Pre-Calculus For Dummies

Mary Jane Sterling aught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies.

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