Algebra II Workbook For Dummies

Mary Jane Sterling taught junior high and high school math before embarking on her career as an instructor at Bradley University, where she taught for more than 35 years.

Introduction

About This Book 1

Foolish Assumptions 2

Icons Used in This Book 2

Beyond the Book 3

Where to Go from Here 3

Part 1: Getting Started With Algebra II 5

Chapter 1: Going Beyond Beginning Algebra 7

Good Citizenship: Following the Order of Operations and Other Properties 7

Specializing in Products and FOIL 10

Variables on the Side: Solving Linear Equations 11

Dealing with Linear Absolute Value Equations 12

Greater Math Skills: Equalizing Linear Inequalities 14

Answers to Problems on Going Beyond Beginning Algebra 16

Chapter 2: Handling Quadratic (and Quadratic-Like) Equations and Inequalities 21

Finding Reasonable Solutions with Radicals 22

UnFOILed Again! Successfully Factoring for Solutions 23

Your Bag of Tricks: Factoring Multiple Ways 25

Keeping Your Act Together: Factoring by Grouping 26

Resorting to the Quadratic Formula 27

Solving Quadratics by Completing the Square 29

Working with Quadratic-Like Equations 30

Checking Out Quadratic Inequalities 32

Answers to Problems on Quadratic (and Quadratic-Like) Equations and Inequalities 34

Chapter 3: Rooting Out the Rational, the Radical, and the Negative 43

Doing Away with Denominators with an LCD 44

Simplifying and Solving Proportions 46

Wrangling with Radicals 48

Changing Negative Attitudes toward Negative Exponents 49

Divided Powers: Solving Equations with Fractional Exponents 51

Answers to Problems on Rooting Out the Rational, the Radical, and the Negative 53

Chapter 4: Graphing for the Good Life 61

Coordinating Axes, Coordinates of Points, and Quadrants 62

Crossing the Line: Using Intercepts and Symmetry to Graph 64

Graphing Lines Using Slope-Intercept and Standard Forms 67

Graphing Basic Polynomial Curves 69

Grappling with Radical and Absolute Value Functions 71

Enter the Machines: Using a Graphing Calculator 73

Answers to Problems on Graphing for the Good Life 77

Part 2: Functions 89

Chapter 5: Formulating Functions 91

Evaluating Functions 91

Determining the Domain and Range of a Function 93

Recognizing Even, Odd, and One-to-One Functions 94

Composing Functions and Simplifying the Difference Quotient 96

Solving for Inverse Functions 99

Answers to Problems on Formulating Functions 101

Chapter 6: Specializing in Quadratic Functions 107

Finding Intercepts and the Vertex of a Parabola 108

Applying Quadratics to Real-Life Situations 109

Graphing Parabolas 111

Answers to Problems on Quadratic Functions 113

Chapter 7: Plugging in Polynomials 119

Finding Basic Polynomial Intercepts 120

Digging up More-Difficult Polynomial Roots with Factoring 122

Determining Where a Function Is Positive or Negative 123

Graphing Polynomials 125

Possible Roots and Where to Find Them: The Rational Root Theorem and Descartes’s Rule 127

Getting Real Results with Synthetic Division and the Remainder Theorem 130

Connecting the Factor Theorem with a Polynomial’s Roots 132

Answers to Problems on Plugging in Polynomials 134

Chapter 8: Acting Rationally with Functions 143

Determining Domain and Intercepts of Rational Functions 144

Introducing Vertical and Horizontal Asymptotes 145

Getting a New Slant with Oblique Asymptotes 147

Removing Discontinuities 148

Going the Limit: Limits at a Number and Infinity 149

Graphing Rational Functions 151

Answers to Problems on Rational Functions 156

Chapter 9: Exposing Exponential and Logarithmic Functions 163

Evaluating e-Expressions and Powers of e 164

Solving Exponential Equations 165

Making Cents: Applying Compound Interest and Continuous Compounding 167

Checking out the Properties of Logarithms 169

Presto-Chango: Expanding and Contracting Expressions with Log Functions 171

Solving Logarithmic Equations 173

They Ought to Be in Pictures: Graphing Exponential and Logarithmic Functions 175

Answers to Problems on Exponential and Logarithmic Functions 179

Part 3: Conics And Systems Of Equations 189

Chapter 10: Any Way You Slice It: Conic Sections 191

Putting Equations of Parabolas in Standard Form 192

Shaping Up: Determining the Focus and Directrix of a Parabola 194

Back to the Drawing Board: Sketching Parabolas 196

Writing the Equations of Circles and Ellipses in Standard Form 198

Determining Foci and Vertices of Ellipses 201

Rounding Out Your Sketches: Circles and Ellipses 203

Hyperbola: Standard Equations and Foci 205

Determining the Asymptotes and Intercepts of Hyperbolas 206

Sketching the Hyperbola 208

Answers to Problems on Conic Sections 211

Chapter 11: Solving Systems of Linear Equations 221

Solving Two Linear Equations Algebraically 221

Using Cramer’s Rule to Defeat Unruly Fractions 223

A Third Variable: Upping the Systems to Three Linear Equations 225

A Line by Any Other Name: Writing Generalized Solution Rules 227

Decomposing Fractions Using Systems 229

Answers to Problems on Systems of Equations 231

Chapter 12: Solving Systems of Nonlinear Equations and Inequalities 237

Finding the Intersections of Lines and Parabolas 237

Crossing Curves: Finding the Intersections of Parabolas and Circles 239

Appealing to a Higher Power: Dealing with Exponential Systems 240

Solving Systems of Inequalities 242

Answers to Problems on Solving Systems of Nonlinear Equations and Inequalities 245

Part 4: Other Good Stuff: Lists, Arrays, And Imaginary Numbers 251

Chapter 13: Getting More Complex with Imaginary Numbers 253

Simplifying Powers of i 254

Not Quite Brain Surgery: Doing Operations on Complex Numbers 255

“Dividing” Complex Numbers with a Conjugate 256

Solving Equations with Complex Solutions 257

Answers to Problems on Imaginary Numbers 259

Chapter 14: Getting Squared Away with Matrices 263

Describing Dimensions and Types of Matrices 263

Adding, Subtracting, and Doing Scalar Multiplication on Matrices 265

Trying Times: Multiplying Matrices by Each Other 267

The Search for Identity: Finding Inverse Matrices 268

Using Matrices to Solve Systems of Equations 272

Answers to Problems on Matrices 274

Chapter 15: Going Out of Sequence with Sequences and Series 279

Writing the Terms of a Sequence 279

Differences and Multipliers: Working with Special Sequences 282

Backtracking: Constructing Recursively Defined Sequences 283

Using Summation Notation 284

Finding Sums with Special Series 286

Answers to Problems on Sequences and Series 289

Chapter 16: Everything You Ever Wanted to Know about Sets and Counting 293

Writing the Elements of a Set from Rules or Patterns 294

Get Together: Combining Sets with Unions, Intersections, and Complements 295

Multiplication Countdowns: Simplifying Factorial Expressions 297

Checking Your Options: Using the Multiplication Property 298

Counting on Permutations When Order Matters 300

Mixing It Up with Combinations 301

Raising Binomials to Powers: Investigating the Binomial Theorem 303

Answers to Problems on Sets and Counting 304

Part 5: The Part Of Tens 309

Chapter 17: Basic Graphs 311

Putting Polynomials in Their Place 311

Lining Up Front and Center 312

Being Absolutely Sure with Absolute Value 313

Graphing Reciprocals of x and x2 .313

Rooting Out Square Root and Cube Root .314

Growing Exponentially with a Graph 315

Logging In on Logarithmic Graphing 316

Chapter 18: Ten Special Sequences and Their Sums 317

Adding as Easy as One, Two, Three 317

Summing Up the Squares 318

Finding the Sum of the Cubes 318

Not Being at Odds with Summing Odd Numbers 319

Evening Things Out by Adding Up Even Numbers 319

Adding Everything Arithmetic 319

Geometrically Speaking 320

Easing into a Sum for e 320

Signing In on the Sine 321

Powering Up on Powers of 2 322

Adding Up Fractions with Multiples for Denominators 322

Index 323