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Table of Contents
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
