Explorations in College Algebra was developed by the College Algebra Consortium based at the University of Massachusetts, Boston and funded by a grant from the National Science Foundation. The materials were developed in the spirit of the reform movement and reflect the guidelines issued by the various professional mathematics societies (including AMATYC, MAA, and NCTM).
This fourth edition shifts the focus away from mechanical rules, computation, and basic skills to emphasize concepts and modeling. It is written in a way that gets readers to explore how algebra is used in the world around them. The authors include the most varied and compelling set of applications available in the market. Readers will also find a problem-solving approach that motivates them to learn the material.
1. Making Sense of Data and Functions
1.1 Describing Single-Variable Data 2
Visualizing Single-Variable Data 2
Mean and Median: What is "Average" Anyway? 6
An Introduction to Algebra Aerobics 7
1.2 Describing Relationships between Two Variables 13
Visualizing Two-Variable Data 13
Constructing a "60-Second Summary" 14
Using Equations to Describe Change 16
1.3 An Introduction to Functions 22
What is a Function? 22
Representing Functions in Multiple Ways 23
Independent and Dependent Variables 24
When is a Relationship Not a Function? 24
1.4 The Language of Functions 29
Function Notation 29
Domain and Range 33
1.5 Visualizing Functions 39
Is There a Maximum or Minimum Value? 39
Is the Function Increasing or Decreasing? 40
Is the Graph Concave Up or Concave Down? 40
Getting the Big Idea 42
Chapter Summary 49
Check Your Understanding 50
Chapter 1 Review: Putting it all Together 52
Exploration 1.1 Collecting, Representing, and Analyzing Data 58
Exploration 1.2 Picturing Functions 61
Exploration 1.3 Deducing Formulas to Describe Data 63
2. Rates of change adn Linear Functions
2.1 Average Rates of Change 66
Describing Change in the U.S. Population over Time 66
Defining the Average Rate of Change 67
Limitations of the Average Rate of Change 68
2.2 Change in teh Average Rate of Change 71
2.3 The Average Rate of Change is a Slope 76
Calculating Slopes 76
2.4 Putting a Slant on Data 82
Slanting the Slope: Choosing Different End POints 82
Slanting the Data with Words and Graphs 83
2.5 Linear Functions: When Rates of Change are Constant 87
What is the U.S. Population Had Grown at a Constant Rate? 87
Real Examples of a Constant Rate of Change 87
The General Equation for a Linear Function 90
2.6 Visualizing Linear Functions 94
The Effect of b 94
The Effect of m 94
2.7 Finding Graphs and Equations of Linear Functions 99
Finding the Graph 99
Finding the Equation 100
2.8 Special Cases 108
Direct Proportionality 108
Horizontal and Vertical Lines 110
Parallel and Perpendicular Lines 112
Piecewise Linear Functions 114'
The absolute vaule function 115
Step functions 117
2.9 Constructing Linear Models for Data 122
Fitting a Line to Data: The Kalama Study 123
Reinitializing the Independent Variable 125
Interpolation and Extrapolation: Making Predictions 126
Chapter SUmmary 131
Check Your Understanding 132
Chapter 2 Review: Putting it all Together 134
Exploration 2.1 Having it Your Way 139
Exploration 2.2A Looking at Lines with the Course Software 141
Exploration 2.2B Looking at Lines with a Graphing Calculator 142
An Extended Exploration: Looking for Links between Education and Earnings
Using U.S. Census Data 146
Summarizing the Data: Regression Lines 148
Is there a Relationship between Education adn Earnings? 148
Regression Lines: How Good a Fit? 151
Interpreting Regression Lines: Correlation vs. Causation 153
Reaising More Questions 154
Do Earnings Depend on Age? 155
Do Earnings Depend upon Gender? 155
How Good are teh Data? 157
Hoe Good is the Analysis? 157
Exploring on your Own 157
Exercises 159
3. When Lines Meet: Linear Systems
3.1 Systems of Linear Equations 166
An Economic Comparison of Solar vs. Conventional Heating Systems 166
3.2 Finding Solutions to Systems of Linear Equations 171
Visualizing Solutions 171
Strategies for Finding Solutions 172
Linear Systems in Economics: Supply and Demand 176
3.3 Reading between the Lines: Linear Inequalities 183
Above and Below th Line 183
Manipulating Inequalities 184
Reading between the Lines 185
Breakeven Points: Regions of Profit or Loss 187
3.4 Systems with Piecewise Linear Functions: Tax Plans 193
Graduated vs. Flat Income Tax 193
Comparing the Two Tax Models 195
The Case of Massachusetts 196
Chapter Summary 201
Check Your Understanding 202
Chapter 3 Review: Putting it all Together 204
Explorationg 3.1 Flat vs. Graduated Income Tax: Who Benefits? 209
4. The Laws of Exponents and Logarithms: Measuring the Universe
4.1 The Numbers of Science: Measuring Time and Space 212
Powers of 10 and the Metrict System 212
Scientific Notation 214
4.2 Positive Integer Exponents 218
Exponent Rules 219
Common Errors 221
Estimating Answers 223
4.3 Negative Integer Exponents 226
Evaluating (a/b) -n 227
4.4 Converting Units 230
Converting Units within the Metric Systems 230
Converting between the Metrict and English Systems 231
Using Multiple Conversion Factors 231
4.5 Fractional Exponents 235
Square Roots: Expressions of the Form a^1/2 235
nth Roots: Expressions of the Form a^1/2
Rules for Radicals 238
Fractional Powers: Expressions of the Form a^m/n
4.6 Orders of Magnitude 242
Comparing Numbers of Widely Differing Sizes 242
Orders of Magnitude 242
Graphing Numbers of Widely Differing Sizes: Log Scales 244
4.7 Logarithms Bas 10 248
Finding the Logarithms of Powers of 10 248
Finding the Logarithm of Any Positive Number 250
Plotting Numbers on a Logarithmic Scale 251
Chapter Summary 255
Check Your Understanding 256
Chapter 4 Review: Putting it all Together 257
Exploration 4.1 The Scale and the Tale of the Universe 260
Exploration 4.2 Patterns in the Positions and Motions of the Planets 262
5. Growth and Decay: An Introduction to Exponential Functions
5.1 Exponential Growth 266
The Growth of E. coli Bacteria 266
The General Exponential Growth Function 267
Looking at Real Growth Data for E. coli Bacteria 268
5.2 Linear vs. Exponential Growth Functions 271
Linear vs. Exponential Growth 271
Comparing the Average Rates of Change 273
A Linear vs. and Exponential Model through Two Ponts 274
Identifying Linear vs. Exponential Functions in a Data Table 275
5.3 Exponential Decay 279
The Decay of Iodine-131 279
The General Exponential Decay Function 279
5.4 Visualizing Exponential Functions 284
The Effect of the Base a 284
The Effect of the Initial Value C 285
Horizontal Asymptotes 287
5.5 Exponential Functions: A Constant Percent Change 290
Exponential Growth: Increasing by a Constant Percent 290
Exponential Decay: Decreasing by a Constant Percent 291
Revisiting Linear vs. Exponential Functions 293
5.6 Examples of Exponential Growth and Decay 298
Half-Lfe and Doubling TIme 299
The "rule of 70" 301
Compound Interest Rates 304
The Malthusian Dilemma 308
Forming a Fractal Tree 309
5.7 Semi-log Plots of Exponential Functions 316
Chapter SUmmary 320
Check Your Understanding 321
Chapter 5 Review: Putting it all Together 322
Exploration 5.1 Properties of Exponential Functions 237
6. Logarithmic Links: Logarithmic and Exponential Functions
6.1 Using Logarithms to Solve Exponential Equations 330
Estimating Solutions to Exponential Equations 330
Rules for Logarithms 331
Solving Exponential Equations 336
6.2 Base e adn Continuous Compouding 340
What is e? 340
Continuous Compounding 341
Exponential Functions Base e 344
6.3 The Natural Logarithm 349
6.4 Logarthmic Functions 352
The Graphs of Logarithmic Functions 353
The Relationship between Logarithmic and Exponential Functions 354
Logarithmic vs. exponential growth 354
Logarithmic and exponential functions are inverses of each other 355
Applications of Logarithmic Functions 357
Measuring acidity: The pH scale 357
Measuring noise: The decibel scale
6.5 Transforming Exponential Functions to Base e 363
Converting a to e^k 364
6.6 Using Semi-log Plots to Construct Exponential Models for Data 369
Why Do Semi-Log Plots of Exponential Functions Produce Straight Lines? 369
Chapter Summary 374
Check Your Understanding 375
Chapter 6 Review: Putting it all Together 377
Exploration 6.1 Properties of Logarithmic Functions 380
7. Power Functions
7.1 The Tension between Surface Area and Volume 384
Scaling Up a Cube 385
Size and Shape 386
7.2 Direct Proportionality: Power Functions with Positive Powers 389
Direct Proportionality 390
Properties of Direct Proportionality 390
Direct Proportionality with more than one Variable 393
7.3 Visualizing Positive Integer Powers 397
The Graphs of f(x)=x^2 and g(x)=x^3 397
Odd vs. Even Powers 399
Symmetry 400
The Effect of the Coefficient k 400
7.4 Comparing Power and Exponential Functions 405
Which Eventually Grows Faster, a Power Function or an Exponential Function? 405
7.5 Inverse Proportionality: Power Functions with Negative Integer Powers 409
Inverse Proportionality 410
Properties of Inverse Proportionality 411
Inverse Square Laws 415
7.6 Visualizing Negative Integer Power Functions 420
The Graphs of f(x)=x^-1 and g(x)=x^-2 420
Odd vs. Even Powers 422
Asymptotes 423
Symmetry 423
The Effect of the Coefficient k 423
7.7 Using Logarithmic Scales to Find the Best Functional Model 429
Looking for Lines 429
Why is a Log-Log Plot of a Power Function a Straight Line? 430
Translating Power Functions into Equivalent Logarithmic Functions 430
Analyzing Weight and Height Data 431
Using a standard plot 431
Using a semi-log plot 431
Using a log-log plot 432
Allometry: The Effect of Scale 434
Chapter Summary 442
Check Your Understanding 443
Chapter 7 Review: Putting it all Together 444
Exploration 7.1 Scaling Objects 448
Exploration 7.2 Predicting Properties of Power Functions 450
Exploration 7.3 Visualizing Power Functions with Negative Integer Powers 451
8. Quadratics, Polynomials, and Beyond
8.1 An Introduction to Quadratic Functions 454
The Simplest Quadratic 454
Designing parabolic devices 455
The General Quadratic 456
Properties of Quadratic Functions 457
Estimating the Vertex and Horizontal Intercepts 459
8.2 Finding the Vertex: Transformations of y=x^2 463
Stretching and Compressing Vertically 464
Reflections across the Horizontal Axis 464
Shifting Vertically and Horizontally 465
Using Transformations to Get the Vertex Form 468
Finding the Vertext from the Standard Form 470
Converting between Standard and Vertex Forms 472
8.3 Finding the Horizontal Intercepts 480
Using Factoring to Find the Horizontal Intercepts 481
Factoring Quadratics 482
Using the Quadratic Formula to Find the Horizontal Intercepts 484
The discriminant 485
Imaginary and complex numbers 487
The Factored Form 488
8.4 The Average Rate of Change of a Quadratic Function 493
8.5 An Introduction to Polynomial Functions 498
Defining a Polynomial Function 498
Visualizing Polynomial Functions 500
Finding the Vertical Intercept 502
Finding the Horizontal Intercepts 503
8.6 New Functions from Old 510
Transforming a Function 510
Stretching, compressing and shifting 510
Reflections 511
Symmetry 512
8.7 Combining Two Functions 521
The Algebra of Functions 521
Rational Functions: The Quotient of Two Polynomials 524
Visualizing Rational Functions 525
Chapter Summary 547
Check Your Understanding 548
Chapter 8 Review: Putting it all Together 550
Exploration 8.1 How Fast Are You? Using a Ruler to Make a Reaction Timer 555
An Extended Exploration: The Mathematics of Motion
The Scientific Method 560
The Free-Fall Experiment 560
Interpreting Data from a Free-Fall Experiment 561
Deriving an Equation Relating Distance and Time 563
Returning to Galileo's Question 565
Velocity: Change in Distance over Time 565
Acceleration: Change in Velocity over Time 566
Deriving an Equation for the Height of an Object in Free Fall 568
Working with an Initial Upward Velocity 569
Collecting and Analyzing Data from a Free Fall Experiment 570
Exercises 573
Appendix: Student Data Tables for Exploration 2.1 579
Solutions 583
Index 692