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Algebra: Form and Function, 2nd Edition

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Algebra: Form and Function, 2nd Edition

William G. McCallum, Eric Connally, Deborah Hughes-Hallett

ISBN: 978-1-119-03225-0 November 2014 464 Pages

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Description

Algebra: Form and Function, 2nd Edition offers a fresh approach to algebra that focuses on teaching readers how to truly understand the principles, rather than viewing them merely as tools for other forms of mathematics. Meant for a College Algebra course, Algebra: Form and Function, 2nd Edition is an introduction to one of the fundamental aspects of modern society. Algebraic equations describe the laws of science, the principles of engineering, and the rules of business. The power of algebra lies in the efficient symbolic representation of complex ideas, which also presents the main difficulty in learning it. Most students rely on surface knowledge of algebraic manipulations without understanding the underlying structure of algebra that allows them to see patterns and apply it to multiple situations; McCallum focuses on the structure from the start.

Related Resources

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 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

  • A new organization that moves quickly through early material. The former chapters 2 and 3 on rules for expressions and rules for equations, and the former chapter 6 on exponent rules, has been moved to appendices, complete with exercises, for those instructors who still want to cover this material.
  • Problems organized according to the three aspects of algebraic knowledge—recognizing building blocks, fluency, and modeling encourage students first to look inside and then to think ahead.
  • Suggested by current users, a simplified presentation throughout the text and clarifications of wording in text, examples, and exercises help student comprehension of the materials being presented.
  • A Balanced Approach: Form, Function, and Fluency. To use algebra in later courses, students need not only manipulative skill, but fluency in the language of algebra, including an ability to recognize algebraic form and an understanding of the purpose of different forms. Algebra achieves this through unique examples and exercises.
  • Restoring Meaning to Expressions and Equations. After introducing each type of function--linear, power, quadratic, exponential, polynomial--the text encourages students to pause and examine the basic forms of expressions for that function, see how they are constructed, and consider the different properties of the function that the different forms reveal.
  • Maintaining Manipulative Skills: Review and Practice. Acquiring the skills to perform basic algebraic manipulations is as important as recognizing algebraic forms. Algebra: Form and Function provides sections reviewing the rules of algebra, and the reasons for them, throughout the book, numerous exercises to reinforce skills in each chapter, and a section of drill problems on solving equations at the end of the chapters on linear, power, and quadratic functions.
  • Students with Varying Backgrounds. Algebra: Form and Function is thought-provoking for well-prepared students while still accessible to students with weaker backgrounds, making it understandable to students of all ability levels.

 

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