Pre-Calculus: 1001 Practice Problems For Dummies (+ Free Online Practice)

Practice your way to a better grade in pre-calc

Pre-Calculus: 1001 Practice Problems For Dummies gives you 1,001 opportunities to practice solving problems from all the major topics in Pre-Calculus—in the book and online! Get extra help with tricky subjects, solidify what you’ve already learned, and get in-depth walk-throughs for every problem with this useful book. These practice problems and detailed answer explanations will turn you into a pre-calc problem-solving machine, no matter what your skill level. Thanks to Dummies, you have a resource to help you put key concepts into practice.

• Work through practice problems on all Pre-Calculus topics covered in school classes
• Read through detailed explanations of the answers to build your understanding
• Access practice questions online to study anywhere, any time
• Improve your grade and up your study game with practice, practice, practice

The material presented in Pre-Calculus: 1001 Practice Problems For Dummies is an excellent resource for students, as well as for parents and tutors looking to help supplement Pre-Calculus instruction.

Pre-Calculus: 1001 Practice Problems For Dummies (9781119883623) was previously published as 1,001 Pre-Calculus Practice Problems For Dummies (9781118853320). While this version features a new Dummies cover and design, the content is the same as the prior release and should not be considered a new or updated product.

Mary Jane Sterling is the author of Algebra I For Dummies, Algebra Workbook For Dummies, Algebra II For Dummies, Algebra II Workbook For Dummies, and oodles of other Dummies titles. She was a Professor of Mathematics at Bradley University in Peoria, Illinois, for more than 30 years.

Introduction 1

What You’ll Find 1

How This Workbook Is Organized 2

Part 1: The Questions 2

Part 2: The Answers 3

Beyond the Book 3

Where to Go for Additional Help 4

Part 1: The Questions 5

Chapter 1: Getting Started with Algebra Basics 7

The Problems You’ll Work On 7

What to Watch Out For 7

Identifying Which System or Systems a Number Belongs To 8

Recognizing Properties of Number Systems 9

Simplifying Expressions with the Order of Operations 9

Graphing Inequalities 10

Using Graphing Formulas 11

Applying Graphing Formulas 11

Chapter 2: Solving Some Equations and Inequalities 13

The Problems You’ll Work On 13

What to Watch Out For 13

Using Interval and Inequality Notation 14

Solving Linear Inequalities 15

Solving Quadratic Inequalities 15

Solving Absolute Value Inequalities 15

Working with Radicals and Fractional Notation 16

Performing Operations Using Fractional Exponents 16

Factoring Using Fractional Notation 16

Solving Radical Equations 17

Rationalizing Denominators 18

Chapter 3: Function Basics 19

The Problems You’ll Work On 19

What to Watch Out For 19

Using Function Notation to Evaluate Function Values 20

Determining the Domain and Range of a Function 20

Recognizing Even Functions 21

Identifying Odd Functions 21

Ruling Out Even and Odd Functions 21

Recognizing One-to-One Functions from Given Relations 22

Identifying One-to-One Functions from Equations 23

Recognizing a Function’s Inverse 23

Determining a Function’s Inverse 24

Executing Operations on Functions 24

Performing Function Composition 25

Doing More Function Composition 25

Using the Difference Quotient 26

Chapter 4: Graphing and Transforming Functions 27

The Problems You’ll Work On 27

What to Watch Out For 28

Functions and Their Inverses 28

Sketching Quadratic Functions from Their Equations 29

Writing Equations from Graphs of Parabolas 29

Investigating and Graphing Radical Functions 30

Investigating Absolute Value Functions 31

Investigating the Graphs of Polynomial Functions 31

Investigating Rational Functions 32

Transformation of Functions 32

Transforming Selected Points Using Functions 33

Sketching Graphs Using Basic Functions and Transformations 33

Sketching More Graphs Using Basic Functions and Transformations 34

Chapter 5: Polynomials 35

The Problems You’ll Work On 35

What to Watch Out For 36

Using Factoring to Solve Quadratic Equations 36

Solving Quadratic Equations by Using the Quadratic Formula 37

Using Completing the Square to Solve Quadratic Equations 37

Solving Polynomial Equations for Intercepts 37

Using Factoring by Grouping to Solve Polynomial Equations 38

Applying Descartes’s Rule of Signs 38

Listing Possible Roots of a Polynomial Equation 39

Dividing Polynomials 39

Using Synthetic Division to Divide Polynomials 39

Checking for Roots of a Polynomial by Using Synthetic Division 40

Writing Polynomial Expressions from Given Roots 40

Writing Polynomial Expressions When Given Roots and a Point 40

Graphing Polynomials 41

Writing Equations from Graphs of Polynomials 41

Chapter 6: Exponential and Logarithmic Functions 43

The Problems You’ll Work On 43

What to Watch Out For 44

Understanding Function Notation 44

Graphing Exponential Functions 44

Solving Exponential Equations 45

Using the Equivalence b y b y x x log = to Rewrite Expressions 46

Using the Equivalence logb y x b x y to Rewrite Expressions 46

Rewriting Logarithmic Expressions 46

Rewriting Logs of Products and Quotients as Sums and Differences 47

Solving Logarithmic Equations 47

Applying Function Transformations to Log Functions 48

Applying Logarithms to Everyday Life 49

Chapter 7: Trigonometry Basics 51

The Problems You’ll Work On 51

What to Watch Out For 52

Using Right Triangles to Determine Trig Functions 52

Solving Problems by Using Right Triangles and Their Functions 53

Working with Special Right Triangles 54

Changing Radians to Degrees 55

Changing Degrees to Radians 55

Finding Angle Measures (in Degrees) in Standard Position 55

Determining Angle Measures (in Radians) in Standard Position 55

Identifying Reference Angles 56

Determining Trig Functions by Using the Unit Circle 56

Calculating Trig Functions by Using Other Functions and Terminal Side Positions 56

Using the Arc Length Formula 57

Evaluating Inverse Functions 57

Solving Trig Equations for x in Degrees 58

Calculating Trig Equations for x in Radians 58

Chapter 8: Graphing Trig Functions 59

The Problems You’ll Work On 59

What to Watch Out For 60

Recognizing Basic Trig Graphs 60

Graphing Sine and Cosine 62

Applying Function Transformations to Graphs of Trig Functions 62

Writing New Trig Functions Using Transformations 62

Graphing Tangent and Cotangent 63

Interpreting Transformations of Trig Functions 63

Graphing Secant and Cosecant 64

Interpreting Transformations from Function Rules 64

Chapter 9: Getting Started with Trig Identities 65

The Problems You’ll Work On 65

What to Watch Out For 66

Proving Basic Trig Identities 66

Returning to Basic Sine and Cosine to Solve Identities 67

Using Multiplication by a Conjugate to Solve Identities 68

Solving Identities After Raising a Binomial to a Power 68

Solving Identities After Factoring out a Common Function 69

Solving Identities After Combining Fractions 69

Performing Algebraic Processes to Make Identities More Solvable 69

Chapter 10: Continuing with Trig Identities 71

The Problems You’ll Work On 71

What to Watch Out For 71

Using Identities That Add or Subtract Angle Measures 72

Confirming Double-Angle Identities 72

Using Identities That Double the Size of the Angle 72

Confirming the Statements of Multiple-Angle Identities 72

Creating Half-Angle Identities from Double-Angle Identities 73

Creating a Half-Angle Identity for Tangent 73

Using Half-Angle Identities to Simplify Expressions 73

Creating Products of Trig Functions from Sums and Differences 73

Using Product-to-Sum Identities to Evaluate Expressions 74

Using Sum-to-Product Identities to Evaluate Expressions 74

Applying Power-Reducing Identities 74

Using Identities to Determine Values of Functions at Various Angles 75

Working through Identities Using Multiple Methods 75

Chapter 11: Working with Triangles and Trigonometry 77

The Problems You’ll Work On 77

What to Watch Out For 78

Applying the Law of Sines to Find Sides 78

Utilizing the Law of Sines to Find Angles 78

Using the Law of Sines for Practical Applications 79

Investigating the Ambiguous Case of the Law of Sines 79

Determining All Angles and Sides of a Triangle 80

Finding Side Measures by Using the Law of Cosines 80

Using the Law of Cosines to Determine an Angle 80

Applying the Law of Cosines to Real-World Situations 81

Finding Areas of Triangles by Using the Sine 81

Applying the Trig Formula for Area of a Triangle 82

Using the Trig Formula for Area in Various Situations 82

Solving Area Problems Needing Additional Computations 83

Finding Areas of Triangles by Using Heron’s Formula 84

Applying Heron’s Formula 84

Practical Applications Using Heron’s Formula 85

Tackling Practical Applications by Using Triangular Formulas 85

Chapter 12: Complex Numbers and Polar Coordinates 87

The Problems You’ll Work On 87

What to Watch Out For 88

Writing Powers of i in Their Simplest Form 88

Adding and Subtracting Complex Numbers 88

Multiplying Complex Numbers 89

Using Multiplication to Divide Complex Numbers 90

Solving Quadratic Equations with Complex Solutions 90

Graphing Complex Numbers 91

Identifying Points with Polar Coordinates 92

Identifying Points Whose Angles Have Negative Measures 93

Converting Polar to Rectangular Coordinates 93

Converting Rectangular to Polar Coordinates 94

Recognizing Polar Curves 94

Chapter 13: Conic Sections 95

The Problems You’ll Work On 95

What to Watch Out For 95

Identifying Conics from Their Equations 96

Rewriting Conic Equations in Standard Form 96

Writing Equations for Circles 97

Determining Foci and Axes of Symmetry of Parabolas 97

Finding the Vertices and Directrixes of Parabolas 98

Writing Equations of Parabolas 98

Determining Centers and Foci of Ellipses 98

Writing Equations of Ellipses 99

Determining Asymptotes of Hyperbolas 99

Writing Equations of Hyperbolas 99

Changing Equation Format from Trig Functions to Algebraic 100

Changing Equation Format from Algebraic to Trig 100

Chapter 14: Systems of Equations and Inequalities 101

The Problems You’ll Work On 101

What to Watch Out For 102

Using Substitution to Solve Systems of Linear Equations with Two Variables 102

Using Elimination to Solve Systems of Linear Equations with Two Variables 102

Solving Systems of Equations Involving Nonlinear Functions 103

Solving Systems of Linear Equations 103

Solving Systems of Linear Equations with Four Variables 104

Graphing Systems of Inequalities 104

Decomposition of Fractions 104

Operating on Matrices 105

Changing Matrices to the Echelon Form 105

Solving Systems of Equations Using Augmented Matrices 106

Solving Systems of Equations Using the Inverse of the Coefficient Matrix 106

Applying Cramer’s Rule to Solve Systems of Equations 107

Chapter 15: Sequences and Series 109

The Problems You’ll Work On 109

What to Watch Out For 109

Finding Terms of Sequences 110

Determining Rules for Sequences 110

Working with Recursively Defined Sequences 111

Adding Terms in an Arithmetic Series 111

Summing Terms of a Series 111

Finding Rules and Summing Terms of a Series 112

Calculating the Sum of a Geometric Series 112

Determining Formulas and Finding Sums 112

Counting Items by Using Combinations 113

Constructing Pascal’s Triangle 113

Applying Pascal’s Triangle 114

Utilizing the Binomial Theorem 114

Chapter 16: Introducing Limits and Continuity 115

The Problems You’ll Work On 115

What to Watch Out For 116

Determining Limits from Graphs 116

Determining One-Sided Limits 117

Determining Limits from Function Values 118

Determining Limits from Function Rules 119

Applying Laws of Limits 121

Investigating Continuity 121

Part 2: The Answers 123

Chapter 17: Answers 125

Index 539

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