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CliffsQuickReview®

Precalculus

By W. Michael Kelley

01 539841 FM.qxd 1/26/04 2:47 PM Page i

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CliffsQuickReview®

Precalculus

By W. Michael Kelley

01 539841 FM.qxd 1/26/04 2:47 PM Page i

About the Author

Mike Kelley has been a high school and college

math instructor. He currently works as an Aca-

demic Technology Coordinator for the College

of Education at the University of Maryland. He

has written several books and owns the Web site

www.calculus-help.com.

Publisher’s Acknowledgments

Editorial

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Development Editor: Ted Lorenzen

Copy Editor: Elizabeth Welch

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Composition

Indexer: Tom Dinse

Proofreader: Ethel M. Winslow

Wiley Publishing, Inc. Composition Services

CliffsQuickReview®Precalculus

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Table of Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1

Why You Need This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

How to Use This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Visit Our Web Site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Chapter 1: Precalculus Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

Classifying Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Interval Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Bounded intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Unbounded intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Algebraic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

The associative property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

The commutative property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

The distributive property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Identity elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Inverse properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Exponential Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Radical Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Properties of radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Simplifying radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Operations with radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Rationalizing expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Polynomial Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Classifying polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Adding and subtracting polynomials . . . . . . . . . . . . . . . . . . . . . . . 15

Multiplying polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Adding and subtracting rational expressions . . . . . . . . . . . . . . . . . 17

Multiplying rational expressions . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Simplifying complex fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Equations and Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Solving equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Solving linear inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Solving absolute value inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 21

Special inequality cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Finding Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Chapter 2: Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26

Relations vs. Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Understanding relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Deﬁning functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Writing functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Function Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

The vertical line test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Finding symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

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Calculating intercepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Determining domain and range . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Eight Key Function Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Basic Function Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Vertical and horizontal shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Reﬂections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Stretching and squishing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Multiple transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Combining and Composing Functions . . . . . . . . . . . . . . . . . . . . . . . 41

Arithmetic combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

The composition of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

What is an inverse function? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Graphs of inverse functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Finding inverse functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Chapter 3: Polynomial and Rational Functions . . . . . . . . . . . . . . . . . . .47

Factoring Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Greatest common factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Factoring by grouping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Factoring quadratic trinomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Special factor patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Solving Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Factoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

The quadratic formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Completing the square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Polynomial Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Long division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Synthetic division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Important Root-Finding Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 56

The remainder theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

The factor theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Calculating Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

The Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . . . . . . 58

Descartes’ Rule of Signs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

The Rational Root Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Determining roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Advanced Graphing Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

The Leading Coefﬁcient Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Finding rational asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Chapter 4: Exponential and Logarithmic Functions . . . . . . . . . . . . . . .67

Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Natural exponential function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Graphs of exponential functions . . . . . . . . . . . . . . . . . . . . . . . . . . 68

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Table of Contents v

Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Natural and common logs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Inverse relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Graphs of logarithmic functions . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Change of base formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Properties of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Solving Exponential and Logarithmic Equations . . . . . . . . . . . . . . . . 76

Exponential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Logarithmic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Exponential Word Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Compound interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Growth and decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Chapter 5: Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .83

Measuring Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Characteristics of angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Degrees and radians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Angle pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Coterminal angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

The Unit Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Right Triangle Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Oblique Triangle Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Reference angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Calculating trigonometric ratios . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Graphs of Sine and Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Periodic graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Transforming sine and cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Other Trigonometric Function Graphs . . . . . . . . . . . . . . . . . . . . . . 101

Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Chapter 6: Analytic Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . .106

Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Four types of identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Simplifying expressions with identities . . . . . . . . . . . . . . . . . . . . 108

Proving Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Solving Trigonometric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Simple equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Quadratic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Equations requiring identities . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Equations requiring squaring . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Functions of multiple angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Sum and Difference Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Additional Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Double-angle formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Half-angle formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

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Sum-product formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Product-sum formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Oblique Triangle Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Law of Sines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Law of Cosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Calculating Triangle Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Given Side-Angle-Side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Given side-side-side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Chapter 7: Vectors and the Trigonometry of Complex Numbers . . .128

Vectors in the Coordinate Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Standard form of a vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Unit vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Basic vector operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Dot Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Properties of the dot product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Measuring angles between vectors . . . . . . . . . . . . . . . . . . . . . . . . 135

Orthogonal vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Complex Numbers and Trigonometry . . . . . . . . . . . . . . . . . . . . . . . 137

Basic operations with complex numbers . . . . . . . . . . . . . . . . . . . 138

Trigonometric form of a complex number . . . . . . . . . . . . . . . . . . 140

Multiplying and dividing complex numbers . . . . . . . . . . . . . . . . 141

Roots and Powers of Complex Numbers . . . . . . . . . . . . . . . . . . . . . 142

DeMoivre’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

Calculating nth roots of complex numbers . . . . . . . . . . . . . . . . . 143

Chapter 8: Analytic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .145

Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Parabolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Ellipses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Standard form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Eccentricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

Hyperbolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Standard form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Graphing hyperbolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

Equations of asymptote lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

Identifying Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Parametric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Graphing parametric equations . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Rewriting parametric equations . . . . . . . . . . . . . . . . . . . . . . . . . . 164

Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

Converting between polar and rectangular coordinates . . . . . . . . 167

Converting between polar and rectangular equations . . . . . . . . . 168

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Chapter 9: Matrices and Systems of Equations . . . . . . . . . . . . . . . . .171

Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Two-Variable Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

Substitution method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

Elimination method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

Nonlinear Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

Characteristics of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

Basic Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

Adding matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

Scalar multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Subtracting matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Multiplying matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

Solving Systems of Equations with Matrices . . . . . . . . . . . . . . . . . . . 180

Gaussian and Gauss-Jordan elimination . . . . . . . . . . . . . . . . . . . 181

Matrix row operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Systems with inﬁnitely many solutions . . . . . . . . . . . . . . . . . . . . 184

Inverse Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Calculating inverse matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

Solving matrix equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Determinants of 2 ×2 matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Minors and cofactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Determinants of square matrices . . . . . . . . . . . . . . . . . . . . . . . . . 190

Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

Graphs of Two-Variable Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 194

Single inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

Systems of inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

Chapter 10: Additional Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .201

Binomial Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Pascal’s Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

Factorials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

The Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

Ordered Number Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

CQR Resource Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .208

Glossary for CQR Precalculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

Table of Contents vii

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