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# Explorations in College Algebra, 5th Edition

**by**Linda Almgren Kime (University of Massachusetts, Boston); Judy Clark (University of Massachusetts-Boston ); Beverly K. Michael (University of Pittsburgh)

5th

### 9780470466445

0470466448

Paperback

1/1/2011

Wiley

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## Summary

Explorations in College Algebra, 5th Edition is designed to make algebra interesting and relevant to the student. The text adopts a problem-solving approach that motivates readers to grasp abstract ideas by solving real-world problems. The problems lie on a continuum from basic algebraic drills to open-ended, non-routine questions. The focus is shifted from learning a set of discrete mathematical rules to exploring how algebra is used in the social, physical, and life sciences. The goal of Explorations in College Algebra, 5th Edition is to prepare students for future advanced mathematics or other quantitatively based courses, while encouraging them to appreciate and use the power of algebra in answering questions about the world around us.

## Table of Contents

An Introduction to Data and Functions | |

Describing Single-Variable Data | |

Visualizing Single-Variable Data | |

Numerical Descriptors: What is "Average" Anyway? | |

An Introduction to Algebra Aerobics | |

An Introduction to Explore and Extend | |

Describing Relationships between Two Variables | |

Visualizing Two-Variable Data | |

Constructing a "60-Second Summary" | |

Using Equations to Describe Change | |

An Introduction to Functions | |

What is a Function? | |

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

Input and Output: Independent and Dependent Variables | |

When is a Relationship Not a Function? | |

The Language of Functions | |

Function Notation | |

Domain and Range | |

Visualizing Functions | |

Is There a Maximum or Minimum Value? | |

When is the Output of the Function Positive, Negative or Zero? | |

Is the Function Increasing or Decreasing? | |

Is the Graph Concave Up or Concave Down? | |

Getting the Big Idea | |

Chapter Summary | |

Check Your Understanding | |

Chapter 1 Review: Putting it all Together | |

Exploration 1.1 Collecting, Representing, and Analyzing Data | |

Rates of Change and Linear Function | |

Average Rates of Change | |

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

Defining the Average Rate of Change | |

Limitations of the Average Rate of Change | |

Change in the Average Rate of Change | |

The Average Rate of Change is a Slope | |

Calculating Slopes | |

Putting a Slant on Data | |

Slanting the Slope: Choosing Different End Points | |

Slanting the Data with Words and Graphs | |

Linear Functions: When Rates of Change are Constant | |

What if the U.S. Population Had Grown at a Constant Rate? | |

Real Examples of a Constant Rate of Change | |

The General Equation for a Linear Function | |

Visualizing Linear Functions | |

The Effect of b | |

The Effect of m | |

Finding Graphs and Equations of Linear Functions | |

Finding the Graph | |

Finding the Equation | |

Special Cases | |

Direct Proportionality | |

Horizontal and Vertical Lines | |

Parallel and Perpendicular Lines | |

Breaking the Line: Piecewise Linear Functions | |

Piecewise Linear Functions | |

The absolute value function | |

Step functions | |

Constructing Linear Models for Data | |

Fitting a Line to Data: The Kalama Study | |

Reinitializing the Independent Variable | |

Interpolation and Extrapolation: Making Predictions | |

Looking for Links between Education and Earnings: Using Regression Lines | |

Using U.S. Census Data | |

Summarizing the Data: Regression Lines | |

Regression Line: How good a fit? | |

Interpreting Regression Lines: Correlation vs. Causation | |

Raising More Questions: Going Deeper | |

Chapter Summary | |

Check Your Understanding | |

Chapter 2 Review: Putting it all Together | |

Having it Your Way | |

A Case Study on Education and Earnings | |

When Lines Meet: Linear Systems | |

Interpreting Intersection Points: Linear and Non-linear Systems | |

When Curves Collide: Non-linear Systems | |

When Lines Meet: Linear System | |

Visualizing and Solving Linear Systems | |

Visualizing Linear Systems | |

Strategies for Solving Linear Systems | |

Linear Systems in Economics: Supply and Demand | |

Reading between the Lines: Linear Inequalities | |

Above and Below the Line | |

Reading between the Lines | |

Manipulating Inequalities | |

Breakeven Points: Regions of Profit or Loss | |

Systems with Piecewise Linear Functions: Tax Plans | |

Graduated vs. Flat Income Tax | |

Comparing the Two Tax Models: Flat vs. Graduated Plans | |

Chapter Summary | |

Check Your Understanding | |

Chapter 3 Review: Putting it all Together | |

Flat vs. Graduated Income Tax: Who Benefits? | |

A Comparison of Hybrid and Conventional Automobiles | |

The Laws of Exponents and Logarithms: Measuring the Universe | |

The Numbers of Science: Measuring Time and Space | |

Powers of 10 and the Metric System | |

Scientific Notation | |

Positive Integer Exponents | |

Exponent Rules | |

Common Errors | |

Estimating Answers | |

Zero, Negative and Fractional Exponents | |

Zero and Negative Exponents | |

Fractional Exponents | |

Converting Units | |

Converting Units within the Metric Systems | |

Converting between the Metric and English Systems | |

Using Multiple Conversion Factors | |

Orders of Magnitude | |

Comparing Numbers of Widely Differing Sizes | |

Orders of Magnitude | |

Graphing Numbers of Widely Differing Sizes: Log Scales | |

Logarithms Base 10 | |

Finding the Logarithms of Powers of 10 | |

Finding the Logarithm of Any Positive Number | |

Plotting Numbers on a Logarithmic Scale | |

Chapter Summary | |

Check Your Understanding | |

Chapter 4 Review: Putting it all Together | |

The Scale and the Tale of the Universe | |

Growth and Decay: An Introduction to Exponential Functions | |

Exponential Growth | |

The Growth of E. coli Bacteria | |

The General Exponential Growth Function | |

Doubling Time | |

Looking at Real Growth Data for E. coli Bacteria | |

Linear vs. Exponential Functions | |

General Forms: Linear and Exponential Function | |

Data Tables: Identifying Linear and Exponential | |

Finding the Equation for an Exponential Function | |

Comparing the Average Rates of Change | |

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

Exponential Decay | |

The Decay of Iodine-131 | |

The General Exponential Decay Function | |

Half-Lives | |

Visualizing Exponential Functions | |

The Graphs of Exponential Functions | |

Horizontal Asymptotes | |

Exponential Functions: A Constant Percent Change | |

Exponential Growth: Increasing by a Constant Percent | |

Exponential Decay: Decreasing by a Constant Percent | |

Revisiting Linear vs. Exponential Functions | |

More Interesting Examples of Exponential Growth and Decay | |

Fitting a Curve | |

Doubling Time and Half-Life: Translating between Forms | |

The "Rule of 70" | |

The Malthusian Dilemma | |

Forming a Fractal Tree | |

Compound Interest and the Number e | |

Compounding at Different Intervals | |

Continuous Compounding Using e | |

Exponential Functions Base e | |

Converting ek into a | |

Semi-log Plots of Exponential Functions | |

Chapter Summary | |

Check Your Understanding | |

Chapter 5 Review: Putting it all Together | |

Computer Viruses | |

Logarithmic Links: Logarithmic and Exponential Functions | |

Using Logarithms to Solve Exponential Equations | |

Estimating Solutions to Exponential Equations | |

Rules for Logarithms | |

Solving Exponential Equations | |

Solving for Doubling Times and Half-Lives | |

Using Natural Logarithms to Solve Exponential Equations Base e | |

The Natural Logarithm | |

Returning to Doubling Times and Half-Lives | |

Visualizing and Applying Logarithmic Functions | |

The Graphs of Logarithmic Functions | |

Logarithmic Growth | |

Explore & Extend 6.3: Visualizing Logarithmic Functions | |

Stretching, Compressing and Reflecting | |

The Relationship between Logarithmic and Exponential Functions | |

Logarithmic vs. exponential growth | |

Logarithmic and exponential functions are inverses of each other | |

Applications of Logarithmic Functions | |

Measuring acidity: The pH scale | |

Using Semi-log Plots to Construct Exponential Models for Data | |

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

Chapter Summary | |

Check Your Understanding | |

Chapter 6 Review: Putting it all Together | |

Properties of Logarithmic Functions | |

Power Functions | |

The Tension between Surface Area and Volume | |

Scaling Up a Cube | |

Size and Shape | |

Direct Proportionality: Power Functions with Positive Powers | |

Direct Proportionality | |

Properties of Direct Proportionality | |

Direct Proportionality with more than one Variable | |

Visualizing Positive Integer Powers | |

The Graphs of f(x)=x 2 and g(x)=x 3 | |

Explore & Extend 7.3: Visualizing Power Functions | |

Odd vs. Even Powers | |

Symmetry | |

The Effect of the Coefficient k | |

Comparing Power and Exponential Functions | |

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

Inverse Proportionality: Power Functions with Negative Integer Powers | |

Inverse Proportionality | |

Properties of Inverse Proportionality | |

Explore & Extend 7.5: Designing Stringed Instruments | |

Inverse Square Laws | |

Visualizing Negative Integer Power Functions | |

The Graphs of f(x)=x^-1 and g(x)=x^-2 | |

Odd vs. Even Powers | |

Asymptotes | |

Symmetry | |

Explore & Extend 7.6: Finding Symmetries | |

The Effect of the Coefficient k | |

Using Logarithmic Scales to Find the Best Functional Model | |

Looking for Lines | |

Why is a Log-Log Plot of a Power Function a Straight Line? | |

Translating Power Functions into Equivalent Logarithmic Functions | |

Analyzing Weight and Height Data | |

Using a standard plot | |

Using a semi-log plot | |

Using a log-log plot | |

Explore & Extend 7.7: Constructing Functions from Log-Log Plots | |

Allometry: The Effect of Scale | |

Chapter Summary | |

Check Your Understanding | |

Chapter 7 Review: Putting it all Together | |

Scaling Objects | |

Quadratics and the Mathematics of Motion | |

An Introduction to Quadratic Functions: The Standard Form | |

The Simplest Quadratic | |

Designing parabolic devices | |

The Standard Form of a Quadratic | |

Properties of Quadratic Functions | |

Estimating the Vertex and Horizontal Intercepts | |

Visualizing Quadratics: The Vertex Form | |

Stretching and Compressing Vertically | |

Reflections across the Horizontal Axis | |

Shifting Vertically and Horizontally | |

Using Transformations to Get the Vertex Form | |

The Standard Form vs. the Vertex Form | |

Finding the Vertex from the Standard Form | |

Converting between Standard and Vertex Forms | |

Finding the Horizontal Intercepts: The Factored Form | |

Using Factoring to Find the Horizontal Intercepts | |

Factoring Quadratics | |

Using the Quadratic Formula to Find the Horizontal Intercepts | |

The discriminant | |

Imaginary and complex numbers | |

The Factored Form | |

The Mathematics of Motion | |

The Scientific Method | |

Deriving an Equation Relating Distance and Time | |

Velocity: Change in Distance over Time | |

Acceleration: Change in Velocity over Time | |

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

Working with an Initial Upward Velocity | |

The Average Rate of Change of a Quadratic Function | |

Chapter Summary | |

Check Your Understanding | |

Chapter 8 Review: Putting it all Together | |

How Fast are You? Using a Ruler to Make a Reaction Timer | |

New Functions from Old | |

Transforming a Function | |

Transformations | |

Stretching, compressing and shifting | |

Reflections | |

Symmetry | |

Combining Two Functions | |

The Algebra of Functions | |

Polynomial Functions | |

Rational Functions | |

A Final Example | |

Chapter Summary | |

Check Your Understanding | |

Chapter 9 Review: Putting it all Together | |

Appendix: Student Data Tables for Exploration 2.1 | |

Data Dictionary for FAM1000 Data | |

Solutions | |

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