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.