Mathematics in Action: Algebraic, Graphical, and Trigonometric Problem Solving

by Consortium for Foundation Mathematics

• ISBN13: 9780321149206
• ISBN10: 0321149203
• Edition: 2nd
• Format: Package
• Copyright: 2004-01-01
• Publisher: Addison Wesley

Summary

This intermediate algebra text, based on standards in the AMATYC Crossroads document, motivates college math students to develop mathematical literacy and a solid foundation for future study in mathematics and other disciplines. This third book of a three-book series presents mathematical concepts and skills through relevant activities derived from real-life situations. These activities are meaningful to students because they illustrate how mathematics arises naturally from real-world situations and problems. The Mathematics in Action series is based on the assumption that students learn mathematics best by doing mathematics in a meaningful context. Therefore, the text takes a collaborative approach to learning. Students take an active role in their own learning by working in groups, thereby developing communication skills, a sense of independence, and a can-do attitude about mathematics. Technology is integrated throughout the book so that students learn to interpret real-life data numerically, symbolically, and graphically. Regardless of their level of preparation for the course, students can use this text to increase their knowledge of mathematics, their problem-solving skills, and their overall confidence in their ability to learn.

Table of Contents

Preface	xiii
Function Sense	1	(180)
Modeling with Functions	1	(42)
Parking Problems	1	(12)
Distinguish between input and output
Define a function
Represent a function numerically and graphically
Write a function using function notation
Fill 'er Up	13	(10)
Objectives: Determine the equation (symbolic representation) that defines a function
Write the equation to define a function
Determine the domain and range of a function
Identify the independent and the dependent variables of a function
Stopping Short	23	(9)
Use a function as a mathematical model
Determine when a function is increasing, decreasing, or constant
Use the vertical line test to determine if a graph represents a function
Graphs Tell Stories	32	(4)
Describe in words what a graph tells you about a given situation
Sketch a graph that best represents the situation described in words
What Have I Learned?	36	(1)
How Can I Practice?	37	(6)
Linear Functions	43	(53)
Walking for Fitness	43	(8)
Determine the average rate of change
Depreciation	51	(12)
Interpret slope as an average rate of change
Use the formula to determine slope
Discover the practical meaning of vertical and horizontal intercepts
Develop the slope/intercept form of an equation of a line
Use the slope/intercept formula to determine vertical and horizontal intercepts
A Visit to the Architect	63	(10)
Determine the slope-intercept form of a line given the slope and vertical intercept
Determine the slope-intercept form of a line given two points on the line
Compare the slopes of parallel lines
Skateboard Heaven	73	(9)
Write an equation of a line in general form Ax + By = C
Write the slope-intercept form of a linear equation given the general form
College Tuition	82	(6)
Determine a line of best fit with a straightedge
Determine the equation of a regression line using a graphing calculator
Use the regression equation to interpolate and extrapolate
What Have I Learned?	88	(2)
How Can I Practice?	90	(6)
Systems of Linear Equations, Inequalities, and Absolute Value Functions	96	(85)
Ride for Less	96	(13)
Solve a system of 2 X 2 linear equations numerically and graphically
Solve a system of 2 X 2 linear equations using the substitution method
Solve an equation of the form ax + b = cx + d for x
Healthy Lifestyle	109	(7)
Solve a 2 X 2 linear system algebraically using the substitution method and the addition method
Solve equations containing parentheses
Sam's Cafe	116	(5)
Solve a 3 X 3 linear system of equations
How Long Can You Live?	121	(13)
Solve linear inequalities numerically and graphically
Use properties of inequalities to solve linear inequalities algebraically
Solve compound inequalities algebraically and graphically
Long Distance by Phone	134	(10)
Objectives: Graph a piecewise linear function
Write a piecewise linear function to represent a given situation
Graph a function defined by y = |x -- c|
How Much Can You Tolerate?	144	(12)
Write a compound inequality to represent a given statement
Determine the error
Solve an equation involving absolute value using a number line
Solve an inequality involving absolute value using a number line
Solve absolute value equations and inequalities using a graphing approach
Interpret absolute value as distance
Graph an absolute value function
What Have I Learned?	156	(2)
How Can I Practice?	158	(5)
Chapter 1 Summary	163	(6)
Chapter 1 Gateway Review	169	(12)
The Algebra of Functions	181	(102)
Addition, Subtraction, and Multiplication of Polynomial Functions	181	(43)
Spending and Earning Money	181	(12)
Identify a polynomial expression
Identify a polynomial function
Add and subtract polynomial expressions
Add and subtract polynomial functions
Viewing the Algebra of Functions	193	(5)
Explore adding and subtracting functions graphically
How Does Your Garden Grow?	198	(10)
Multiply two binomials using the FOIL method
Multiply two polynomial functions
Apply property of exponents to multiply powers having the same base
Stargazing	208	(9)
Convert scientific notation to decimal notation
Convert decimal notation to scientific notation
Apply the property of exponents to divide powers having the same base
Apply the property of exponents a0 = 1, where a ≠ 0
Apply the property of exponents a-n = 1/an, where a ≠ 0, and n is any real number
What Have I Learned?	217	(1)
How Can I Practice	218	(6)
Composition and Inverse of Functions	224	(59)
Inflated Balloons	224	(6)
Determine the composition of two functions
Explore the relationship between f(g(x)) and g(f(x))
Finding a Bargain	230	(5)
Solve problems using the composition of functions
The Square of a Cube	235	(9)
Objectives: Apply the property of exponents to simplify an expression involving a power to a power
Apply the property of exponents to expand the power of a product
Determine the nth root of a real number
Write a radical as a power having a rational exponent and as a rational exponent having a power
Study Time	244	(7)
Determine the inverse of a function represented by a table of values
Use the notation f-1 to represent an inverse function
Use the property f(f-1(x)) = f-1(f(x)) = x to recognize inverse functions
Determine the domain and range of a function and its inverse
Temperature Conversions	251	(11)
Determine the equation of the inverse of a function represented by an equation
Describe the relationship between the graphs of inverse functions
Determine the graph of the inverse of a function represented by a graph
Use the graphing calculator to produce graphs of an inverse function
What Have I Learned?	262	(2)
How Can I Practice?	264	(7)
Chapter 2 Summary	271	(4)
Chapter 2 Gateway Review	275	(8)
Exponential and Logarithmic Functions	283	(128)
Exponential Functions	283	(62)
The Summer Job	283	(14)
Determine the growth or decay factor of an exponential function
Identify the properties of the graph of an exponential function defined by y = bx, where b > 0 and b ≠ 1
Graph an exponential function
Cellular Phones	297	(11)
Determine the growth and decay factor for an exponential function represented by a table of values or an equation
Graph exponential functions defined by y = abx, where b > 0 and b ≠ 1
Determine the doubling and halving time
Population Growth	308	(8)
Determine annual growth or decay rate of an exponential function represented by a table of values or an equation
Graph an exponential function having equation y = a(1 + r)x
Photocopying Machines	316	(3)
Generate data given the growth or decay rate of an exponential function
Write exponential functions given the growth or decay rate
Graph exponential functions from data
Determine doubling and halving times from exponential functions
Compound Interest	319	(12)
Apply the compound interest and continuous compounding formulas to a given situation
Graph base e exponential functions
Solve problems involving continuous growth and decay models
College Graduates	331	(6)
Determine the equation of an exponential function that best fits the given data
Make predictions using an exponential regression equation
Determine whether a linear or exponential model best fits the data
What Have I Learned?	337	(1)
How Can I Practice?	338	(7)
Logarithmic Functions	345	(66)
The Diameter of Spheres	345	(9)
Define logarithm
Write an exponential statement in logarithmic form
Write a logarithmic statement in exponential form
Determine log and In using the calculator
Walking Speed of Pedestrians	354	(7)
Determine the inverse of the exponential function
Identify properties of the graph of y = log x
Identify the properties of the graph of a logarithmic function
Graph the natural logarithmic function
Walking Speed of Pedestrians, continued	361	(9)
Compare the average rate of change of increasing logarithmic, linear, and exponential functions
Determine the regression equation of a natural logarithmic function that best fits a set of data
The Elastic Ball	370	(11)
Apply the log of a product property
Apply the log of a quotient property
Apply the log of a power property
Discover change of base formula
Prison Growth	381	(7)
Solve exponential equations both graphically and algebraically
Frequency and Pitch	388	(4)
Solve logarithmic equations both graphically and algebraically
What Have I Learned?	392	(2)
How Can I Practice	394	(5)
Chapter 3 Summary	399	(4)
Chapter 3 Gateway Review	403	(8)
Quadratic and Higher-Order Polynomial Functions	411	(118)
Introduction to Quadratic Functions	411	(60)
Baseball and the Sears Tower	411	(10)
Objectives: Identify functions of the form f(x) = ax2 + bx + c as quadratic functions
Explore the role of a as it relates to the graph of f(x) = ax2 + bx + c
Explore the role of b as it relates to the graph of f(x) = ax2 + bx + c
Explore the role of c as it relates to the graph of f(x) = ax2 + bx + c
The Shot Put	421	(13)
Determine the vertex or turning point of a parabola
Determine the axis of symmetry of a parabola
Identify the domain and range
Determine the vertical intercept of a parabola
Determine the horizontal-intercept(s) of a parabola graphically
Per Capita Personal Income	434	(7)
Solve quadratic equations numerically
Solve quadratic equations graphically
Solve quadratic inequalities graphically
Sir Isaac Newton	441	(7)
Factor expressions by removing the greatest common factor
Factor trinomials using trial and error
Use the zero-product principle to solve equations
Solve quadratic equations by factoring
Motorcycle Deaths	448	(9)
Solve quadratic equations by the quadratic formula
Air Quality in Atlanta	457	(6)
Determine quadratic regression models using the graphing calculator
Solve problems using quadratic regression models
What Have I Learned?	463	(2)
How Can I Practice?	465	(6)
Complex Numbers and Problem Solving Using Quadratic Functions	471	(18)
Complex Numbers	471	(8)
Identify the imaginary unit i = √-1
Identify a complex number
Determine the value of the discriminant b2 - 4ac
Determine the types of solutions to a quadratic equation
Solve a quadratic equation in the complex number system
Airfare	479	(4)
Build a quadratic model as a product of linear models
Analyze a model contextually
Chemical-Waste Holding Region	483	(2)
Solving problems using quadratic functions
What Have I Learned?	485	(1)
How Can I Practice?	486	(3)
Curve Fitting and Higher Order Polynomial Functions	489	(40)
The Power of Power Functions	489	(8)
Identify a direct variation function
Determine the constant of variation
Identify the properties of graphs of power functions defined by y = kxn, where n is a positive integer, k ≠ 0
Hot Air Balloon	497	(7)
Identify equations that define polynomial functions
Determine the degree of a polynomial function
Determine the intercepts of the graph of a polynomial function
Identify the properties of the graphs of polynomial functions
Stolen Bases	504	(6)
Determine the regression equation of a polynomial function that best fits the data
Distinguish between a discrete function and a continuous function
Finding the Maximum Volume	510	(2)
Problem solving using polynomial functions
What Have I Learned?	512	(2)
How Can I Practice	514	(5)
Chapter 4 Summary	519	(4)
Chapter 4 Gateway Review	523	(6)
Rational and Radical Functions	529	(116)
Rational Functions	529	(68)
Speed Limits	529	(10)
Determine the domain and range of a function defined by y = k/x, k is a nonzero real number
Determine the vertical and horizontal asymptotes of the graph of y = k/x
Sketch a graph of functions of the form y = k/x
Determine the properties of graphs having equation y = k/x
Loudness of a Sound	539	(13)
Graph an inverse variation function defined by an equation of the form y = k/xn, where n is any positive integer
Describe the properties of graphs having equation y = k/x01>Determine the constant of proportionality (also called the constant of variation)
Percent Markup	552	(10)
Determine the domain of a rational function defined by an equation of the form y = k/g(x), where k is a nonzero constant and g(x) is a polynomial expression
Identify the vertical and horizontal asymptotes of y = k/g(x)
Sketch a graph of rational functions defined by y = k/g(x)
Blood-Alcohol Levels	562	(14)
Solve an equation involving a rational expression using an algebraic approach
Solve an equation involving a rational expression using a graphing approach
Determine horizontal asymptotes of the graph of y = f(x)/g(x), where f(x) and g(x) are first-degree polynomials
Traffic Flow	576	(8)
Determine the least common denominator (LCD) of two or more rational expressions
Solve an equation involving rational expressions using an algebraic approach
Solve a formula for a specific variable
Electrical Circuits	584	(6)
Add and subtract rational expressions
Simplify a complex fraction
What Have I Learned?	590	(1)
How Can I Practice?	591	(6)
Radical Functions	597	(48)
Hang Time	597	(14)
Determine the domain of a radical function defined by where g(x) is a polynomial
Graph functions having equation y = √g(x) and y = √g(x)
Identify the properties of the graph of y = √g(x) and y = √g(x)
Falling Objects	611	(10)
Solve an equation involving a radical expression using a graphical and algebraic approach
Propane Tank	621	(9)
Determine the domain of a function defined by an equation of the form y = n√g(x), where n is a positive integer and g(x) is a polynomial
Graph y = n√g(x)
Identify the properties of graphs of y = n√g(x)
Solve radical equations that contain radical expressions with an index other than 2
What Have I Learned?	630	(1)
How Can I Practice?	631	(2)
Chapter 5 Summary	633	(4)
Chapter 5 Gateway Review	637	(8)
Introduction to the Trigonometric Functions	645	(1)
Introducing the Sine, Cosine, and Tangent Functions	645	(41)
The Leaning Tower of Pisa	645	(10)
Identify the sides and corresponding angles of a right triangle
Determine the length of the sides of similar right triangles using proportions
Determine the sine, cosine, and tangent of an angle using a right triangle
Determine the sine, cosine, and tangent of an acute angle by using the graphing calculator
A Gasoline Problem	655	(6)
Identify complementary angles
Demonstrate that the sine and cosine of complementary angles are equal
The Sidewalks of New York	661	(6)
Determine the inverse tangent of a number
Determine the inverse sine and cosine of a number using the graphing calculator
Identify the domain and range of the inverse sine, cosine, and tangent functions
Solving a Murder	667	(5)
Determine the measure of all sides and all angles of a right triangle
How Stable Is That Tower?	672	(4)
Solve problems using right-triangle trigonometry
Seeing Abraham Lincoln	676	(4)
Solve optimization problems using right-triangle trigonometry and by analyzing graphs
What Have I Learned?	680	(3)
How Can I Practice?	683	(3)
Why Are the Trigonometric Functions Called Circular Functions?	686	(1)
Learn Trig or Crash!	686	(11)
Determine the coordinates of points on a unit circle using sine and cosine functions
Sketch a graph of y = sin x and y = cos
Identify the properties of the graphs of the sine and cosine functions
It Won't Hertz	697	(7)
Convert between degree and radian measure
Identify the period and frequency of a function defined by y = a sin (bx) or y = a cos(bx) using the graph
Get in Shape	704	(6)
Determine the amplitude of the graph of y = a sin (bx) or y = a cos (bx)
Speeding Up	710	(5)
Determine the period of the graph of y = a sin (bx) and y = a cos (bx) using a formula
Running with a Friend	715	(5)
Determine the displacement of y = a sin (bx + c) and y = a cos(bx + c) using a formula
What Have I Learned?	720	(1)
How Can I Practice?	721	(4)
Chapter 6 Summary	725	(4)
Chapter 6 Gateway Review	729
APPENDIXES
Appendix A Concept Review	1	(22)
Appendix B Trigonometry	23	(10)
Appendix C The TI-83 Plus Graphing Calculator	33	(14)
Selected Answers	47	(24)
Glossary	71	(4)
Index	75