Explorations in College Algebra, 2nd Edition,9780471371946

Explorations in College Algebra, 2nd Edition

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Edition: 2nd
ISBN13: 9780471371946
ISBN10: 0471371947
Format: Paperback
Publisher(s): Wiley
List Price: $113.60

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Summary

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).

Table of Contents

Making Sense of Data and Functions
1(57)
An Introduction to Single Variable Data
2(5)
Describing Data
2(1)
Numerical Descriptors: Measures of Central Tendency
2(1)
The Mean
2(1)
The median
3(1)
Numerical Descriptors: Frequency and Relative Frequency
4(1)
Percentage of what?
5(1)
Visualizing Single Variable Data
5(2)
An Introduction to Algebra Aerobics
7(1)
Writing about Data
7(4)
Constructing a ``60 Second Summary''
8(3)
Visualizing Two Variable Data
11(2)
Interpreting Equations and Their Graphs
13(3)
An Introduction to Functions
16(3)
What is a Function?
16(1)
Representing Functions with Words, Tables, Graphs, and Equations
16(1)
When Is a Relationship Not a Functions?
17(1)
How to tell if a graph represents a function: The vertical line test
18(1)
The Language of Functions
19(38)
Independent and Dependent Variables
19(2)
Domain and Range
21(4)
Putting Equations into ``Function Form''
25(2)
Function Notation
27(3)
Chapter Summary
30(1)
Exercises
31(19)
Collecting, Representing, and Analyzing Data
50(3)
Picturing Functions
53(2)
Deducing Formulas to Describe Data
55(2)
Rates of Change and Linear Functions
57(64)
Average Rates of Change
58(4)
Describing Change in the U.S. Population over Time
58(2)
Defining the Average Rate of Change
60(1)
Limitations of the Average Rate of Change
61(1)
The average rate of change is an average
61(1)
The average rate of change depends on the end points
61(1)
Change in the Average Rate of Change
62(2)
The Average Rate of Change is a Slope
64(4)
A note about calculating slopes: It doesn't matter which point is first
65(3)
Putting a Slant on Data
68(3)
Slanting the Slope: Choosing Different End Points
68(1)
Slanting the Data with Words and Graphs
69(2)
When Rates of Change Are Constant
71(4)
What if the U.S. Population Had Grown at a Constant Rate? A Hypothetical Example
71(1)
A Real Example of a Constant Rate of Change
71(2)
Finding an Equation to Model the Relationship between Female Infant Weight and Age
73(1)
Looking at this function in the abstract
74(1)
Linear Functions
75(5)
The General Linear Equations
75(1)
Finding the Graph of a Linear Function
76(1)
Finding the Equation of a Linear Function
76(1)
Finding the equation from a graph
77(1)
Finding the equation from words
77(2)
Finding the equation from a table
79(1)
Special Cases
80(5)
Direct Proportionality
80(1)
How to recognize direct proportionality
80(2)
Horizontal and Vertical Lines
82(2)
Parallel and Perpendicular Lines
84(1)
Why does this relationship hold for perpendicular lines?
84(1)
Finding Linear Models for Data
85(36)
Fitting a Line to Data
85(2)
Fitting a Line to a ``Cloud'' of Data Points
87(2)
Reinitializing the Independent Variable
89(2)
Chapter Summary
91(1)
Exercises
92(23)
Having It Your Way
115(2)
A Looking at Lines
117(1)
B Looking at Lines with a Graphing Calculator
118(3)
An Extended Exploration: Looking for Links Between Education and Income 121(328)
Using U.S. Census Data
122(4)
Going Further: How Good Are the Data?
126(1)
Summarizing the Data: Regression Lines
126(5)
What is the Relationship between Education and Income?
126(4)
Regression Lines: How Good a Fit?
130(1)
On Your Own: Interpret the correlation coefficient
130(1)
Interpreting Regression Lines: Correlation vs. Causation
131(1)
Next Steps: Raising More Questions
132(2)
Does Income Depend on Age?
132(1)
Does Income Depend on Gender?
133(1)
Going Deeper: Asking More Questions
134(1)
Exploring On Your Own
134(2)
Exercises
136(15)
When Lines Meet: Linear Systems
151(32)
An Economic Comparison of Solar vs. Conventional Heating Systems
152(3)
Finding Solutions to Systems of Linear Equations
155(6)
Visualizing Solutions to Systems of Linear Equations
155(1)
Using Equations to Find Solutions
155(1)
Substitution Method
156(2)
Elimination Method
158(2)
Special Cases: How Can You Tell if There Is No Unique Intersection Point?
160(1)
Intersection Points Representing Equilibrium
161(2)
Supply and Demand Curves
161(2)
Graduated vs. Flat Income Tax: Using Piecewise Linear Functions
163(20)
Simple Tax Models
163(1)
A flat tax model
164(1)
A graduated tax model: A piecewise linear function
164(1)
Comparing the two tax models
165(2)
The Case of Massachusetts
167(2)
Chapter Summary
169(1)
Exercises
170(11)
Flat vs. Graduated Income Tax: Who Benefits?
181(2)
The Laws of Exponents and Logarithms: Measuring the Universe
183(50)
Measuring Time and Space: The Numbers of Science
184(5)
The Metric System and Power of Ten
184(1)
Deep space
184(2)
Summary of Powers of Ten
186(1)
Scientific Notation
187(1)
Deep time
188(1)
Simplifying Expressions with Positive Integer Exponents
189(6)
Exponent Rules
190(1)
Using Ratios to Compare Sizes of Objects
191(2)
Common Errors
193(1)
Estimating Answers
194(1)
Simplifying Expressions with Negative Integer Exponents
195(2)
Converting Units
197(2)
Converting Units within the Metric System
197(1)
Converting between the Metric and English Systems
198(1)
Using Multiple Conversion Factors
199(1)
Simplifying Expressions with Fractional Exponents
199(5)
Square Roots: Expressions of the Form a1/2
200(1)
Estimating square roots
200(1)
nth Roots: Expressions of the Form a1/n
200(1)
Using a Calculator
201(1)
Rules for Computations with Radicals
201(2)
Fractional Powers: Expressions of the Form am/n
203(1)
Orders of Magnitude
204(4)
Comparing Numbers of Widely Differing Sizes
204(1)
A Measurement Scale Based on Orders of Magnitude: The Richter Scale
205(2)
Graphing Numbers of Widely Differing Sizes
207(1)
Logarithms Base 10
208(25)
Finding the Logarithms of Powers of 10
208(2)
Finding the Logarithms of Numbers between 1 and 10
210(1)
Finding the Logarithm of Any Positive Number
211(1)
Using logarithms to write any positive number as a power of 10
211(1)
Chapter Summary
212(1)
Exercises
213(16)
The Scale and the Tale of the Universe
229(2)
Patterns in the Positions and Motions of the Planets
231(2)
Growth and Decay: An Introduction to Exponential Functions
233(54)
Exponential Growth
234(7)
The Growth of E. coli bacteria
234(1)
A mathematical model for E. coli growth
235(1)
The General Exponential Growth Function
235(1)
Linear vs. Exponential Growth
236(2)
Comparing the Average Rates of Change of Linear and Exponential Functions
238(1)
Looking at Real Growth Data for E. coli Bacteria
239(1)
Limitations of the Model
240(1)
Exponential Decay
241(2)
The Decay of Iodine-131
241(2)
The Graphs of Exponential Functions
243(3)
The Effect of the Base a
243(1)
When a > 1: Exponential growth
243(1)
When 0 < a <1: Exponential decay
243(1)
The Effect of the Coefficient C
244(1)
Horizontal Asymptotes
244(2)
Exponential Growth or Decay Expressed in Percentages
246(2)
Examples of Exponential Growth and Decay
248(15)
Medicare Costs
248(2)
A Linear vs. an Exponential Model through Two Points
250(1)
Linear model: The population increases by a fixed amount each year
250(1)
Exponential model: The population increases by a fixed percentage each year
250(2)
Radioactive Decay
252(1)
The ``Rule of 70'': A Rule of Thumb for Calculating Doubling or Halving Times
253(2)
White Blood Cell Counts
255(1)
Compound Interest
256(1)
Short-term returns
256(1)
Long-term returns
257(1)
Too good to be true?
258(1)
Inflation and the Diminishing Dollar
258(2)
Musical Pitch
260(1)
The Malthusian Dilemma
260(1)
Trees
261(2)
Semi-log Plots of Exponential Functions
263(24)
Chapter Summary
265(1)
Exercises
266(17)
Properties of Exponential Functions
283(2)
Recognizing Exponential Patterns in Data Tables
285(2)
Logarithmic Links: Logarithmic and Exponential Functions
287(54)
Using Logarithms to Solve Exponential Equations
288(9)
Estimating Solutions to Exponential Equations
288(1)
Properties of Logarithms
289(5)
Answering Our Original Question: Using Logarithms to Solve Exponential Equations
294(3)
Base e and Continuous Compounding
297(5)
A Brief Introduction to e
297(1)
Continuous Compounding
298(2)
Generalizing Our Results
300(2)
The Natural Logarithm
302(3)
Logarithmic Functions
305(5)
Measuring Acidity: The pH Scale
306(1)
Measuring Noise: The Decibel Scale
307(3)
Writing Exponential Functions Using Base e
310(4)
Translating from Base a to Base e
310(2)
Determining the Equation of an Exponential Function through Two Points
312(2)
An Introduction to Composition and Inverse Functions
314(27)
Comparing y = log x and y = 10x
314(2)
The Composition of Two Functions
316(1)
Inverse Functions
317(1)
Converting between dollars and yen
317(3)
Determining if a function has an inverse
320(1)
Finding the formula for the inverse of a function
320(3)
Chapter Summary
323(2)
Exercises
325(12)
Properties of Logarithmic Functions
337(4)
Power Functions
341(56)
The Tension between Surface Area and Volume
342(4)
Scaling Up a Cube
342(1)
Surface area of a cube
342(1)
Volume of a cube
343(1)
Surface area/volume
344(1)
Size and Shape
344(2)
Power Functions with Positive Powers
346(5)
Direct Proportionality
347(2)
Direct Proportionality with More Than One Variable
349(2)
Visualizing Positive Integer Powers
351(3)
Moving to the Abstract
351(1)
What happens when x > 0 and approaches + ∞
351(1)
What happens when x < 0 and approaches - ∞
351(1)
Odd vs. Even Powers
352(1)
Symmetry of the graphs
353(1)
The Effect of the Coefficient k
353(1)
k > 0: Comparing y = kxp to y = xp when k is positive
353(1)
k < 0: Comparing y = kxp to y = xp when k is negative
353(1)
Comparing Power and Exponential Functions
354(3)
Visualizing the difference
355(2)
Power Functions with Negative Integer Powers
357(7)
Inverse square laws
358(6)
Visualizing Negative Integer Power Functions
364(3)
Moving to the Abstract
364(1)
What happens when x > 0 and approaches + ∞?
364(1)
What happens when 0 < x < 1?
364(1)
What happens when x < 0?
365(1)
Odd vs. Even Powers
365(2)
Using Logarithmic Scales to Find the Best Function Model
367(6)
Semi-log Plots
367(1)
Log-log Plots
368(1)
Conclusion
369(1)
Using Semi-log and Log-log Plots to Investigate Data
370(3)
Allometry: The Effect of Scale
373(24)
Surface Area vs. Body Mass
373(2)
Metabolic Rate vs. Body Mass
375(1)
Chapter Summary
376(1)
Exercises
377(13)
Scaling Objects
390(2)
Predicting Properties of Power Functions
392(2)
Visualizing Power Functions with Negative Integer Powers
394(3)
Quadratic and Other Polynomial Functions
397(52)
Polynomial Functions
398(8)
Adding Power Functions: Polynomials
399(1)
Visualizing Polynomial Functions
400(1)
Intercepts of Polynomial Functions
401(5)
Properties of Quadratic Functions
406(4)
Visualizing Quadratic Functions
406(2)
Estimating the Vertex and the Intercepts
408(2)
Finding the Horizontal Intercepts of a Quadratic Function
410(8)
Using the Quadratic Formula
411(4)
The Factored Form
415(1)
Factoring Review
415(3)
Finding the Vertex
418(8)
Why the Vertex is Important
418(1)
Using a Formula to Find the Vertex
418(2)
The Vertex Form: The a-h-k Form
420(3)
Converting from the a-h-k Form to the a-b-c Form of a Quadratic
423(1)
Two Strategies for Getting from the a-b-c to the a-h-k Form of a Quadratic
423(2)
Finding an Equation from the Graph of a Parabola
425(1)
Average Rates of Change of Quadratic Function
426(23)
Chapter Summary
429(1)
Exercises
430(16)
Properties of Quadratic Functions
446(3)
An Extended Exploration: The Mathematics of Motion 449(22)
The Scientific Method
450(11)
The Free Fall Experiment
450(1)
Interpreting Data from a Free Fall Experiment
451(1)
Important Questions
452(1)
Deriving an Equation Relating Distance and Time
453(2)
Returning to Galileo's Question
455(3)
Deriving an Equation for the Height of an Object in Free Fall
458(3)
Collecting and Analyzing Data from a Free Fall Experiment
461(10)
Exercises
464(7)
Anthology of Readings 471(62)
Answers to Algebra Aerobics 533(26)
Brief Solutions to Odd-Numbered Problems 559(36)
Index 595

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