Algebra: Form and Function Cloth

Algebra: Form and Function was designed based on the fundamental goal for a student to foster understanding of algebraic structure- that is, an understanding of how the arrangements of symbols allows us to predict, for example, the behavior of a function or the number of solutions to an equation. Mastering algebraic structure enables students to read algebraic expressions and equations in real-life contexts, not just manipulate them, and to choose which form or which operation will best suit the context.  It facilitates being able to translate back and forth between symbolic, graphical, numerical, and verbal representations.

By balancing practice in manipulation and opportunities to see the big picture, Algebra: Form and Function offers a way for teachers to help students achieve real mastery of algebra. 

William G. McCallum is a University Distinguished Professor of Mathematics and Head of the Department of Mathematics at the University of Arizona. Born in Sydney, Australia in 1956, he received his Ph.D. in Mathematics from Harvard University in 1984, under the supervision of Barry Mazur. After spending two years at the University of California, Berkeley, and one at the Mathematical Sciences Research Institute in Berkeley, he joined the faculty at the University of Arizona in 1987. In 1989 he joined the Harvard calculus consortium, and is the lead author of the consortium's multivariable calculus and college algebra texts. In 1993-94 he spent a year at the Institut des Hautes Etudes Scientifiques, and in 1995-96 he spent a year at the Institute for Advanced Study on a Centennial Fellowship from the American Mathematical Society. In 2005 he received the Director's Award for Distinguished Teaching Scholars from the National Science Foundation. In 2006 he founded the Institute for Mathematics and Education at the University of Arizona. He was Director of the Institute until 2009 and now chairs its advisory board. In 2009-2010 he led the work team that developed the NGA/CCSSO Common Core Math Standards. His professional interests include arithmetical algebraic geometry and mathematics education.

1 Functions and Algebraic Structure 1

1.1 What is a Function? 2

1.2 Functions and Expressions 8

1.3 Functions and Equations 15

1.4 Functions and Change 24

1.5 Functions, Modeling, and Proportionality 30

Review Problems 36

2 Linear Functions 41

2.1 Introduction to Linear Functions 42

2.2 Linear Expressions 49

2.3 Linear Equations 57

2.4 Equations for Lines in the Plane 65

2.5 Modeling with Linear Functions 73

2.6 Systems of Linear Equations 80

Review Problems 91

Solving Drill 97

3 Quadratic Functions 99

3.1 Introduction to Quadratic Functions 100

3.2 Quadratic Expressions 103

3.3 Converting to Factored and Vertex Form 111

3.4 Quadratic Equations 116

3.5 Factoring Hidden Quadratics 124

3.6 Complex Numbers 129

Review Problems 134

Solving Drill 138

4 Power Functions 139

4.1 Power Functions: Positive Exponents 140

4.2 Power Functions: Negative and Fractional Exponents 146

4.3 Power Functions and Expressions 152

4.4 Power Functions and Equations 158

4.5 Modeling with Power Functions 165

Review Problems 172

Solving Drill 175

5 More On Functions 177

5.1 Domain and Range 178

5.2 Composing and Decomposing Functions 186

5.3 Shifting and Scaling 191

5.4 Inverse Functions 201

Review Problems 206

6 Exponential Functions 209

6.1 Exponential Functions 210

6.2 Exponential Expressions: Growth Rates 216

6.3 Exponential Expressions: Half-Life and Doubling Time 222

6.4 Equations and Exponential Functions 230

6.5 Modeling with Exponential Functions 237

6.6 Exponential Functions and Base e 243

Review Problems 248

7 Logarithms 253

7.1 Introduction to Logarithms 254

7.2 Solving Equations Using Logarithms 263

7.3 Applications of Logarithms to Modeling 269

7.4 Natural Logarithms and Other Bases 274

Review Problems 283

8 Polynomial Functions 287

8.1 Polynomial Functions 288

8.2 Expressions and Polynomial Functions 292

8.3 Solving Polynomial Equations 299

8.4 Long-Run Behavior of Polynomial Functions 306

Review Problems 314

9 Rational Functions 319

9.1 Rational Functions 320

9.2 Long-Run Behavior of Rational Functions 326

9.3 Putting a Rational Function in Quotient Form 337

Review Problems 343

Appendix A: Expressions 345

A.1 Reordering and Regrouping 346

A.2 The Distributive Law 349

Appendix B: Equations 355

B.1 Using the Operations of Arithmetic to Solve Equations 356

Appendix C: Inequalities 361

C.1 Solving Inequalities 362

Appendix D: Quadratics 367

D.1 Quadratic Expressions 368

D.2 Solving Quadratic Equations 375

Appendix E: Algebraic Fractions 377

E.1 Algebraic Fractions 378

E.2 Equations Involving Algebraic Fractions 385

Appendix F: Absolute Value 387

F.1 Absolute Value 388

F.2 Absolute Value Equations and Inequalities 390

Appendix G: Exponents 395

G.1 Exponents with Integer Powers 396

G.2 Exponents with Fractional Powers 405

10 Summation Notation (Online Only) 10-1

10.1 Using Subscripts and Sigma Notation 10-2

11 Sequences and Series (Online Only) 11-1

11.1 Sequences 11-2

11.2 Arithmetic Series 11-8

11.3 Geometric Sequences and Series 11-13

11.4 Applications of Series 11-19

Review Problems 11-25

12 Matrices and Vectors (Online Only) 12-1

12.1 Matrices 12-2

12.2 Matrix Multiplication 12-5

12.3 Matrices and Vectors 12-10

12.4 Matrices and Systems of Linear Equations 12-18

Review Problems 12-28

13 Probability and Statistics (Online Only) 13-1

13.1 The Mean 13-2

13.2 The Standard Deviation 13-9

13.3 Probability 13-14

Review Problems 13-26

Answers to Odd-Numbered Problems 413

Index 431

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