAPHS CARTESIAN COORDINATE SYSTEMS Each point on a coordinate axis is associated with a number called its coordinate. Each point on a flat, two-dimensional surface, called a coordinate plane or xy-plane, is associated with an ordered pair of numbers called coordinates of the point. Ordered pairs are denoted by a, b, where the real number a is the x-coordinate or abscissa and the real number b is the y-coordinate or ordinate. The coordinates of a point are determined by the point’s position relative to a horizontal coordinate axis called the x-axis and a vertical coordinate axis called the y-axis. The axes intersect at the point 0, 0, called the origin. In Figure 1.12, the axes are labeled such that positive numbers appear to the right of the origin on the x-axis and above the origin on the y-axis. The four regions formed by the axes are called quadrants and are numbered counterclockwise. This two-dimensional coordinate system is referred to as a Cartesian coordinate system in honor of René Descartes.

y

y

M AT H M AT T E R S The concepts of analytic geometry developed over an extended period of time, culminating in 1637 with the publication of two works: Discourse on the Method for Rightly Directing One’s Reason and Searching for Truth in the Sciences by Rene´ Descartes (1596–1650) and Introduction to Plane and Solid Loci by Pierre de Fermat. Each of these works was an attempt to integrate the study of geometry with the study of algebra. Of the two mathematicians, Descartes is usually given most of the credit for developing analytic geometry. In fact, Descartes became so famous in La Haye, the city in which he was born, that it was renamed La HayeDescartes.

Quadrant II Horizontal axis −4

4 2

−2 −2

Quadrant III

−4

4

Quadrant I

4

Quadrant IV

FIGURE 1.12

(4, 3)

(0, 1)

(3, 1)

2

Vertical axis 2 Origin

(1, 3)

(−3, 1) x

−4

−2

2 −2

4

x

(3, −2)

(−2, −3) −4

FIGURE 1.13

To plot a point Pa, b means to draw a dot at its location in the coordinate plane. In Figure 1.13 we have plotted the points 4, 3, 3, 1, 2, 3, 3, 2, 0, 1, 1, 3, and 3, 1. The order in which the coordinates of an ordered pair are listed is important. Figure 1.13 shows that 1, 3 and 3, 1 do not denote the same point. Data often are displayed in visual form as a set of points called a scatter diagram or scatter plot. For instance, the scatter diagram in Figure 1.14 shows the number of Internet virus incidents from 1993 to 2003. The point whose coordinates are approximately 7, 21,000 means that in the year 2000 there were approximately 21,000 Internet virus incidents. The line segments that connect the points in Figure 1.14 help illustrate trends.

Computer virus incidents

1.2

17

A Two-Dimensional Coordinate System and Graphs

150,000 125,000 100,000 75,000 50,000 25,000 1

2

3

4

5

6

7

8

9

10

t

Year (t = 0 corresponds to 1993)

FIGURE 1.14 Source: www.cert.org

take note The notation (a, b) was used earlier to denote an interval on a one-

QUESTION If the trend in Figure 1.14 continues, will the number of virus in-

cidents in 2004 be more or less than 200,000?

dimensional number line. In this section, (a, b) denotes an ordered

In some instances, it is important to know when two ordered pairs are equal.

pair in a two-dimensional plane. This should not cause confusion in future sections because as each

Equality of Ordered Pairs

mathematical topic is introduced,

The ordered pairs a, b and c, d are equal if and only if a  c and b  d.

it will be clear whether a onedimensional or a two-dimensional coordinate system is involved.

For instance, if 3, y  x, 2, then x  3 and y  2.

THE DISTANCE AND MIDPOINT FORMULAS y (1, 2) 2

−2

2 5 −2 (1, −3)

FIGURE 1.15

4

x

The Cartesian coordinate system makes it possible to combine the concepts of algebra and geometry into a branch of mathematics called analytic geometry. The distance between two points on a horizontal line is the absolute value of the difference between the x-coordinates of the two points. The distance between two points on a vertical line is the absolute value of the difference between the y-coordinates of the two points. For example, as shown in Figure 1.15, the distance d between the points with coordinates 1, 2 and 1, 3 is d  2  3  5. If two points are not on a horizontal or vertical line, then a distance formula for the distance between the two points can be developed as follows. The distance between the points P1x1, y1 and P2x2, y2 in Figure 1.16 is the length of the hypotenuse of a right triangle whose sides are horizontal and verti-

ANSWER

More. The increase between 2002 and 2003 was more than 70,000. If this trend continues, the increase between 2003 and 2004 will be at least 70,000 more than 150,000. That is, the number of virus incidents in 2004 will be at least 220,000.

18

Chapter 1

Functions and Graphs cal line segments that measure x2  x1 and y2  y1, respectively. Applying the Pythagorean Theorem to this triangle produces d 2  x2  x12  y2  y12 d  x2  x12  y2  y12

y P1(x1, y1)

• The square root theorem. Because d is nonnegative, the negative root is not listed.

y1

 x2  x12   y2  y12

d

| y2 – y1| y2

Thus we have established the following theorem.

P2(x 2 , y2) x

x2

x1

• Because |x2  x1|2  (x2  x1)2 and |y2  y1|2  (y2  y1)2

| x2 – x 1 |

The Distance Formula The distance d between the points P1x1, y1 and P2x2, y2 is

FIGURE 1.16

d  x2  x12  y2  y12

The distance d between the points whose coordinates are P1x1, y1 and P2x2, y2 is denoted by dP1, P2. To find the distance dP1, P2 between the points P13, 4 and P27, 2, we apply the distance formula with x1  3, y1  4, x2  7, and y2  2. dP1, P2  x2  x12   y2  y12   7  3 2  2  42  104  226  10.2 The midpoint M of a line segment is the point on the line segment that is equidistant from the endpoints P1x1, y1 and P2x2, y2 of the segment. See Figure 1.17. y

P2(x 2 , y2) M (x, y)

The Midpoint Formula The midpoint M of the line segment from P1x1, y1 to P2x2, y2 is given by





x1  x2 y1  y2 , 2 2

P1(x1, y1) x

FIGURE 1.17

The midpoint formula states that the x-coordinate of the midpoint of a line segment is the average of the x-coordinates of the endpoints of the line segment and that the y-coordinate of the midpoint of a line segment is the average of the y-coordinates of the endpoints of the line segment. The midpoint M of the line segment connecting P12, 6 and P23, 4 is M



 

  

x1  x2 y1  y2 2  3 6  4 ,  ,  2 2 2 2

1 ,5 2

1.2

A Two-Dimensional Coordinate System and Graphs

EXAMPLE 1

19

Find the Midpoint and Length of a Line Segment

Find the midpoint and the length of the line segment connecting the points whose coordinates are P14, 3 and P24, 2. Solution Midpoint  

   



x1  x2 y1  y2 , 2 2



4  4 3  2 , 2 2

 0,

1 2

dP1, P2  x2  x12   y2  y12  4  42  2  32  82  52  64  25  89 TRY EXERCISE 6, PAGE 27

GRAPH

OF AN

EQUATION

The equations below are equations in two variables. y  3x 3  4x  2

x 2  y 2  25

y

x x1

The solution of an equation in two variables is an ordered pair x, y whose coordinates satisfy the equation. For instance, the ordered pairs 3, 4, 4, 3, and 0, 5 are some of the solutions of x 2  y 2  25. Generally, there are an infinite number of solutions of an equation in two variables. These solutions can be displayed in a graph.

Graph of an Equation The graph of an equation in the two variables x and y is the set of all points whose coordinates satisfy the equation.

Consider y  2x  1. Substituting various values of x into the equation and solving for y produces some of the ordered pairs of the equation. It is convenient to record the results in a table similar to the one shown on the following page. The graph of the ordered pairs is shown in Figure 1.18.

20

Chapter 1

Functions and Graphs x

y  2x  1

y

(x, y)

2

22  1

5

2, 5

21  1

3

1, 3

0

20  1

1

0, 1

1

21  1

1

1, 1

2

22  1

3

2, 3

1



  

Choosing some noninteger values of x produces more ordered pairs to graph, 3 5 such as  , 4 and , 4 , as shown in Figure 1.19. Using still other values 2 2 of x would result in more and more ordered pairs being graphed. The result would be so many dots that the graph would appear as the straight line shown in Figure 1.20, which is the graph of y  2x  1.

−4

y

y

y

4

4

4

2

2

2

−2

2

x

4

−4

−2

2

x

4

−4

−2

2

4

−2 −4

−4

FIGURE 1.18

−4

FIGURE 1.19

EXAMPLE 2

FIGURE 1.20

Draw a Graph by Plotting Points

Graph: x 2  y  1 y

Solution

(−2, 5)

(2, 5)

Solve the equation for y.

4 (−1, 2)

(1, 2) (0, 1)

−4

−2

2

4

x

y  x2  1

Select values of x and use the equation to calculate y. Choose enough values of x so that an accurate graph can be drawn. Plot the points and draw a curve through them. See Figure 1.21. x

y  x2  1

y

(x, y)

yx 1

2

2

2  1

5

2, 5

FIGURE 1.21

1

12  1

2

1, 2

0

0  1

1

0, 1

1

1  1

2

1, 2

2

2  1

5

2, 5

2

TRY EXERCISE 26, PAGE 28

2 2 2

x

1.2 M AT H M AT T E R S Maria Agnesi (1718–1799) wrote Foundations of Analysis for the Use of Italian Youth, one of the most successful textbooks of the 18th century.The French Academy authorized a translation into French in 1749, noting that “there is no other book, in any language, which would enable a reader to penetrate as deeply, or as rapidly, into the fundamental concepts of analysis.” A curve that she discusses in her text a3 is given by the equation y  2 . x  a2 Unfortunately, due to a translation error from Italian to English, the curve became known as the “witch of Agnesi.”

21

A Two-Dimensional Coordinate System and Graphs

INTEGRATING TECHNOLOGY Some graphing calculators, such as the TI-83, have a TABLE feature that allows you to create a table similar to the one shown in Example 2. Enter the equation to be graphed, the first value for x, and the increment (the difference between successive values of x). For instance, entering y1  x 2  1, an initial value of 2 for x, and an increment of 1 yields a display similar to the one in Figure 1.22. Changing the initial value to 6 and the increment to 2 gives the table in Figure 1.23. Plot1 Plot2 Plot3 \Y 1 = X2+1 \Y2 = TABLE SETUP \Y3 = TblStart=-2 \Y4 = ∆Tbl=1 \Y5 = Indpnt: Auto Ask \Y6 = Depend: Auto Ask \Y7 =

X

Y1

-2 -1 0 1 2 3 4

5 2 1 2 5 10 17

X=-2 y y=

a

3

a x2 + a2

X -2 -1 0 1 2 3 4 X=-2

x

X -6 -4 -2 0 2 4 6 X=-6

Y1 5 2 1 2 5 10 17

FIGURE 1.22

Y1 37 17 5 1 5 17 37

FIGURE 1.23

With some calculators, you may scroll through the table by using the upor down-arrow keys. In this way, you can determine many more ordered pairs of the graph.

Graph by Plotting Points

EXAMPLE 3

Graph: y  x  2 Solution This equation is already solved for y, so start by choosing an x value and using the equation to determine the corresponding y value. For example, if x  3, then y  3  2  5  5. Continuing in this manner produces the following table:

y (−3, 5) 5 (−2, 4) (−1, 3)

(5, 3)

(0, 2)

−2

(1, 1)

(4, 2)

When x is

3

2

1

0

1

2

3

4

5

y is

5

4

3

2

1

0

1

2

3

(3, 1)

(2, 0)

y  x  2

FIGURE 1.24

5

x

Now plot the points listed in the table. Connecting the points forms a V shape, as shown in Figure 1.24. TRY EXERCISE 30, PAGE 28

22

Chapter 1

Functions and Graphs

Graph by Plotting Points

EXAMPLE 4

Graph: y 2  x Solution Solve the equation for y. y2  x y  x y 4 2

(16, 4) (9, 3) (4, 2) (1, 1)

(0, 0) −2 −4

Choose several x values, and use the equation to determine the corresponding y values.

(1, −1) (4, −2)

8

12

0

1

4

9

16

y is

0

1

2

3

4

16 x

(9, − 3) (16, − 4)

y2  x

When x is

Plot the points as shown in Figure 1.25. The graph is a parabola. TRY EXERCISE 32, PAGE 28

FIGURE 1.25

INTEGRATING TECHNOLOGY A graphing calculator or computer graphing software can be used to draw the graphs in Examples 3 and 4. These graphing utilities graph a curve in much the same way as you would, by selecting values of x and calculating the corresponding values of y. A curve is then drawn through the points. If you use a graphing utility to graph y  x  2, you will need to use the absolute value function that is built into the utility. The equation you enter will look similar to Y1=abs(X–2). To graph the equation in Example 4, you will enter two equations. The equations you enter will be similar to Y1= (X) Y2=– (X) The graph of the first equation will be the top half of the parabola; the graph of the second equation will be the bottom half.

INTERCEPTS Any point that has an x- or a y-coordinate of zero is called an intercept of the graph of an equation because it is at these points that the graph intersects the x- or the y-axis.

1.2

23

A Two-Dimensional Coordinate System and Graphs

Definition of x-Intercepts and y-Intercepts If x1, 0 satisfies an equation, then the point x1, 0 is called an x-intercept of the graph of the equation. If 0, y1 satisfies an equation, then the point 0, y1 is called a y-intercept of the graph of the equation.

To find the x-intercepts of the graph of an equation, let y  0 and solve the equation for x. To find the y-intercepts of the graph of an equation, let x  0 and solve the equation for y.

EXAMPLE 5

Find x- and y-Intercepts

Find the x- and y-intercepts of the graph of y  x 2  2x  3. Algebraic Solution

Visualize the Solution

To find the y-intercept, let x  0 and solve for y.

The graph of y  x 2  2x  3 is shown below. Observe that the graph intersects the x-axis at 1, 0 and 3, 0, the x-intercepts. The graph also intersects the y-axis at 0, 3, the y-intercept.

y  02  20  3  3 To find the x-intercepts, let y  0 and solve for x. 0  x 2  2x  3 0  x  3x  1 x  3  0 or x  1  0 x  3 or x  1

y

Because y  3 when x  0, 0, 3 is a y-intercept. Because x  3 or 1 when y  0, 3, 0 and 1, 0 are x-intercepts. Figure 1.26 confirms that these three points are intercepts.

(4, 5) 4

2 (−1, 0)

(3, 0)

−2

2

(0, −3) −4

4

(2, − 3) (1, −4)

y  x 2  2x  3

FIGURE 1.26 TRY EXERCISE 40, PAGE 28

x

24

Chapter 1

Functions and Graphs

INTEGRATING TECHNOLOGY In Example 5 it was possible to find the x-intercepts by solving a quadratic equation. In some instances, however, solving an equation to find the intercepts may be very difficult. In these cases, a graphing calculator can be used to estimate the x-intercepts. The x-intercepts of the graph of y  x 3  x  4 can be estimated using the INTERCEPT feature of a TI-83 calculator. The keystrokes and some sample screens for this procedure are shown below. Press Y= . Now enter X^3+X+4. Press ZOOM and select the standard viewing window.

Press 2nd CALC to access the CALCULATE menu. The y-coordinate of an x-intercept is zero. Therefore, select 2:zero. Press ENTER .

The “Left Bound?” shown on the bottom of the screen means to move the cursor until it is to the left of an x-intercept. Press ENTER . 10