Pre-Calculus Workbook For Dummies

Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of Trigonometry For Dummies and Finite Math For Dummies.

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