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Get ahead in pre-calculus


Pre-calculus courses have become increasingly popular with 35 percent of students in the U.S. taking the course in middle or high school. Often, completion of such a course is a prerequisite for calculus and other upper level mathematics courses.


Pre-Calculus For Dummies is an invaluable resource for students enrolled in pre-calculus courses. By presenting the essential topics in a clear and concise manner, the book helps students improve their understanding of pre-calculus and become prepared for upper level math courses.





Provides fundamental information in an approachable manner

Includes fresh example problems

Practical explanations mirror today's teaching methods

Offers relevant cultural references



Whether used as a classroom aid or as a refresher in preparation for an introductory calculus course, this book is one you'll want to have on hand to perform your very best.

Pre-Calculus For Dummies 3e

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