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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
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