Introduction 1 About This Book 1 Foolish Assumptions 2 Icons Used in This Book 3 Beyond the Book 3 Where to Go from Here 3 Part 1: Getting Started with Pre-Calculus 5 Chapter 1: Pre-Pre-Calculus 7 Pre-Calculus: An Overview 8 All the Number Basics (No, Not How to Count Them!) 9 The multitude of number types: Terms to know 9 The fundamental operations you can perform on numbers 11 The properties of numbers: Truths to remember 11 Visual Statements: When Math Follows Form with Function 12 Basic terms and concepts 13 Graphing linear equalities and inequalities 14 Gathering information from graphs 15 Get Yourself a Graphing Calculator 16 Chapter 2: Playing with Real Numbers 19 Solving Inequalities 19 Recapping inequality how-tos 20 Solving equations and inequalities when absolute value is involved 20 Expressing solutions for inequalities with interval notation 22 Variations on Dividing and Multiplying: Working with Radicals and Exponents 24 Defining and relating radicals and exponents 24 Rewriting radicals as exponents (or, creating rational exponents) 25 Getting a radical out of a denominator: Rationalizing 26 Chapter 3: The Building Blocks of Pre-Calculus Functions 31 Qualities of Special Function Types and Their Graphs 32 Even and odd functions 32 One-to-one functions 32 Dealing with Parent Functions and Their Graphs 33 Linear functions 33 Quadratic functions 33 Square-root functions 34 Absolute-value functions 34 Cubic functions 35 Cube-root functions 36 Graphing Functions That Have More Than One Rule: Piece-Wise Functions 37 Setting the Stage for Rational Functions 38 Step 1: Search for vertical asymptotes 39 Step 2: Look for horizontal asymptotes 40 Step 3: Seek out oblique asymptotes 41 Step 4: Locate the x- and y-intercepts 42 Putting the Results to Work: Graphing Rational Functions 42 Chapter 4: Operating on Functions 49 Transforming the Parent Graphs 50 Stretching and flattening 50 Translations 52 Reflections 54 Combining various transformations (a transformation in itself!) 55 Transforming functions point by point 57 Sharpen Your Scalpel: Operating on Functions 58 Adding and subtracting 59 Multiplying and dividing 60 Breaking down a composition of functions 60 Adjusting the domain and range of combined functions (if applicable) 61 Turning Inside Out with Inverse Functions 63 Graphing an inverse 64 Inverting a function to find its inverse 66 Verifying an inverse 66 Chapter 5: Digging Out and Using Roots to Graph Polynomial Functions 69 Understanding Degrees and Roots 70 Factoring a Polynomial Expression 71 Always the first step: Looking for a GCF 72 Unwrapping the box containing a trinomial 73 Recognizing and factoring special polynomials 74 Grouping to factor four or more terms 77 Finding the Roots of a Factored Equation 78 Cracking a Quadratic Equation When It Won't Factor 79 Using the quadratic formula 79 Completing the square 80 Solving Unfactorable Polynomials with a Degree Higher Than Two 81 Counting a polynomial's total roots 82 Tallying the real roots: Descartes's rule of signs 82 Accounting for imaginary roots: The fundamental theorem of algebra 83 Guessing and checking the real roots 84 Put It in Reverse: Using Solutions to Find Factors 90 Graphing Polynomials 91 When all the roots are real numbers 91 When roots are imaginary numbers: Combining all techniques 95 Chapter 6: Exponential and Logarithmic Functions 97 Exploring Exponential Functions 98 Searching the ins and outs of exponential functions 98 Graphing and transforming exponential functions 100 Logarithms: The Inverse of Exponential Functions 102 Getting a better handle on logarithms 102 Managing the properties and identities of logs 103 Changing a log's base 105 Calculating a number when you know its log: Inverse logs 105 Graphing logs 106 Base Jumping to Simplify and Solve Equations 109 Stepping through the process of exponential equation solving 109 Solving logarithmic equations 112 Growing Exponentially: Word Problems in the Kitchen 113 Part 2: The Essentials of Trigonometry 117 Chapter 7: Circling in on Angles 119 Introducing Radians: Circles Weren't Always Measured in Degrees 120 Trig Ratios: Taking Right Triangles a Step Further 121 Making a sine 121 Looking for a cosine 122 Going on a tangent 124 Discovering the flip side: Reciprocal trig functions 125 Working in reverse: Inverse trig functions 126 Understanding How Trig Ratios Work on the Coordinate Plane 127 Building the Unit Circle by Dissecting the Right Way 129 Familiarizing yourself with the most common angles 129 Drawing uncommon angles 131 Digesting Special Triangle Ratios 132 The 45er: 45 -45 -90 triangle 132 The old 30-60: 30 -60 -90 triangle 133 Triangles and the Unit Circle: Working Together for the Common Good 135 Placing the major angles correctly, sans protractor 135 Retrieving trig-function values on the unit circle 138 Finding the reference angle to solve for angles on the unit circle 142 Measuring Arcs: When the Circle Is Put in Motion 146 Chapter 8: Simplifying the Graphing and Transformation of Trig Functions 149 Drafting the Sine and Cosine Parent Graphs 150 Sketching sine 150 Looking at cosine 152 Graphing Tangent and Cotangent 154 Tackling tangent 154 Clarifying cotangent 157 Putting Secant and Cosecant in Pictures 159 Graphing secant 159 Checking out cosecant 161 Transforming Trig Graphs 162 Messing with sine and cosine graphs 163 Tweaking tangent and cotangent graphs 173 Transforming the graphs of secant and cosecant 176 Chapter 9: Identifying with Trig Identities: The Basics 181 Keeping the End in Mind: A Quick Primer on Identities 182 Lining Up the Means to the End: Basic Trig Identities 182 Reciprocal and ratio identities 183 Pythagorean identities 185 Even/odd identities 188 Co-function identities 190 Periodicity identities 192 Tackling Difficult Trig Proofs: Some Techniques to Know 194 Dealing with demanding denominators 195 Going solo on each side 199 Chapter 10: Advanced Identities: Your Keys to Success 201 Finding Trig Functions of Sums and Differences 202 Searching out the sine of a b 202 Calculating the cosine of a b 206 Taming the tangent of a b 209 Doubling an Angle and Finding Its Trig Value 211 Finding the sine of a doubled angle 212 Calculating cosines for two 213 Squaring your cares away 215 Having twice the fun with tangents 216 Taking Trig Functions of Common Angles Divided in Two 217 A Glimpse of Calculus: Traveling from Products to Sums and Back 219 Expressing products as sums (or differences) 219 Transporting from sums (or differences) to products 220 Eliminating Exponents with Power-Reducing Formulas 221 Chapter 11: Taking Charge of Oblique Triangles with the Laws of Sines and Cosines 223 Solving a Triangle with the Law of Sines 224 When you know two angle measures 225 When you know two consecutive side lengths 228 Conquering a Triangle with the Law of Cosines 235 SSS: Finding angles using only sides 236 SAS: Tagging the angle in the middle (and the two sides) 238 Filling in the Triangle by Calculating Area 240 Finding area with two sides and an included angle (for SAS scenarios) 241 Using Heron's Formula (for SSS scenarios) 241 Part 3: Analytic Geometry and System Solving 243 Chapter 12: Plane Thinking: Complex Numbers and Polar Coordinates 245 Understanding Real versus Imaginary 246 Combining Real and Imaginary: The Complex Number System 247 Grasping the usefulness of complex numbers 247 Performing operations with complex numbers 248 Graphing Complex Numbers 250 Plotting Around a Pole: Polar Coordinates 251 Wrapping your brain around the polar coordinate plane 252 Graphing polar coordinates with negative values 254 Changing to and from polar coordinates 256 Picturing polar equations 259 Chapter 13: Creating Conics by Slicing Cones 263 Cone to Cone: Identifying the Four Conic Sections 264 In picture (graph form) 264 In print (equation form) 266 Going Round and Round: Graphing Circles 267 Graphing circles at the origin 267 Graphing circles away from the origin 268 Writing in center-radius form 269 Riding the Ups and Downs with Parabolas 270 Labeling the parts 270 Understanding the characteristics of a standard parabola 271 Plotting the variations: Parabolas all over the plane 272 The vertex, axis of symmetry, focus, and directrix 273 Identifying the min and max of vertical parabolas 276 The Fat and the Skinny on the Ellipse 278 Labeling ellipses and expressing them with algebra 279 Identifying the parts from the equation 281 Pair Two Curves and What Do You Get? Hyperbolas 284 Visualizing the two types of hyperbolas and their bits and pieces 284 Graphing a hyperbola from an equation 287 Finding the equations of asymptotes 287 Expressing Conics Outside the Realm of Cartesian Coordinates 289 Graphing conic sections in parametric form 290 The equations of conic sections on the polar coordinate plane 292 Chapter 14: Streamlining Systems, Managing Variables 295 A Primer on Your System-Solving Options 296 Algebraic Solutions of Two-Equation Systems 297 Solving linear systems 297 Working nonlinear systems 300 Solving Systems with More than Two Equations 304 Decomposing Partial Fractions 306 Surveying Systems of Inequalities 307 Introducing Matrices: The Basics 309 Applying basic operations to matrices 310 Multiplying matrices by each other 311 Simplifying Matrices to Ease the Solving Process 312 Writing a system in matrix form 313 Reduced row-echelon form 313 Augmented form 314 Making Matrices Work for You 315 Using Gaussian elimination to solve systems 316 Multiplying a matrix by its inverse 320 Using determinants: Cramer's Rule 323 Chapter 15: Sequences, Series, and Expanding Binomials for the Real World 327 Speaking Sequentially: Grasping the General Method 328 Determining a sequence's terms 328 Working in reverse: Forming an expression from terms 329 Recursive sequences: One type of general sequence 330 Difference between Terms: Arithmetic Sequences 331 Using consecutive terms to find another 332 Using any two terms 332 Ratios and Consecutive Paired Terms: Geometric Sequences 334 Identifying a particular term when given consecutive terms 334 Going out of order: Dealing with nonconsecutive terms 335 Creating a Series: Summing Terms of a Sequence 337 Reviewing general summation notation 337 Summing an arithmetic sequence 338 Seeing how a geometric sequence adds up 339 Expanding with the Binomial Theorem 342 Breaking down the binomial theorem 344 Expanding by using the binomial theorem 345 Chapter 16: Onward to Calculus 351 Scoping Out the Differences between Pre-Calculus and Calculus 352 Understanding Your Limits 353 Finding the Limit of a Function 355 Graphically 355 Analytically 356 Algebraically 357 Operating on Limits: The Limit Laws 361 Calculating the Average Rate of Change 362 Exploring Continuity in Functions 363 Determining whether a function is continuous 364 Discontinuity in rational functions 365 Part 4: The Part of Tens 367 Chapter 17: Ten Polar Graphs 369 Spiraling Outward 369 Falling in Love with a Cardioid 370 Cardioids and Lima Beans 370 Leaning Lemniscates 371 Lacing through Lemniscates 372 Roses with Even Petals 372 A rose Is a Rose Is a Rose 373 Limacon or Escargot? 373 Limacon on the Side 374 Bifolium or Rabbit Ears? 374 Chapter 18: Ten Habits to Adjust before Calculus 375 Figure Out What the Problem Is Asking 375 Draw Pictures (the More the Better) 376 Plan Your Attack - Identify Your Targets 377 Write Down Any Formulas 377 Show Each Step of Your Work 378 Know When to Quit 378 Check Your Answers 379 Practice Plenty of Problems 380 Keep Track of the Order of Operations 380 Use Caution When Dealing with Fractions 381 Index 383
