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# College Algebra, 6th Edition

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# College Algebra, 6th Edition

ISBN: 978-1-119-39307-8 January 2018

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Explorations in College Algebra's overarching goal is to reshape the College Algebra course to make it more relevant and accessible to all students. This is achieved by shifting the focus from learning a set of discrete mechanical rules to exploring how algebra is used in social and physical sciences and the world around you. By connecting mathematics to real-life situations, students come to appreciate its power and beauty.

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1 An Introduction to Data and Functions 1

1.1 Describing Single-Variable Data 1

Visualizing Single-Variable Data 2

Numerical Descriptors: What Is “Average” Anyway? 4

An Introduction to Explore & Extend 5

An Introduction to Algebra Aerobics 5

1.2 Describing Relationships Between Two Variables 9

Visualizing Two-Variable Data 9

Constructing a “60-Second Summary” 10

Using Equations to Describe Change 12

1.3 An Introduction to Functions 18

What Is a Function? 18

Representing Functions: Words, Tables, Graphs, and Equations 18

Input and Output: Independent and Dependent Variables 19

When Is a Relationship Not a Function? 20

1.4 The Language of Functions 24

Function Notation 24

Finding Output Values: Evaluating a Function 25

Finding Input Values: Solving Equations 25

Finding Input and Output Values from Tables and Graphs 26

Rewriting Equations Using Function Notation 26

Domain and Range 29

1.5 Visualizing Functions 34

Is There a Maximum or Minimum Value? 34

When Is the Output of the Function Positive, Negative, or Zero? 35

Is the Function Increasing or Decreasing? 35

Is the Graph Concave Up or Concave Down? 36

Getting the Big Idea 37

Chapter Summary 45

Check Your Understanding 46

Chapter 1 Review: Putting It All Together 48

2 Rates of Change and Linear Functions 53

2.1 Average Rates of Change 53

Describing Change in the U.S. Population over Time 53

Defining the Average Rate of Change 55

Limitations of the Average Rate of Change 56

2.2 Change in the Average Rate of Change 59

2.3 The Average Rate of Change Is a Slope 64

Calculating Slopes 64

2.4 Putting a Slant on Data 69

Slanting the Slope: Choosing Different End Points 69

Slanting the Data with Words and Graphs 70

2.5 Linear Functions: When Rates of Change Are Constant 75

What If the U.S. Population Had Grown at a Constant Rate? A Hypothetical Example 75

The General Equation for a Linear Function 78

2.6 Visualizing Linear Functions 81

The Effect of m 82

2.7 Constructing Graphs and Equations of Linear Functions 87

Finding the Graph 87

Finding the Equation 89

2.8 Special Cases 94

Direct Proportionality 94

Horizontal and Vertical Lines 97

Parallel and Perpendicular Lines 99

2.9 Breaking the Line: Piecewise Linear Functions 104

Piecewise Linear Functions 104

2.10 Constructing Linear Models of Data 111

Fitting a Line to Data: The Kalama Study 111

Reinitializing the Independent Variable 113

Interpolation and Extrapolation: Making Predictions 114

2.11 Looking for Links Between Education and Earnings: A Case Study on Using Regression Lines 120

Using U.S. Census Data 120

Summarizing the Data: Regression Lines 121

Interpreting Regression Lines: Correlation vs. Causation 124

Raising More Questions: Going Deeper 125

Chapter Summary 132

Check Your Understanding 132

Chapter 2 Review: Putting It All Together 135

3 When Lines Meet: Linear Systems 141

3.1 Interpreting Intersection Points: Linear and Nonlinear Systems 141

When Curves Collide: Nonlinear Systems 141

When Lines Meet: Linear Systems 144

3.2 Visualizing and Solving Linear Systems 151

Visualizing Linear Systems 151

Strategies for Solving Linear Systems 152

Systems with No Solution or Infinitely Many Solutions 154

Linear Systems in Economics: Supply and Demand 156

3.3 Reading Between the Lines: Linear Inequalities 161

Above and Below the Line 161

Reading Between the Lines 162

Manipulating Inequalities 164

Breakeven Points: Regions of Profit or Loss 165

3.4 Systems with Piecewise Linear Functions: Tax Plans 171

Graduated vs. Flat Income Tax 171

Comparing the Flat and Graduated Tax Plans 174

Chapter Summary 177

Check Your Understanding 178

Chapter 3 Review: Putting It All Together 180

4 The Laws of Exponents and Logarithms: Measuring the Universe 185

4.1 The Numbers of Science: Measuring Time and Space 185

Powers of 10 and the Metric System 185

Scientific Notation 188

4.2 Positive Integer Exponents 192

Exponent Rules 193

Common Errors 195

Estimating Answers 196

4.3 Zero, Negative, and Fractional Exponents 200

Zero and Negative Exponents 200

Evaluating (a/b)-n 201

Fractional Exponents 202

Expressions of the Form a1/2: Square Roots 202

nth Roots: Expressions of the Form a1/n 203

Rules for Radicals 204

Expressions of the Form am/n 206

4.4 Converting Units 210

Converting Units Within the Metric System 210

Converting Between the Metric and English Systems 211

Using Multiple Conversion Factors 211

4.5 Orders of Magnitude 214

Comparing Numbers of Widely Differing Sizes 214

Orders of Magnitude 214

Graphing Numbers of Widely Differing Sizes: Log Scales 215

4.6 Logarithms as Numbers 218

Finding the Logarithms of Powers of 10 219

When is log10x Not Defined? 220

Finding the Logarithm of Any Positive Number 221

Plotting Numbers on a Logarithmic Scale 222

Chapter Summary 226

Check Your Understanding 226

Chapter 4 Review: Putting It All Together 227

5 Growth and Decay: An Introduction to Exponential Functions 231

5.1 Exponential Growth 231

The Growth of E. coli Bacteria 231

The General Exponential Growth Function 233

Doubling Time 234

Looking at Real Growth Data for E. coli Bacteria 236

5.2 Exponential Decay 239

The Decay of Iodine-131 239

The General Exponential Decay Function 240

Half-Life 241

5.3 Comparing Linear and Exponential Functions 245

Linear Functions 246

Exponential Functions 246

Identifying Exponential Functions in a Data Table 246

A Linear vs. an Exponential Model Through Two Points 247

Comparing the Average Rates of Change 249

In the Long Run, Exponential Growth Will Always Outpace Linear Growth 250

5.4 Visualizing Exponential Functions 253

The Graphs of Exponential Functions 253

The Effect of the Base a 253

The Effect of the Initial Value C 254

Horizontal Asymptotes 256

5.5 Exponential Functions: A Constant Percent Change 259

Exponential Growth: Increasing by a Constant Percent 259

Exponential Decay: Decreasing by a Constant Percent 260

Revisiting Linear vs. Exponential Functions 262

5.6 More Examples of Exponential Growth and Decay 267

Returning to Doubling Times and Half-Lives 268

The Malthusian Dilemma 275

Forming a Fractal Tree 276

5.7 Compound Interest and the Number e 283

Compounding at Different Intervals 284

Continuous Compounding Using e 286

Continuous Compounding Formula 286

Exponential Functions Base e 287

Converting ek into a 289

5.8 Semi-Log Plots of Exponential Functions 293

Chapter Summary 297

Check Your Understanding 298

Chapter 5 Review: Putting It All Together 300

6 Logarithmic Links: Logarithmic and Exponential Functions 305

6.1 Using Logarithms to Solve Exponential Equations 305

Estimating Solutions to Exponential Equations 305

Rules for Logarithms 307

Solving Exponential Equations Using Logarithms 310

Solving for Doubling Times and Half-Lives 311

6.2 Using Natural Logarithms to Solve Exponential Equations Base e 315

The Natural Logarithm 315

Returning to Doubling Times and Half-Lives 317

Converting Exponential Functions from Base a to Base e 319

6.3 Visualizing and Applying Logarithmic Functions 324

The Graphs of Logarithmic Functions 324

Logarithmic Growth 324

Inverse Functions: Logarithmic vs. Exponential 326

Applications of Logarithmic Functions 329

6.4 Using Semi-Log Plots to Construct Exponential Models for Data 334

Why Do Semi-Log Plots of Exponential Functions Produce Straight Lines? 335

Chapter Summary 339

Check Your Understanding 340

Chapter 6 Review: Putting It All Together 341

7 Power Functions 345

7.1 The Tension between Surface Area and Volume 345

Scaling Up a Cube 345

Size and Shape 348

7.2 Direct Proportionality: Power Functions with Positive Powers 350

Direct Proportionality 351

Properties of Direct Proportionality 352

Direct Proportionality with More Than One Variable 355

7.3 Visualizing Positive Integer Power Functions 358

The Graphs of f(x) =x2 and g(x) =x3 358

Odd vs. Even Positive Integer Powers 359

The Effect of the Coefficient k 361

7.4 Comparing Power and Exponential Functions 365

Which Eventually Grows Faster, a Power Function or an Exponential Function? 365

7.5 Inverse Proportionality: Power Functions with Negative Powers 369

Inverse Proportionality 370

Properties of Inverse Proportionality 372

Inverse Square Laws 375

7.6 Visualizing Negative Integer Power Functions 380

The Graphs of f(x) =x−1 and g(x) =x−2 380

Odd vs. Even Negative Integer Powers 382

The Effect of the Coefficient k 383

7.7 Using Logarithmic Scales to Find the Best Functional Model 389

Looking for Lines 389

Why Is a Log-Log Plot of a Power Function a Straight Line? 390

Translating Power Functions into Equivalent Logarithmic Functions 390

Analyzing Weight and Height Data 391

Allometry: The Effect of Scale 394

Chapter Summary 402

Check Your Understanding 403

Chapter 7 Review: Putting It All Together 404

8 Quadratics and the Mathematics of Motion 409

8.1 An Introduction to Quadratic Functions: The Standard Form 409

The Simplest Quadratic 409

Designing Parabolic Devices 410

The Standard Form of a Quadratic 411

Properties of Quadratic Functions 412

8.2 Visualizing Quadratics: The Vertex Form 418

Stretching and Compressing Vertically 418

Reflecting Across the Horizontal Axis 418

Shifting Vertically and Horizontally 420

Using Transformations to Get the Vertex Form 423

8.3 The Standard Form vs. the Vertex Form 426

Finding the Vertex from the Standard Form 426

Converting Between Standard and Vertex Forms 428

8.4 Finding the Horizontal Intercepts: The Factored Form 435

Using Factoring to Find the Horizontal Intercepts 435

Factoring Quadratics 436

Using the Quadratic Formula to Find the Horizontal Intercepts 439

The Factored Form 442

Standard, Factored and Vertex Forms 445

8.5 The Average Rate of Change of a Quadratic Function 448

Generalizing to All Quadratic Functions 449

8.6 The Mathematics of Motion 453

The Scientific Method 454

The Free Fall Experiment 454

Deriving an Equation Relating Distance and Time 454

Velocity: Change in Distance over Time 456

Acceleration: Change in Velocity over Time 458

Deriving an Equation for the Height of an Object in Free Fall 460

Working with an Initial Upward Velocity 463

Chapter Summary 468

Check Your Understanding 469

Chapter 8 Review: Putting It All Together 470

9 New Functions from Old 473

9.1 Transformations 473

Transforming a Function 473

9.2 The Algebra of Functions 485

9.3 Polynomials: The Sum of Power Functions 492

Defining a Polynomial Function 493

Visualizing Polynomial Functions 495

Finding the Vertical Intercept 498

Finding the Horizontal Intercepts 499

9.4 Rational Functions: The Quotient of Polynomials 505

Building a Rational Function: Finding the Average Cost of an MRI Machine 505

Defining a Rational Function 506

Visualizing Rational Functions 507

9.5 Composition and Inverse Functions 514

Composing Two Functions 514

Composing More Than Two Functions 516

Inverse Functions: Returning the Original Value 518

9.6 Exploring, Extending & Expanding 528

Chapter Summary 531

Check Your Understanding 532

Chapter 9 Review: Putting It All Together 534

ANSWERS For all Algebra Aerobics and Check Your Understanding problems; for odd numbered problems in the Exercises and Chapter Reviews.
All answers are grouped by chapter. ANS-1

Index I-1