Explorations in College Algebra, 2nd Edition

by Linda Almgren Kime (Univ. of Massachusetts, Boston); Judy Clark (Univ. of Massachusetts, Boston)

• ISBN13: 9780471371946
• ISBN10: 0471371947
• Edition: 2nd
• Format: Paperback
• Publisher: Wiley

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