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