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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.
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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