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

785 Pages·2011·4.34 MB·English

by  - - Consortium

Mathematics
in Action
Algebraic, Graphical, and Trigonometric
Problem Solving
Fourth Edition
The Consortium for Foundation Mathematics
Ralph Bertelle Columbia-Greene Community College
Judith Bloch University of Rochester
Roy Cameron SUNY Cobleskill
Carolyn Curley Erie Community College—South Campus
Ernie Danforth Corning Community College
Brian Gray Howard Community College
Arlene Kleinstein SUNY Farmingdale
Kathleen Milligan Monroe Community College
Patricia Pacitti SUNY Oswego
Rick Patrick Adirondack Community College
Renan Sezer LaGuardia Community College
Patricia Shuart Polk State College—Winter Haven,Florida
Sylvia Svitak Queensborough Community College
Assad J.Thompson LaGuardia Community College
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Mathematics in action: Algebraic, graphical, and trigonometric problem
these restrictions
solving / the Consortium for Foundation Mathematics. — 4th ed.
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p. cm.
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Includes index.
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ISBN-13: 978-0-321-69861-2 (student ed.) needs of other
ISBN-10: 0-321-69861-4 (student ed.) instructors who rely
ISBN-13: 978-0-321-69290-0 (instructor ed.) on these materials.
ISBN-10: 0-321-69290-X (instructor ed.)
1. Algebra—Textbooks. I. Consortium for Foundation Mathematics.
II. Title: Algebraic, graphical, and trigonometric problem solving.
QA152.3.M38 2012
512—dc22 2009052062
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1 2 3 4 5 6 7 8 9 10—EB—14 13 12 11 10
Contents
Preface xiv
To the Student xx
CHAPTER 1 Function Sense 1
Cluster 1 Modeling with Functions 1
Activity 1.1 Parking Problems 1
Objectives: 1.Identify inputand outputin situations involving two variable quantities.
2.Identify a functional relationship between two variables.
3.Identify the independentand dependentvariables.
4.Use a table to numerically representa functional relationship between
two variables.
5.Write a function using function notation.
Activity 1.2 Fill ’er Up 11
Objectives: 1.Determine the equation (symbolic representation) thatdefines
a function.
2.Determine the domain and range of a function.
3.Identify the independentand the dependentvariables of a function.
Activity 1.3 Graphically Speaking 18
Objectives: 1.Representa function verbally,symbolically,numerically,and graphically.
2.Distinguish between a discrete function and a continuous function.
3.Graph a function using technology.
Activity 1.4 Stopping Short 28
Objectives: 1.Use a function as a mathematical model.
2.Determine when a function is increasing,decreasing,or constant.
3.Use the vertical line testto determine if a graph represents a function.
Activity 1.5 Graphs Tell Stories 37
Objectives: 1.Describe in words whata graph tells you abouta given situation.
2.Sketch a graph thatbestrepresents the situation described in words.
iii
iv Contents
3.Identify increasing,decreasing,and constantparts of a graph.
4.Identify minimum and maximum points on a graph.
WhatHave I Learned? 44
How Can I Practice? 46
Cluster 2 Linear Functions 51
Activity 1.6 Walking for Fitness 51
Objective: 1.Determine the average rate of change.
Activity 1.7 Depreciation 58
Objectives: 1.Interpretslope as an average rate of change.
2.Use the formula to determine slope.
3.Discover the practical meaning of vertical and horizontal intercepts.
4.Develop the slope-interceptform of an equation of a line.
5.Use the slope-interceptformula to determine vertical and
horizontal intercepts.
6.Determine the zeros of a function.
Activity 1.8 A New Computer 69
Objectives: 1.Write a linear equation in the slope-interceptform,given the initial
value and the average rate of change.
2.Write a linear equation given two points,one of which is the
vertical intercept.
3.Use the point-slope form to write a linear equation given two points,
neither of which is the vertical intercept.
4.Compare slopes of parallel lines.
Activity 1.9 Skateboard Heaven 80
Objectives: 1.Write an equation of a line in standard form Ax + By = C.
2.Write the slope-interceptform of a linear equation given the
standard form.
3.Determine the equation of a horizontal line.
4.Determine the equation of a vertical line.
Activity 1.10 College Tuition 88
Objectives: 1.Constructscatterplots from sets of data pairs.
2.Recognize when patterns of points in a scatterplothave a linear form.
3.Recognize when the pattern in the scatterplotshows thatthe two
variables are positively related or negatively related.
4.Estimate and draw a line of bestfitthrough a setof points in a
scatterplot.
5.Use a graphing calculator to determine a line of bestfitby the
least-squares method.
Contents v
6.Measure the strength of the correlation (association) by a
correlation coefficient.
7.Recognize thata strong correlation does notnecessarily imply a linear
or a cause-and-effectrelationship.
WhatHave I Learned? 99
How Can I Practice? 100
Cluster 3 Systems of Linear Equations,Inequalities,and Absolute
Value Functions 105
Activity 1.11 Moving Out 105
Objectives: 1.Solve a system of 2 * 2linear equations numerically and graphically.
2.Solve a system of 2 * 2linear equations using the substitution method.
3.Solve an equation of the form ax + b = cx + dfor x.
Activity 1.12 Healthy Lifestyle 117
Objectives: 1.Solve a 2 * 2linear system algebraically using the substitution
method and the addition method.
2.Solve equations containing parentheses.
Activity 1.13 Manufacturing Cell Phones 124
Objective: 1.Solve a 3 * 3linear system of equations.
Activity 1.14 Earth Week 129
Objective: 1.Solve a linear system of equations using matrices.
Activity 1.15 How Long Can You Live? 136
Objectives: 1.Solve linear inequalities in one variable numerically
and graphically.
2.Use properties of inequalities to solve linear inequalities in one
variable algebraically.
3.Solve compound inequalities algebraically.
4.Use interval notation to representa setof real numbers described
by an inequality.
Activity 1.16 Sales Commission 147
Objectives: 1.Graph a piecewise linear function.
2.Write a piecewise linear function to representa given situation.
3.Graph a function defined by y = ƒx - cƒ.
WhatHave I Learned? 157
How Can I Practice? 158
Chapter 1 Summary 162
Chapter 1 Gateway Review 167
vi Contents
CHAPTER 2 The Algebra of Functions 177
Cluster 1 Addition,Subtraction,and Multiplication of Polynomial Functions 177
Activity 2.1 Spending and Earning Money 177
Objectives: 1.Identify a polynomial expression.
2.Identify a polynomial function.
3.Add and subtractpolynomial expressions.
4.Add and subtractpolynomial functions.
Activity 2.2 The Dormitory Parking Lot 188
Objectives: 1.Multiply two binomials using the FOIL method.
2.Multiply two polynomial functions.
3.Apply the property of exponents to multiply powers having the same base.
Activity 2.3 Stargazing 199
Objectives: 1.Convertscientific notation to decimal notation.
2.Convertdecimal notation to scientific notation.
3.Apply the property of exponents to divide powers having the same base.
4.Apply the property of exponents a0 = 1,where a Z 0.
5.Apply the property of exponents a-n = 1 ,where a Z 0.and nis any
n
a
real number.
Activity 2.4 The Cube of a Square 207
Objectives: 1.Apply the property of exponents to simplify an expression involving
a power to a power.
2.Apply the property of exponents to expand the power of a product.
3.Determine the nth rootof a real number.
4.Write a radical as a power having a rational exponentand write a base
to a rational exponentas a radical.
WhatHave I Learned? 216
How Can I Practice? 217
Cluster 2 Composition and Inverses of Functions 223
Activity 2.5 Inflated Balloons 223
Objectives: 1.Determine the composition of two functions.
2.Explore the relationship between f1g1x22and g1 f1x22.
Activity 2.6 Finding a Bargain 228
Objective: 1.Solve problems using the composition of functions.
Activity 2.7 Study Time 232
Objectives: 1.Determine the inverse of a function represented by a table of values.
2.Use the notation f -1to representan inverse function.
Contents vii
3.Use the property f1 f -11x22 = f -1f1x22 = xto recognize inverse functions.
4.Determine the domain and range of a function and its inverse.
Activity 2.8 Temperature Conversions 238
Objectives: 1.Determine the equation of the inverse of a function represented
by an equation.
2.Describe the relationship between graphs of inverse functions.
3.Determine the graph of the inverse of a function represented by a graph.
4.Use the graphing calculator to produce graphs of an inverse function.
WhatHave I Learned? 247
How Can I Practice? 248
Chapter 2 Summary 252
Chapter 2 Gateway Review 255
CHAPTER 3 Exponential and Logarithmic Functions 261
Cluster 1 Exponential Functions 261
Activity 3.1 The Summer Job 261
Objectives: 1.Determine the growth factor of an exponential function.
2.Identify the properties of the graph of an exponential function
defined by y = bx,where b 7 1.
3.Graph an increasing exponential function.
Activity 3.2 Half-Life of Medicine 269
Objectives: 1.Determine the decay factor of an exponential function.
2.Graph a decreasing exponential function.
3.Identify the properties of an exponential function defined by y = bx,
where b 7 0and b Z 1.
Activity 3.3 Cell Phones 277
Objectives: 1.Determine the growth and decay factor for an exponential function
represented by a table of values or an equation.
2.Graph an exponential function defined by y = abx,where b 7 0
and b Z 1,a Z 0.
3.Determine the doubling and halving time.
Activity 3.4 Population Growth 289
Objectives: 1.Determine the annual growth or decay rate of an exponential function
represented by a table of values or an equation.
2.Graph an exponential function having equation y = a11 + r2x Z 0.
Activity 3.5 Time Is Money 297
Objective: 1.Apply the compound interestand continuous compounding formulas
to a given situation.
viii Contents
Activity 3.6 Continuous Growth and Decay 305
Objectives: 1.Discover the relationship between the equations of exponential
functions defined by y = abtand the equations of continuous growth
and decay exponential functions defined by y = aekt.
2.Solve problems involving continuous growth and decay models.
3.Graph base eexponential functions.
Activity 3.7 Bird Flu 314
Objectives: 1.Determine the regression equation of an exponential function thatbest
fits the given data.
2.Make predictions using an exponential regression equation.
3.Determine whether a linear or exponential model bestfits the data.
WhatHave I Learned? 322
How Can I Practice? 323
Cluster 2 Logarithmic Functions 329
Activity 3.8 The Diameter of Spheres 329
Objectives: 1.Define logarithm.
2.Write an exponential statementin logarithmic form.
3.Write a logarithmic statementin exponential form.
4.Determine log and ln values using a calculator.
Activity 3.9 Walking Speed of Pedestrians 337
Objectives: 1.Determine the inverse of the exponential function.
2.Identify the properties of the graph of a logarithmic function.
3.Graph the natural logarithmic function.
Activity 3.10 Walking Speed of Pedestrians,continued 344
Objectives: 1.Compare the average rate of change of increasing logarithmic,linear,
and exponential functions.
2.Determine the regression equation of a natural logarithmic function
having the equation y = a + bln xthatbestfits a setof data.
Activity 3.11 The Elastic Ball 354
Objectives: 1.Apply the log of a productproperty.
2.Apply the log of a quotientproperty.
3.Apply the log of a power property.
4.Discover change-of-base formula.
Activity 3.12 Prison Growth 363
Objective: 1.Solve exponential equations both graphically and algebraically.
WhatHave I Learned? 370
How Can I Practice? 372
Chapter 3 Summary 375
Chapter 3 Gateway Review 378
Contents ix
CHAPTER 4 Quadratic and Higher-Order
Polynomial Functions 385
Cluster 1 Introduction to Quadratic Functions 385
Activity 4.1 Baseball and the Willis Tower 385
Objectives: 1.Identify functions of the form f1x2 = ax2 + bx + cas quadratic
functions.
2.Explore the role of cas itrelates to the graph of f1x2 = ax2 + bx + c.
3.Explore the role of aas itrelates to the graph of f1x2 = ax2 + bx + c.
4.Explore the role of bas itrelates to the graph of f1x2 = ax2 + bx + c.
Note:a Z 0in Objectives 1–4.
Activity 4.2 The ShotPut 395
Objectives: 1.Determine the vertex or turning pointof a parabola.
2.Identify the vertex as the maximum or minimum.
3.Determine the axis of symmetry of a parabola.
4.Identify the domain and range.
5.Determine the y-interceptof a parabola.
6.Determine the x-intercept(s) of a parabola using technology.
7.Interpretthe practical meaning of the vertex and intercepts in a
given problem.
Activity 4.3 Per Capita Personal Income 406
Objectives: 1.Solve quadratic equations graphically.
2.Solve quadratic equations numerically.
3.Solve quadratic inequalities graphically.
Activity 4.4 Sir Isaac Newton 412
Objectives: 1.Factor expressions by removing the greatestcommon factor.
2.Factor trinomials using trial and error.
3.Use the Zero-Productprinciple to solve equations.
4.Solve quadratic equations by factoring.
Activity 4.5 Price of Gold 419
Objective: 1.Solve quadratic equations by the quadratic formula.
Activity 4.6 HeatIndex 428
Objectives: 1.Determine quadratic regression models using a graphing calculator.
2.Solve problems using quadratic regression models.
Activity 4.7 Complex Numbers 433
Objectives: 1.Identify the imaginary uniti = 1-1.
2.Identify a complex number.
3.Determine the value of the discriminantb2 - 4ac.