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Table of Contents
Introduction 1
About This Book 1
Foolish Assumptions 2
Icons Used in This Book 2
Beyond the Book 3
Where to Go from Here 3
Part 1: Setting the Foundation: The Nuts And Bolts of Pre-Calculus 5
Chapter 1: Preparing for Pre-Calculus 7
Reviewing Order of Operations: The Fun in Fundamentals 8
Keeping Your Balance While Solving Equalities 10
When Your Image Really Counts: Graphing Equalities and Inequalities 12
Graphing with two points 12
Graphing by using the slope-intercept form 13
Graphing inequalities 14
Using Graphs to Find Distance, Midpoint, and Slope 15
Finding the distance 15
Calculating the midpoint 16
Discovering the slope 16
Answers to Problems on Fundamentals 19
Chapter 2: Real Numbers Come Clean 25
Solving Inequalities 25
Expressing Inequality Solutions in Interval Notation 28
Radicals and Exponents — Just Simplify! 30
Getting Out of a Sticky Situation, or Rationalizing 33
Answers to Problems on Real Numbers 35
Chapter 3: Controlling Functions by Knowing Their Function 39
Using Both Faces of the Coin: Even and Odd 40
Leaving the Nest: Transforming Parent Graphs 42
Quadratic functions 42
Square root functions 42
Absolute value functions 43
Cubic functions 43
Cube root functions 44
Steeper or flatter 44
Translations 46
Reflections 46
Combinations of transformations 46
Graphing Rational Functions 49
Piecing Together Piecewise Functions 52
Combining Functions 54
Evaluating Composition of Functions 55
Working Together: Domain and Range 57
Unlocking the Inverse of a Function: Turning It Inside Out 59
Answers to Problems on Functions 61
Chapter 4: Searching for Roots 75
Factoring a Factorable Quadratic 75
Solving a Quadratic Polynomial Equation 78
Completing the square 78
Quadratic formula 79
Solving High-Order Polynomials 80
Factoring by grouping 80
Determining positive and negative roots: Descartes’ Rule of Signs 81
Counting on imaginary roots 81
Getting the rational roots 81
Finding roots through synthetic division 82
Using Roots to Create an Equation 84
Graphing Polynomials 85
Answers to Problems on Roots and Degrees 89
Chapter 5: Exponential and Logarithmic Functions 95
Working with Exponential Functions 95
Eagerly Engaging Edgy Logarithmic Solutions 98
Making Exponents and Logs Work Together 101
Using Exponents and Logs in Practical Applications 103
Answers to Problems on Exponential and Logarithmic Functions 106
Part 2: Trig is the Key: Basic Review, The Unit Circle, and Graphs 113
Chapter 6: Basic Trigonometry and the Unit Circle 115
Finding the Six Trigonometric Ratios 115
Solving Word Problems with Right Triangles 118
Unit Circle and the Coordinate Plane: Finding Points and Angles 121
Finding Ratios from Angles on the Unit Circle 124
Solving Trig Equations 127
Making and Measuring Arcs 129
Answers to Problems on Basic Trig and the Unit Circle 131
Chapter 7: Graphing and Transforming Trig Functions 137
Getting a Grip on Periodic Graphs 137
Parent Graphs and Transformations: Sine and Cosine 138
Tangent and Cotangent: More Family Members 141
Generations: Secant and Cosecant 143
Answers to Problems on Graphing and Transforming Trig Functions 147
Part 3: Digging Into Advanced Trig: Identities, Theorems, and Applications 155
Chapter 8: Basic Trig Identities 157
Using Reciprocal Identities to Simplify Trig Expressions 157
Simplifying with Pythagorean Identities 159
Discovering Even-Odd Identities 160
Simplifying with Co-Function Identities 162
Moving with Periodicity Identities 163
Tackling Trig Proofs (Identities) 165
Answers to Problems on Basic Trig Identities 167
Chapter 9: Advanced Trig Identities 175
Simplifying with Sum and Difference Identities 175
Using Double-Angle Identities 178
Reducing with Half-Angle Identities 180
Changing Products to Sums 181
Expressing Sums as Products 182
Powering Down: Power-Reducing Formulas 184
Answers to Problems on Advanced Trig Identities 186
Chapter 10: Solving Oblique Triangles 193
Solving a Triangle with the Law of Sines: ASA and AAS 194
Tackling Triangles in the Ambiguous Case: SSA 195
Conquering a Triangle with the Law of Cosines: SAS and SSS 197
Using Oblique Triangles to Solve Practical Applications 198
Figuring Area 201
Answers to Problems on Solving Triangles 202
Part 4: Polar Coordinates, Cones, Solutions, Sequences, and Finding Your Limits 209
Chapter 11: Exploring Complex Numbers and Polar Coordinates 211
Performing Operations with and Graphing Complex Numbers 212
Round a Pole: Graphing Polar Coordinates 215
Changing to and from Polar 217
Graphing Polar Equations 220
Archimedean spiral 220
Cardioid 220
Rose 220
Circle 220
Lemniscate 220
Limaçon 221
Answers to Problems on Complex Numbers and Polar Coordinates 223
Chapter 12: Conquering Conic Sections 229
A Quick Conic Review 230
Going Round and Round with Circles 230
The Ups and Downs: Graphing Parabolas 232
Standing tall: Vertical parabolas 233
Lying down on the job: Horizontal parabolas 235
The Fat and the Skinny: Graphing Ellipses 237
Short and fat: The horizontal ellipse 237
Tall and skinny: The vertical ellipse 239
No Caffeine Required: Graphing Hyperbolas 241
Horizontal hyperbolas 241
Vertical hyperbolas 244
Identifying Conic Sections 246
Conic Sections in Parametric Form and Polar Coordinates 248
Parametric form for conic sections 248
Changing from parametric form to rectangular form 250
Conic sections on the polar coordinate plane 251
Answers to Problems on Conic Sections 253
Chapter 13: Finding Solutions for Systems of Equations 265
A Quick-and-Dirty Technique Overview 266
Solving Two Linear Equations with Two Variables 266
The substitution method 267
The elimination method 268
Not-So-Straight: Solving Nonlinear Systems 269
One equation that’s linear and one that isn’t 269
Two nonlinear equations 270
Systems of rational equations 271
Systems of More Than Two Equations 272
Graphing Systems of Inequalities 274
Breaking Down Decomposing Partial Fractions 276
Working with a Matrix 278
Getting It in the Right Form: Simplifying Matrices 281
Solving Systems of Equations Using Matrices 283
Gaussian elimination 283
Inverse matrices 285
Cramer’s Rule 287
Answers to Problems on Systems of Equations 289
Chapter 14: Spotting Patterns in Sequences and Series 301
General Sequences and Series: Determining Terms 301
Working Out the Common Difference: Arithmetic Sequences and Series 303
Simplifying Geometric Sequences and Series 305
Expanding Polynomials Using the Binomial Theorem 308
Answers to Problems on Sequences, Series, and Binomials 310
Chapter 15: Previewing Calculus 315
Finding Limits: Graphically, Analytically, and Algebraically 316
Graphically 316
Analytically 318
Algebraically 319
Knowing Your Limits 321
Calculating the Average Rate of Change 322
Determining Continuity 323
Answers to Problems on Calculus 326
Part 5: The Part of Tens 329
Chapter 16: Ten Plus Parent Graphs 331
Squaring Up with Quadratics 331
Cueing Up for Cubics 332
Rooting for Square Roots and Cube Roots 333
Graphing Absolutely Fabulous Absolute Value Functions 334
Flipping over Rational Functions 334
Exploring Exponential Graphs and Logarithmic Graphs 335
Seeing the Sine and Cosine 336
Covering Cosecant and Secant 337
Tripping over Tangent and Cotangent 338
Lining Up and Going Straight with Lines 339
Chapter 17: Ten Missteps to Avoid 341
Going Out of Order (of Operations) 341
FOILing Binomials Incorrectly 342
Breaking Up Fractions Badly 342
Combining Terms That Can’t Be Combined 342
Forgetting to Flip the Fraction 342
Losing the Negative (Sign) 343
Oversimplifying Roots 343
Executing Exponent Errors 343
Ignoring Extraneous 344
Misinterpreting Trig Notation 344
Index 345
