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Pre-Calculus For Dummies, 3rd Edition



Pre-Calculus For Dummies, 3rd Edition

Mary Jane Sterling

ISBN: 978-1-119-50871-7 October 2018 416 Pages

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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.

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

Limaçon or Escargot? 373

Limaçon 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