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.
Table of Contents
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