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4. Intermediate Algebra  # Mathematics in Action: Algebraic, Graphical, and Trigonometric Problem Solving - NASTA, 5th edition • Consortium for Foundation

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ISBN-13: 9780134380100

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## Overview

In Mathematics in Action series, students discover mathematical concepts through activities and applications that demonstrate how math applies to their everyday lives. Different from most math books, this series teaches through activities—encouraging students to learn by constructing, reflecting on, and applying the mathematical concepts. The user-friendly approach instills confidence in even the most reticent math students and shows them how to interpret data algebraically, numerically, symbolically, and graphically. The active style develops mathematical literacy and critical thinking skills. Updated examples, brand-new exercises, and a clearer presentation make the Fifth Edition of this text more relevant than ever to today’s students.

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1. Function Sense

1.1 Parking Problems

1.1.1 Identify input and output in situations involving two variable quantities

1.1.2 Identify a functional relationship between two variables.

1.1.3 Identify the independent and dependent variables

1.1.4 Use a table to numerically represent a functional relationship between two variables

1.1.5 Write a function using function notation

1.2 Fill'er Up

1.2.1 Determine the equation (symbolic representation) that defines a function

1.2.2 Determine the domain and range of a function

1.3 Graphically Speaking

1.3.1 Represent a function verbally, symbolically, numerically, and graphically

1.3.2 Distinguish between a discrete function and a continuous function

1.3.3 Graph a function using technology

1.4 Stopping Short

1.4.1 Use a function as a mathematical model

1.4.2 Determine when a function is increasing, decreasing, or constant

1.4.3 Use the vertical line test to determine whether a graph represents a function

1.5 Graphs Tell Stories

1.5.1 Describe in words what a graph tells you about a given situation

1.5.2 Sketch a graph that best represents the situation described in words

1.5.3 Identify increasing, decreasing, and constant parts of a graph

1.5.4 Identify minimum and maximum points on a graph

1.6 Walking for Fitness

1.6.1 Determine the average rate of change

1.7 Depreciation

1.7.1 Interpret slope as an average rate of change

1.7.2 Use the formula to determine slope

1.7.3 Discover the practical meaning of vertical and horizontal intercepts

1.7.4 Develop the slope-intercept form of an equation of a line

1.7.5 Use the slope-intercept formula to determine vertical and horizontal intercepts

1.7.6 Determine the zeroes of a function

1.8 A New Camera

1.8.1 Write a linear equation in the slope-intercept form given the initial value and the average rate of change

1.8.2 Write a linear equation given two points, one of which is the vertical intercept

1.8.3 Use the point-slope form to write a linear equation given two points, neither of which is the vertical intercept

1.8.4 Compare slopes of parallel lines

1.9 Skateboard Heaven

1.9.1 Write an equation of a line in general form Ax + By = C

1.9.2 Write the slope-intercept form of a linear equation given the general form

1.9.3 Determine the equation of a horizontal line

1.9.4 Determine the equation of a vertical line

1.10 College Tuition

1.10.1 Recognize when patterns of points in a scatterplot have a linear form

1.10.2 Recognize when the pattern in the scatterplot shows that the two variables are positively related or negatively related

1.10.3 Estimate and draw a line of best fit through a set of points in a scatterplot

1.10.4 Use a graphing calculator to determine a line of best fit by the least-squares method

1.10.5 Measure the strength of the correlation (association) by a correlation coefficient

1.10.6 Recognize that a strong correlation does not necessarily imply a linear or a cause-and-effect relationship

1.11 Moving Out

1.11.1 . Solve a system of 2 x 2 linear equations numerically and graphically

1.11.2 Solve a system of 2 x 2 linear equations using the substitution method

1.11.3 . Solve an equation of the form ax + b = cx + d for x

1.12 Fireworks

1.12.1 Solve a 2 x 2 linear system algebraically using the substitution method and the addition method

1.12.2 Solve equations containing parentheses

1.13 Manufacturing Smartphones

1.13.1 Solve a 3 x 3 linear system of equations

1.14 Earth Week

1.14.1 Solve a linear system of equations using matrices

1.15 How Long Can You Live?

1.15.1 Solve linear inequalities in one variable numerically and graphically

1.15.2 Use properties of inequalities to solve linear inequalities in one variable algebraically

1.15.3 Solve compound inequalities algebraically

1.15.4 Use interval notation to represent a set of real numbers described by an inequality

1.16 Working Overtime

1.16.1 Graph a piecewise linear function

1.16.2 Write a piecewise linear function to represent a given situation

1.16.3 Graph a function defined by y =  |x - c|

2. The Algebra of Functions

2.1 Spending and Earning Money

2.1.1 Identify a polynomial expression

2.1.2 Identify a polynomial function

2.1.3 Add and subtract polynomial expressions

2.1.4 Add and subtract polynomial functions

2.2 The Dormitory Parking Lot

2.2.1 Multiply two binomials using the FOIL method

2.2.2 Multiply two polynomial functions

2.2.3 Apply the property of exponents to multiply powers having the same base

2.3 Stargazing

2.3.1 Convert scientific notation to decimal notation

2.3.2 Convert decimal notation to scientific notation

2.3.3 Apply the property of exponents to divide powers having the same base

2.3.4 Apply the definition of exponents a0 = 1, where a ≠ 0

2.3.5 Apply the definition of exponents a-n = 1 / (an), where a ≠ 0 and n is any real number

2.4 The Cube of a Square

2.4.1 Apply the property of exponents to simplify an expression involving a power to a power

2.4.2 Apply the property of exponents to expand the power of a product

2.4.3 Determine the nth root of a real number.

2.4.4 Write a radical as a power having a rational exponent, and write a base to a rational exponent as a radical

2.5 Inflated Balloons

2.5.1 Determine the composition of two functions

2.5.2 Explore the relationship between f(g(x)) and g(f(x))

2.6 Finding a Bargain

2.6.1 Solve problems using the composition of functions

2.7 Study Time

2.7.1 Determine the inverse of a function represented by a table of values

2.7.2 Use the notation f-1 to represent an inverse function

2.7.3 Use the property f(f-1(x)) = f-1(f(x)) = x to recognize inverse functions

2.7.4 Determine the domain and range of a function and its inverse

2.8 Temperature Conversions

2.8.1 Determine the equation of the inverse of a function represented by an equation

2.8.2 Describe the relationship between the graphs of inverse functions

2.8.3 Determine the graph of the inverse of a function represented by a graph

2.8.4 Use the graphing calculator to produce graphs of an inverse function

3. Exponential and Logarithmic Functions

3.1 Prince George and Dracula

3.1 Determine the growth factor of an exponential function

3.2 Identify the properties of the graph of an exponential function defined by y = bx, where b > 1.

3.3 Graph an increasing exponential function

3.2 Half-Life of Medicine

3.2.1 Determine the decay factor of an exponential function

3.2.2 Graph a decreasing exponential function

3.2.3 Identify the properties of an exponential functions defined by y = bx, where b > 0 and b ≠ 1

3.3 National Debt

3.3.1 Determine the growth and decay factor for an exponential function represented by a table of values or an equation

3.3.2 Graph exponential functions defined by y = abx, where b > 0 and b ≠ 1, a ≠ 0

3.3.3 Determine the doubling and halving time

Population Growth

3.4.1 Determine the annual growth or decay rate of an exponential function represented by a table of values or an equation

3.4.2 Graph an exponential function having equation y = a(1 + r)x

3.5 Time is Money

3.5.1 Apply the compound interest and continuous compounding formulas to a given situation

3.6 Continuous Growth and Decay

3.6.1 Discover the relationship between the equations of exponential functions defined by y = abt and the equations of continuous growth and decay exponential functions defined by y = aekt

3.6.2 Solve problems involving continuous growth and decay models

3.6.3 Graph base e exponential functions

3.7 Bird Flu

3.7.1 Determine the regression equation of an exponential function that best fits the given data

3.7.2 Make predictions using an exponential regression equation

3.7.3 Determine whether a linear or exponential model best fits the data

3.8 The Diameter of Spheres

3.8.1 Define logarithm

3.8.2 Write an exponential statement in logarithmic form

3.8.3 Write a logarithmic statement in exponential form

3.8.4 Determine log and ln values using a calculator

3.9 Walking Speed of Pedestrians

3.9.1 Determine the inverse of the exponential function

3.9.2 Identify the properties of the graph of a logarithmic function

3.9.3 Graph the natural logarithmic function

3.10 Walking Speed of Pedestrians, continued

3.10.1 Compare the average rate of change of increasing logarithmic, linear, and exponential functions

3.10.2 Determine the regression equation of a natural logarithmic function having equation y = a + b ln(x) that best fits a set of data

3.11 The Elastic Ball

3.11.1 Apply the log of a product property

3.11.2 Apply the log of a quotient property

3.11.3 Apply the log of a power property

3.11.4 Discover the change of base formula

3.12 Changing Demographics

3.12.1 Solve exponential equations both graphically and algebraically

4. Quadratic and Higher-Order Polynomial Functions

4.1 Baseball and the Willis Tower

4.1.1 Identify functions of the form f(x) = ax2 + bx + c as quadratic functions

4.1.2 Explore the role of c as it relates to the graph of f(x) = ax2 + bx + c

4.1.3 Explore the role of a as it relates to the graph of f(x) = ax2 + bx + c

4.1.4 Explore the role of b as it relates to the graph of f(x) = ax2 + bx + c

4.2 The Shot Put

4.2.1 Determine the vertex, or turning point, of a parabola

4.2.2 Identify the vertex as a maximum or minimum

4.2.3 Determine the axis of symmetry of a parabola

4.2.4 Identify the domain and range

4.2.5 Determine the y-intercept of a parabola

4.2.6 Determine the x-intercept(s) of a parabola using technology

4.2.7 Interpret the practical meaning of the vertex and intercepts in a given problem

4.3 Per Capita Personal Income

4.4 Sir Isaac Newton

4.4.1 Use the zero-product property to solve equations

4.4.2 factor expressions by removing the greatest common factor

4.4.3 factor trinomials using trial and error

4.4.4 Solve quadratic equations by factoring

4.5 Price of Gold

4.6 Heat Index

4.6.1 Determine quadratic regression models using a graphing calculator

4.6.2 Solve problems using quadratic regression models

4.7 Complex Numbers

4.7.1 Identify the imaginary unit i = √-1

4.7.2 Identify complex numbers, a + bi

4.7.3 Determine the value of the discriminant b2 - 4ac used in the quadratic formula

4.7.4 Solve quadratic equations in the complex number system

4.7.5 Determine the types of solutions to quadratic equations

4.8 The Power of Power Functions

4.8.1 Identify a direct variation function

4.8.2 Determine the constant of variation

4.8.3 Identify the properties of graphs of power functions defined by y = kxn, where n is a positive integer, k ≠ 0

4.9 Volume of a Storage Tank

4.9.1 Identify equations that define polynomial functions

4.9.2 Determine the degree of a polynomial function

4.9.3 Determine the intercepts of the graph of a polynomial function

4.9.4 Identify the properties of the graphs of polynomial functions

4.10 Federal Prison Population

4.10.1 Determine the regression equation of a polynomial function that best fits the data

Cluster 1: Rational Functions

5.1 Speed Limits

5.1.1 Determine the domain and range of functions defined by y = k/x, where k is a nonzero real number

5.1.2 Determine the vertical and horizontal asymptotes of graphs of y = k/x

5.1.3 Sketch a graph of functions of the form y = k/x

5.1.4 Determine the properties of graphs having equation y = k/x

5.2 Loudness of a Sound

5.2.1 Graph functions defined by an equation of the form y = k/xn, where n is a positive integer and k is a nonzero real number, x ≠ 0.

5.2.2 Describe the properties of graphs having equation y = k/xn, x ≠ 0

5.2.3 Determine the constant of proportionality (also called the constant of variation)

5.3 Percent Markup

5.3.1 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 first-degree polynomial

5.3.2 Identify the vertical and horizontal asymptotes of y = k /g(x)

5.3.3 Sketch graphs of rational functions defined by y=k/g(x)

5.4 Blood-Alcohol Levels

5.4.1 Solve an equation involving a rational expression using an algebraic approach

5.4.2 Solve an equation involving a rational expression using a graphing approach

5.4.3 Determine horizontal asymptotes of the graph of y = f(x)/g(x), where f(x) and g(x) are first-degree polynomials

5.5 Traffic Flow

5.5.1 Determine the least common denominator (LCD) of two or more rational expressions

5.5.2 Solve an equation involving rational expressions using an algebraic approach

5.5.3 Solve a formula for a specified variable

5.6 Electrical Circuits

5.6.1 Multiply and divide rational expressions

5.6.2 Add and subtract rational expressions

5.6.3 Simplify complex fractions

5.7 Skydiving

5.7.1 Determine the domain of a radical function defined by y = √g(x), where g(x) is a polynomial

5.7.2 Graph functions having equation y = √g(x) and y = -√g(x)

5.7.3 Identify the properties of the graph of y = √g(x) and y = √g(x)

5.8 Falling Objects

5.8.1 Solve an equation involving a radical expression using a graphical and algebraic approach

5.9 Propane Tank

5.9.1 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

5.9.2 Graph y = n√g(x)

5.9.3 Identify the properties of graphs of y = n√g(x)

5.9.4 Solve radical equations that contain radical expressions with an index other than 2

6. Introduction to Trigonometric Functions

6.1 The Leaning Tower of Pisa

6.1.1 Identify the sides and corresponding angles of a right triangle

6.1.2 Determine the length of the sides of similar right triangles using proportions

6.1.3 Determine the sine, cosine, and tangent of an angle within a right triangle

6.1.4 Determine the sine, cosine, and tangent of an acute angle by use of the graphing calculator

6.2 A Gasoline Problem

6.2.1 Identify complimentary angles

6.2.2 Demonstrate that the sine of one of the complementary angles equals the cosine of the other

6.3 The Sidewalks of New York

6.3.1 Determine the inverse tangent of a number

6.3.2 Determine the inverse sine and cosine of a number using the graphing calculator

6.3.3 Identify the domain and range of the inverse sine, cosine, and tangent functions

6.4 Solving a Murder

6.4.1 Determine the measure of all sides and all angles of a right triangle

6.5 How Stable is that Tower

6.5.1 Solve problems using right triangle trigonometry

6.5.2 Solve optimization problems using right triangle trigonometry with a graphing approach

6.6 Learn Trig or Crash!

6.6.1 Determine the coordinates of points on a unit circle using sine and cosine functions

6.6.2 Sketch the graph of y = sin x and y = cos x

6.6.3 Identify the properties of the graphs of the sine and cosine functions

6.7 It Won't Hertz

6.7.1 Convert between degree and radian measure

6.7.2 Identify the period and frequency of a function defined by y = a sin(bx) or y = a cos(bx) using the graph

6.8 Get in Shape

6.8.1 Determine the amplitude of the graph of y = a sin(bx) and y = a cos(bx)

6.8.2 Determine the period of the graph of y = a sin(bx) and y = a cos(bx) using a formula.

6.9 The Carousel

6.9.1 Determine the displacement of y = a sin (bx + c) and y = a cos (bx + c) using a formula

6.10 Texas Temperatures

6.10.1 Determine the equation of a sine function that best fits the given data

6.10.2 Make predictions using a sine regression equation

Appendices

A. Concept Review

B. Trigonometry

C. Getting Started with the TI-84 Plus Family of Calculators