Table of Contents

Introduction 1

What You’ll Find 1

How This Workbook Is Organized 2

Part I: The Questions 2

Part II: The Answers 3

Beyond the Book 3

What you’ll find online 3

How to register 4

Where to Go for Additional Help 4

Part I: The Questions 7

Chapter 1: Getting Started with Algebra Basics 9

The Problems You’ll Work On 9

What to Watch Out For 9

Identifying Which System or Systems a Number Belongs To 10

Recognizing Properties of Number Systems 10

Simplifying Expressions with the Order of Operations 11

Graphing Inequalities 12

Using Graphing Formulas 13

Applying Graphing Formulas 13

Chapter 2: Solving Some Equations and Inequalities 15

The Problems You’ll Work On 15

What to Watch Out For 15

Using Interval and Inequality Notation 16

Solving Linear Inequalities 17

Solving Quadratic Inequalities 17

Solving Absolute Value Inequalities 17

Working with Radicals and Fractional Notation 18

Performing Operations Using Fractional Exponents 18

Factoring Using Fractional Notation 19

Solving Radical Equations 19

Rationalizing Denominators 20

Chapter 3: Function Basics 21

The Problems You’ll Work On 21

What to Watch Out For 21

Using Function Notation to Evaluate Function Values 22

Determining the Domain and Range of a Function 22

Recognizing Even Functions 23

Identifying Odd Functions 23

Ruling Out Even and Odd Functions 23

Recognizing One-to-One Functions from Given Relations 23

Identifying One-to-One Functions from Equations 25

Recognizing a Function’s Inverse 25

Determining a Function’s Inverse 26

Executing Operations on Functions 26

Performing Function Composition 27

Doing More Function Composition 27

Using the Difference Quotient 28

Chapter 4: Graphing and Transforming Functions 29

The Problems You’ll Work On 29

What to Watch Out For 29

Functions and Their Inverses 30

Sketching Quadratic Functions from Their Equations 30

Writing Equations from Graphs of Parabolas 31

Investigating and Graphing Radical Functions 32

Investigating Absolute Value Functions 33

Investigating the Graphs of Polynomial Functions 33

Investigating Rational Functions 34

Transformation of Functions 34

Transforming Selected Points Using Functions 34

Sketching Graphs Using Basic Functions and Transformations 35

Sketching More Graphs Using Basic Functions and Transformations 35

Chapter 5: Polynomials 37

The Problems You’ll Work On 37

What to Watch Out For 37

Using Factoring to Solve Quadratic Equations 38

Solving Quadratic Equations by Using the Quadratic Formula 38

Using Completing the Square to Solve Quadratic Equations 39

Solving Polynomial Equations for Intercepts 39

Using Factoring by Grouping to Solve Polynomial Equations 40

Applying Descartes’s Rule of Signs 40

Listing Possible Roots of a Polynomial Equation 40

Dividing Polynomials 41

Using Synthetic Division to Divide Polynomials 41

Checking for Roots of a Polynomial by Using Synthetic Division 41

Writing Polynomial Expressions from Given Roots 42

Writing Polynomial Expressions When Given Roots and a Point 42

Graphing Polynomials 43

Writing Equations from Graphs of Polynomials 43

Chapter 6: Exponential and Logarithmic Functions 45

The Problems You’ll Work On 45

What to Watch Out For 45

Understanding Function Notation 46

Graphing Exponential Functions 46

Solving Exponential Equations 47

Using the Equivalence bx = y ⇔ logby = x to Rewrite Expressions 48

Using the Equivalence logby = x ⇔ bx = y to Rewrite Expressions 48

Rewriting Logarithmic Expressions 48

Rewriting Logs of Products and Quotients as Sums and Differences 49

Solving Logarithmic Equations 49

Applying Function Transformations to Log Functions 50

Applying Logarithms to Everyday Life 51

Chapter 7: Trigonometry Basics 53

The Problems You’ll Work On 53

What to Watch Out For 53

Using Right Triangles to Determine Trig Functions 54

Solving Problems by Using Right Triangles and Their Functions 55

Working with Special Right Triangles 56

Changing Radians to Degrees 57

Changing Degrees to Radians 57

Finding Angle Measures (in Degrees) in Standard Position 57

Determining Angle Measures (in Radians) in Standard Position 58

Identifying Reference Angles 58

Determining Trig Functions by Using the Unit Circle 58

Calculating Trig Functions by Using Other Functions and Terminal Side Positions 59

Using the Arc Length Formula 59

Evaluating Inverse Functions 60

Solving Trig Equations for x in Degrees 60

Calculating Trig Equations for x in Radians 60

Chapter 8: Graphing Trig Functions 61

The Problems You’ll Work On 61

What to Watch Out For 61

Recognizing Basic Trig Graphs 62

Graphing Sine and Cosine 64

Applying Function Transformations to Graphs of Trig Functions 64

Writing New Trig Functions Using Transformations 64

Graphing Tangent and Cotangent 65

Interpreting Transformations of Trig Functions 65

Graphing Secant and Cosecant 66

Interpreting Transformations from Function Rules 66

Chapter 9: Getting Started with Trig Identities 67

The Problems You’ll Work On 67

What to Watch Out For 67

Proving Basic Trig Identities 68

Returning to Basic Sine and Cosine to Solve Identities 69

Using Multiplication by a Conjugate to Solve Identities 70

Solving Identities After Raising a Binomial to a Power 70

Solving Identities After Factoring out a Common Function 70

Solving Identities After Combining Fractions 71

Performing Algebraic Processes to Make Identities More Solvable 71

Chapter 10: Continuing with Trig Identities 73

The Problems You’ll Work On 73

What to Watch Out For 73

Using Identities That Add or Subtract Angle Measures 74

Confirming Double-Angle Identities 74

Using Identities That Double the Size of the Angle 74

Confirming the Statements of Multiple-Angle Identities 74

Creating Half-Angle Identities from Double-Angle Identities 75

Creating a Half-Angle Identity for Tangent 75

Using Half-Angle Identities to Simplify Expressions 75

Creating Products of Trig Functions from Sums and Differences 75

Using Product-to-Sum Identities to Evaluate Expressions 75

Using Sum-to-Product Identities to Evaluate Expressions 76

Applying Power-Reducing Identities 76

Using Identities to Determine Values of Functions at Various Angles 76

Working through Identities Using Multiple Methods 77

Chapter 11: Working with Triangles and Trigonometry 79

The Problems You’ll Work On 79

What to Watch Out For 79

Applying the Law of Sines to Find Sides 80

Utilizing the Law of Sines to Find Angles 80

Using the Law of Sines for Practical Applications 81

Investigating the Ambiguous Case of the Law of Sines 81

Determining All Angles and Sides of a Triangle 82

Finding Side Measures by Using the Law of Cosines 82

Using the Law of Cosines to Determine an Angle 82

Applying the Law of Cosines to Real-World Situations 83

Finding Areas of Triangles by Using the Sine 83

Applying the Trig Formula for Area of a Triangle 84

Using the Trig Formula for Area in Various Situations 84

Solving Area Problems Needing Additional Computations 85

Finding Areas of Triangles by Using Heron’s Formula 86

Applying Heron’s Formula 86

Practical Applications Using Heron’s Formula 87

Tackling Practical Applications by Using Triangular Formulas 87

Chapter 12: Complex Numbers and Polar Coordinates 89

The Problems You’ll Work On 89

What to Watch Out For 89

Writing Powers of i in Their Simplest Form 90

Adding and Subtracting Complex Numbers 90

Multiplying Complex Numbers 91

Using Multiplication to Divide Complex Numbers 91

Solving Quadratic Equations with Complex Solutions 92

Graphing Complex Numbers 92

Identifying Points with Polar Coordinates 94

Identifying Points Whose Angles Have Negative Measures 94

Converting Polar to Rectangular Coordinates 95

Converting Rectangular to Polar Coordinates 95

Recognizing Polar Curves 96

Chapter 13: Conic Sections 97

The Problems You’ll Work On 97

What to Watch Out For 97

Identifying Conics from Their Equations 98

Rewriting Conic Equations in Standard Form 98

Writing Equations for Circles 98

Determining Foci and Axes of Symmetry of Parabolas 99

Finding the Vertices and Directrixes of Parabolas 99

Writing Equations of Parabolas 100

Determining Centers and Foci of Ellipses 100

Writing Equations of Ellipses 100

Determining Asymptotes of Hyperbolas 101

Writing Equations of Hyperbolas 101

Changing Equation Format from Trig Functions to Algebraic 101

Changing Equation Format from Algebraic to Trig 102

Chapter 14: Systems of Equations and Inequalities 103

The Problems You’ll Work On 103

What to Watch Out For 104

Using Substitution to Solve Systems of Linear Equations with Two Variables 104

Using Elimination to Solve Systems of Linear Equations with Two Variables 104

Solving Systems of Equations Involving Nonlinear Functions 105

Solving Systems of Linear Equations 105

Solving Systems of Linear Equations with Four Variables 106

Graphing Systems of Inequalities 106

Decomposition of Fractions 107

Operating on Matrices 107

Changing Matrices to the Echelon Form 108

Solving Systems of Equations Using Augmented Matrices 108

Solving Systems of Equations Using the Inverse of the Coefficient Matrix 109

Applying Cramer’s Rule to Solve Systems of Equations 110

Chapter 15: Sequences and Series 111

The Problems You’ll Work On 111

What to Watch Out For 111

Finding Terms of Sequences 112

Determining Rules for Sequences 112

Working with Recursively Defined Sequences 112

Adding Terms in an Arithmetic Series 113

Summing Terms of a Series 113

Finding Rules and Summing Terms of a Series 113

Calculating the Sum of a Geometric Series 114

Determining Formulas and Finding Sums 114

Counting Items by Using Combinations 114

Constructing Pascal’s Triangle 115

Applying Pascal’s Triangle 115

Utilizing the Binomial Theorem 115

Chapter 16: Introducing Limits and Continuity 117

The Problems You’ll Work On 117

What to Watch Out For 117

Determining Limits from Graphs 118

Determining One-Sided Limits 119

Determining Limits from Function Values 120

Determining Limits from Function Rules 121

Applying Laws of Limits 122

Investigating Continuity 123

Part II: The Answers 125

Chapter 17: Answers 127

Index 527

About the Author

Mary Jane Sterling is the author of several books, including Algebra I For Dummies, Algebra II For Dummies, Trigonometry For Dummies, and Linear Algebra For Dummies. She taught at Bradley University in Peoria, Illinois, for more than 35 years.