Cliff Notes - QuickReview Precalculus
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CliffsQuickReview®
Precalculus
By W. Michael Kelley
01 539841 FM.qxd 1/26/04 2:47 PM Page i
01 539841 FM.qxd 1/26/04 2:47 PM Page viii
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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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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01 539841 FM.qxd 1/26/04 2:47 PM Page ii
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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iv CliffsQuickReview Precalculus
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
vi CliffsQuickReview Precalculus
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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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01 539841 FM.qxd 1/26/04 2:47 PM Page viii
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CliffsQuickReview®PrecalculusByW.MichaelKelley01539841FM.qxd1/26/042:47PMPagei01539841FM.qxd1/26/042:47PMPageviiiCliffsQuickReview®PrecalculusByW.MichaelKelley01539841FM.qxd1/26/042:47PMPageiAbouttheAuthorMikeKelleyhasbeenahighschoolandcollegemathinstructor.HecurrentlyworksasanAca-demicTechnologyCoo...
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