Explorations in College Algebra, 5th Edition

COVER......Page 1
Geometric Formulas......Page 3
TITLE PAGE......Page 5
COPYRIGHT......Page 6
Letter from a Student......Page 8
PREFACE......Page 9
TABLE OF CONTENTS......Page 17
CHAPTER 1 AN INTRODUCTION TO DATA AND FUNCTIONS......Page 25
Visualizing Single-Variable Data......Page 26
Numerical Descriptors: What Is “Average” Anyway?......Page 28
Visualizing Two-Variable Data......Page 34
Constructing a “60-Second Summary”......Page 35
Using Equations to Describe Change......Page 37
What Is a Function?......Page 43
Representing Functions: Words, Tables, Graphs, and Equations......Page 44
When Is a Relationship Not a Function?......Page 45
Function Notation......Page 50
Finding Input Values: Solving Equations......Page 51
Rewriting Equations Using Function Notation......Page 52
Domain and Range......Page 55
Is There a Maximum or Minimum Value?......Page 60
Is the Function Increasing or Decreasing?......Page 61
Is the Graph Concave Up or Concave Down?......Page 62
Getting the Big Idea......Page 63
CHAPTER SUMMARY......Page 72
CHECK YOUR UNDERSTANDING......Page 73
CHAPTER 1 REVIEW: PUTTING IT ALL TOGETHER......Page 75
EXPLORATION 1.1 COLLECTING, REPRESENTING, AND ANALYZING DATA......Page 80
CHAPTER 2 RATES OF CHANGE AND LINEAR FUNCTIONS......Page 83
Describing Change in the U.S. Population over Time......Page 84
Defining the Average Rate of Change......Page 85
Limitations of the Average Rate of Change......Page 86
2.2 Change in the Average Rate of Change......Page 91
Calculating Slopes......Page 96
Slanting the Slope: Choosing Different End Points......Page 102
Slanting the Data with Words and Graphs......Page 103
What If the U.S. Population Had Grown at a Constant Rate? A Hypothetical Example......Page 109
Real Examples of a Constant Rate of Change......Page 110
The General Equation for a Linear Function......Page 112
2.6 Visualizing Linear Functions......Page 116
The Effect of m......Page 117
Finding the Graph......Page 123
Finding the Equation......Page 125
Direct Proportionality......Page 131
Horizontal and Vertical Lines......Page 134
Parallel and Perpendicular Lines......Page 136
Piecewise Linear Functions......Page 141
Fitting a Line to Data: The Kalama Study......Page 148
Reinitializing the Independent Variable......Page 151
Interpolation and Extrapolation: Making Predictions......Page 152
Using U.S. Census Data......Page 158
Summarizing the Data: Regression Lines......Page 159
Interpreting Regression Lines: Correlation vs. Causation......Page 162
Raising More Questions: Going Deeper......Page 163
CHAPTER SUMMARY......Page 170
CHECK YOUR UNDERSTANDING......Page 171
CHAPTER 2 REVIEW: PUTTING IT ALL TOGETHER......Page 173
EXPLORATION 2.1 HAVING IT YOUR WAY......Page 178
EXPLORATION 2.2 A CASE STUDY ON EDUCATION AND EARNINGS IN THE U.S.......Page 180
CHAPTER 3 WHEN LINES MEET: LINEAR SYSTEMS......Page 183
When Curves Collide: Nonlinear Systems......Page 184
When Lines Meet: Linear Systems......Page 187
Strategies for Solving Linear Systems......Page 195
Systems with No Solution or Infinitely Many Solutions......Page 198
Linear Systems in Economics: Supply and Demand......Page 199
Above and Below the Line......Page 205
Reading between the Lines......Page 206
Manipulating Inequalities......Page 208
Breakeven Points: Regions of Profit or Loss......Page 209
Graduated vs. Flat Income Tax......Page 217
Comparing the Flat and Graduated Tax Plans......Page 219
CHAPTER SUMMARY......Page 223
CHECK YOUR UNDERSTANDING......Page 224
CHAPTER 3 REVIEW: PUTTING IT ALL TOGETHER......Page 226
EXPLORATION 3.1 FLAT VS.GRADUATED INCOME TAX: WHO BENEFITS?......Page 231
EXPLORATION 3.2 A COMPARISON OF HYBRID AND CONVENTIONAL AUTOMOBILES......Page 233
CHAPTER 4 THE LAWS OF EXPONENTS AND LOGARITHMS: MEASURING THE UNIVERSE......Page 235
Powers of 10 and the Metric System......Page 236
Scientific Notation......Page 238
4.2 Positive Integer Exponents......Page 242
Exponent Rules......Page 243
Common Errors......Page 245
Estimating Answers......Page 246
Zero and Negative Exponents......Page 250
Evaluating (a/b)−ⁿ......Page 251
Expressions of the Form a½: Square Roots......Page 252
nth Roots: Expressions of the Form a 1/n......Page 253
Rules for Radicals......Page 254
Expressions of the Form a m/n......Page 256
Converting Units within the Metric System......Page 261
Using Multiple Conversion Factors......Page 262
Orders of Magnitude......Page 266
Graphing Numbers of Widely Differing Sizes: Log Scales......Page 267
Finding the Logarithms of Powers of 10......Page 271
Finding the Logarithm of Any Positive Number......Page 273
Plotting Numbers on a Logarithmic Scale......Page 274
CHAPTER SUMMARY......Page 279
CHAPTER 4 REVIEW: PUTTING IT ALL TOGETHER......Page 280
EXPLORATION 4.1 THE SCALE AND THE TALE OF THE UNIVERSE......Page 284
CHAPTER 5 GROWTH AND DECAY: AN INTRODUCTION TO EXPONENTIAL FUNCTIONS......Page 287
The Growth of E. coli Bacteria......Page 288
The General Exponential Growth Function......Page 289
Doubling Time......Page 290
Looking at Real Growth Data for E. coli Bacteria......Page 292
The Decay of Iodine-131......Page 295
The General Exponential Decay Function......Page 296
Half-Life......Page 297
Exponential Functions......Page 302
Identifying Exponential Functions in a Data Table......Page 303
A Linear vs. an Exponential Model through Two Points......Page 304
Comparing the Average Rates of Change......Page 306
In the Long Run, Exponential Growth Will Always Outpace Linear Growth......Page 307
The Effect of the Base a......Page 310
The Effect of the Initial Value C......Page 311
Horizontal Asymptotes......Page 313
Exponential Growth: Increasing by a Constant Percent......Page 316
Exponential Decay: Decreasing by a Constant Percent......Page 317
Revisiting Linear vs. Exponential Functions......Page 319
5.6 More Examples of Exponential Growth and Decay......Page 325
Returning to Doubling Times and Half-Lives......Page 326
The Malthusian Dilemma......Page 333
Forming a Fractal Tree......Page 335
5.7 Compound Interest and the Number e......Page 342
Compounding at Different Intervals......Page 343
Continuous Compounding Using e......Page 345
Continuous Compounding Formula......Page 347
Exponential Functions Base e......Page 348
Converting eĸ into a......Page 349
5.8 Semi-Log Plots of Exponential Functions......Page 355
CHAPTER SUMMARY......Page 359
CHECK YOUR UNDERSTANDING......Page 360
CHAPTER 5 REVIEW: PUTTING IT ALL TOGETHER......Page 362
EXPLORATION 5.1 COMPUTER VIRUSES......Page 366
CHAPTER 6 LOGARITHMIC LINKS: LOGARITHMIC AND EXPONENTIAL FUNCTIONS......Page 369
Estimating Solutions to Exponential Equations......Page 370
Rules for Logarithms......Page 371
Solving Exponential Equations Using Logarithms......Page 376
Solving for Doubling Times and Half-Lives......Page 377
The Natural Logarithm......Page 381
Returning to Doubling Times and Half-Lives......Page 383
Converting exponential functions from base a to base e......Page 385
6.3 Visualizing and Applying Logarithmic Functions......Page 390
Logarithmic Growth......Page 391
Inverse Functions: Logarithmic vs. Exponential......Page 394
Applications of Logarithmic Functions......Page 396
Why Do Semi-Log Plots of Exponential Functions Produce Straight Lines?......Page 403
CHAPTER SUMMARY......Page 408
CHECK YOUR UNDERSTANDING......Page 409
CHAPTER 6 REVIEW: PUTTING IT ALL TOGETHER......Page 410
EXPLORATION 6.1 CHANGING BASES......Page 414
CHAPTER 7 POWER FUNCTIONS......Page 417
Scaling Up a Cube......Page 418
Size and Shape......Page 420
7.2 Direct Proportionality: Power Functions with Positive Powers......Page 423
Direct Proportionality......Page 424
Properties of Direct Proportionality......Page 425
Direct Proportionality with More Than One Variable......Page 428
The Graphs of ƒ(x) = x² and g(x) = x³......Page 432
Odd vs. Even Positive Integer Powers......Page 434
The Effect of the Coefficient k......Page 435
Which Eventually Grows Faster, a Power Function or an Exponential Function?......Page 440
Inverse Proportionality......Page 445
Properties of Inverse Proportionality......Page 447
Inverse Square Laws......Page 450
7.6 Visualizing Negative Integer Power Functions......Page 456
The Graphs of f(x) = x-¹ and g(x) = x-²......Page 457
Odd vs. Even Negative Integer Powers......Page 458
The Effect of the Coefficient k......Page 460
Looking for Lines......Page 467
Why Is a Log-Log Plot of a Power Function a Straight Line?......Page 468
Translating Power Functions into Equivalent Logarithmic Functions......Page 469
Analyzing Weight and Height Data......Page 471
Allometry: The Effect of Scale......Page 474
CHAPTER SUMMARY......Page 482
CHECK YOUR UNDERSTANDING......Page 483
CHAPTER 7 REVIEW: PUTTING IT ALL TOGETHER......Page 484
EXPLORATION 7.1 SCALING OBJECTS......Page 488
CHAPTER 8 QUADRATICS AND THE MATHEMATICS OF MOTION......Page 491
The Simplest Quadratic......Page 492
Designing Parabolic Devices......Page 493
The Standard Form of a Quadratic......Page 494
Properties of Quadratic Functions......Page 495
Estimating the Vertex and Horizontal Intercepts......Page 497
Reflecting across the Horizontal Axis......Page 502
Shifting Vertically and Horizontally......Page 504
Using Transformations to Get the Vertex Form......Page 507
Finding the Vertex from the Standard Form......Page 511
Converting between Standard and Vertex Forms......Page 513
Using Factoring to Find the Horizontal Intercepts......Page 520
Factoring Quadratics......Page 521
Using the Quadratic Formula to Find the Horizontal Intercepts......Page 524
The Factored Form......Page 528
Standard, Factored and Vertex Forms......Page 530
8.5 The Average Rate of Change of a Quadratic Function......Page 534
8.6 The Mathematics of Motion......Page 539
Deriving an Equation Relating Distance and Time......Page 540
Velocity: Change in Distance over Time......Page 542
Acceleration: Change in Velocity over Time......Page 544
Deriving an Equation for the Height of an Object in Free Fall......Page 546
Working with an Initial Upward Velocity......Page 549
CHAPTER SUMMARY......Page 555
CHECK YOUR UNDERSTANDING......Page 556
CHAPTER 8 REVIEW: PUTTING IT ALL TOGETHER......Page 557
EXPLORATION 8.1 HOW FAST ARE YOU? USING A RULER TO MAKE A REACTION TIMER......Page 560
CHAPTER 9 NEW FUNCTIONS FROM OLD......Page 563
Transforming a Function......Page 564
9.2 The Algebra of Functions......Page 577
9.3 Polynomials: The Sum of Power Functions......Page 585
Defining a Polynomial Function......Page 586
Visualizing Polynomial Functions......Page 588
Finding the Vertical Intercept......Page 591
Finding the Horizontal Intercepts......Page 592
Building a Rational Function: Finding the Average Cost of an MRI Machine......Page 599
Defining a Rational Function......Page 600
Visualizing Rational Functions......Page 601
Composing Two Functions......Page 609
Composing More Than Two Functions......Page 612
Inverse Functions: Returning the Original Value......Page 613
9.6 Exploring, Extending & Expanding......Page 623
CHAPTER SUMMARY......Page 634
CHECK YOUR UNDERSTANDING......Page 635
CHAPTER 9 REVIEW: PUTTING IT ALL TOGETHER......Page 636
Student Data Tables for Exploration 2.1......Page 641
Data Dictionary for FAM1000 Data......Page 645
SOLUTIONS......Page 647
INDEX......Page 761