Moreland MATH - HOME · 56. Each group member should consult an almanac, newspaper, magazine, or the Internet to find data that can be modeled by linear, exponential, logarithmic,

1/9/2018 Thinking Mathematically, Sixth Edition

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7 Algebra: Graphs, Functions and Linear Systems > Chapter Summary, Review, and Test

50. This work by artist Scott Kim (1955–) has the same kind of symmetry as the graph of a quadratic function.

In Exercises 51–53, the value of a in and the vertex of the parabola are given. How many x-intercepts does the parabola have? Explain how youarrived at this number.

51. vertex at

52. vertex at

53. vertex at

Technology Exercises54. Use a graphing calculator to graph the exponential functions that you graphed by hand in Exercises 1–6. Describe similarities and differences between the graphsobtained by hand and those that appear in the calculator's viewing window.

55. Use a graphing calculator to graph the quadratic functions that you graphed by hand in Exercises 9–14.

Group Exercise56. Each group member should consult an almanac, newspaper, magazine, or the Internet to find data that can be modeled by linear, exponential, logarithmic, orquadratic functions. Group members should select the two sets of data that are most interesting and relevant. Then consult a person who is familiar with graphingcalculators to show you how to obtain a function that best fits each set of data. Once you have these functions, each group member should make one predictionbased on one of the models, and then discuss a consequence of this prediction. What factors might change the accuracy of the prediction?

Chapter Summary, Review, and TestSUMMARY – DEFINITIONS AND CONCEPTS EXAMPLES

7.1 Graphing and Functionsa. The rectangular coordinate system is formed using two number lines that intersect at right angles at their zero points. See Figure 7.1 on page 408. Thehorizontal line is the x-axis and the vertical line is the y-axis. Their point of intersection, (0, 0), is the origin. Each point in the system corresponds to anordered pair of real numbers, (x, y).

Ex. 1, p.409

b. The graph of an equation in two variables is the set of all points whose coordinates satisfy the equation.Ex. 2, p.410; Ex. 3, p.410

c. If an equation in x and y yields one value of y for each value of x, then y is a function of x, indicated by writing for y.Ex. 4, p.412; Ex. 5, p.413

d. The graph of a function is the graph of its ordered pairs. Ex. 6, p.414

e. The Vertical Line Test: If any vertical line intersects a graph in more than one point, the graph does not define y as a function of x. Ex. 7, p.4157.2 Linear Functions and Their Graphsa. A function whose graph is a straight line is a linear function. b. The graph of a linear equation in two variables, is a straight line. The line can be graphed using intercepts and a checkpoint. Tolocate the x-intercept, set and solve for x. To locate the y-intercept, set and solve for y.

Ex. 1, p.421

c. The slope of the line through and isEx. 2, p.422; Ex. 7, p.428

d. The equation is the slope-intercept form of the equation of a line, in which m is the slope and b is the y-intercept.

Ex. 3, p.425; Ex. 4, p.426; Ex. 8, p.429

e. Horizontal and Vertical Lines

1. The graph of is a horizontal line that intersects the y-axis at (0, b).

2. The graph of is a vertical line that intersects the x-axis at (a, 0).

Ex. 5, p.427; Ex. 6, p.427

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y = a + bx + cx2

a = −2; (4, 8)

a = 1; (2, 0)

a = 3; (3, 1)

f (x)

Ax + By = C,y = 0 x = 0

( , )x1 y1 ( , )x2 y2

m = = = .Rise

Run

−y2 y1−x2 x1

−y1 y2−x1 x2

y = mx + b

y = b

x = a

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7 Algebra: Graphs, Functions and Linear Systems > Chapter Summary, Review, and Test

7.3 Systems of Linear Equations in Two Variablesa. Two equations in the form are called a system of linear equations. A solution of the system is an ordered pair that satisfies both equationsin the system.

Ex. 1,p. 433

b. Linear systems with one solution can be solved by graphing. The coordinates of the point of intersection of the lines are the system's solution. Ex. 2,p. 434

c. Systems of linear equations in two variables can be solved by eliminating a variable, using the substitution method (see the box on page 435) or the additionmethod (see the box on page 438).

Ex. 3,p.435; Ex. 4,p.436; Ex. 5,p.438; Ex. 6,p. 439

d. When solving by substitution or addition, if the variable is eliminated and a false statement results, the linear system has no solution. If the variable iseliminated and a true statement results, the system has infinitely many solutions.

Ex. 7,p.440; Ex. 8,p. 441

e. Functions of Business. A company produces and sells x units of a product.

The point of intersection of the graphs of R and C is the break-even point. The x-coordinate of the point reveals the number of units that a company mustproduce and sell so that the money coming in, the revenue, is equal to the money going out, the cost. The y-coordinate gives the amount of money coming inand going out.

Ex. 9,p. 442

7.4 Linear Inequalities in Two Variablesa. A linear inequality in two variables can be written in the form or

b. The procedure for graphing a linear inequality in two variables is given in the box on page 448.

Ex. 1,p.448; Ex. 2,p. 449

c. Some inequalities can be graphed without using test points, including (the half-plane above ), (the half-plane to the right of ), (the half-plane above ), and

Ex. 3,p. 450

d. Graphing Systems of Linear Inequalities

1. Graph each inequality in the system in the same rectangular coordinate system.

2. Find the intersection of the individual graphs.

Ex. 5,p.452; Ex. 6,p. 453

7.5 Linear Programming

a. An objective function is an algebraic expression in three variables describing a quantity that must be maximized or minimized. Ex. 1,p. 456

b. Constraints are restrictions, expressed as linear inequalities.

Ex. 2,p.457; Ex. 3,p. 458

c. Steps for solving a linear programming problem are given in the box on page 458. Ex. 4,p. 4597.6 Modeling Data: Exponential, Logarithmic, and Quadratic Functions

a. The exponential function with base b is defined by or and Ex. 1,p. 463b. All exponential functions of the form where have the shape of the graph in Figure 7.43 on page 463, making this function a good model fordata points in a scatter plot that are increasing more and more rapidly.

Ex. 2,p. 465

c. The irrational number appears in many applied exponential functions. The function is called the natural exponential function. Ex. 3,p. 466d. The logarithmic function with base b is defined by or and means so a logarithm isan exponent.

Ex. 4,p. 468

e. means the common logarithmic function. means the natural logarithmic function. Calculators contain and keys for evaluating these functions.



Ax + By = C

Revenue Function:

Cost Function:

Profit Function:

R (x)

C (x)

P (x)

=

=

=

(price per unit sold) x

fixed cost + (cost per unit produced) x

R (x) − C (x)

Ax + By > C, Ax + By ≥ C, Ax + By < C, Ax + By ≤ C.

y > mx + b y = mx + b y < mx + b, x > ax = a x < a, y > b y = b y < b.

y = bx f (x) = , b > 0bx b ≠ 1.

y = ,bx b > 1,

e, e ≈ 2.72, f (x) = ex

y = xlogb f (x) = x, x > 0, b > 0,logb b ≠ 1. y = xlogb = x,by

y = log x y = x,log10 y = ln x y = x,logeLOG LN

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7 Algebra: Graphs, Functions and Linear Systems > Chapter Summary, Review, and Test > Review Exercises

f. All logarithmic functions of the form where have the shape of the graph in Figure 7.49 on page 468, making this function a goodmodel for data points in a scatter plot that are increasing, but whose rate of increase is slowing down.

Ex. 5, p.468

g. A quadratic function is any function of the form or The graph of a quadratic function is called aparabola.

Figure7.53, p.470

h. The x-coordinate of a parabola's vertex is The y-coordinate is found by substituting the x-coordinate into the parabola's equation. The vertex,or turning point, is the low point when the graph opens upward and the high point when the graph opens downward.

i. The six steps for graphing a parabola are given in the box on page 471. Ex. 6, p.471j. All quadratic functions have the shape of one of the graphs in Figure 7.53 on page 470, making this function a good model for data points in a scatter plotthat are decreasing and then increasing, or vice versa. See Table 7.5 on page 473.

Ex. 7, p.472

Review Exercises7.1

In Exercises 1–4, plot the given point in a rectangular coordinate system.

1.

2.

3.

4.

Graph each equation in Exercises 5–7. Let 0, 1, 2, and 3.

5.

6.

7.

8. If find

9. If find

10. If find

11. If find

In Exercises 12–13, evaluate for the given values of x. Then use the ordered pairs from your table to graph the function.

12.

x

0 2 4 6

13.

x

0 1 2

In Exercises 14–15, use the vertical line test to identify graphs in which y is a function of x.



y = x,logb b > 1,

y = a + bx + cx2 f (x) = a + bx + c, a ≠ 0.x2

x = .−b2a

(2, 5)

(−4, 3)

(−5, −3)

(2, −5)

x = −3, −2, −1,

y = 2x − 2

y = |x| + 2

y = x

f (x) = 4x + 11, f (−2) .

f (x) = −7x + 5, f (−3) .

f (x) = 3 − 5x + 2,x2 f (4) .

f (x) = −3 + 6x + 8,x2 f (−4) .

f (x) (x, f (x))

f (x) = |x|12

f (x) = |x|12

−6−4−2

f (x) = − 2x2

f (x) = − 2x2

−2−1

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14.

15.

16. Whether on the slopes or at the shore, people are exposed to harmful amounts of the sun's skin-damaging ultraviolet (UV) rays. The quadratic function

models the average time in which skin damage begins for burn-prone people, in minutes, where x is the UV index, or measure of the sun's UV intensity. Thegraph of D is shown for a UV index from 1 (low) to 11 (high).

d

Source: National Oceanic and Atmospheric Administration

a. Find and interpret D(1). How is this shown on the graph of D?

b. Find and interpret D(10). How is this shown on the graph of D?

7.2

In Exercises 17–19, use the x- and y-intercepts to graph each linear equation.

17.

18.

19.


D (x) = 0.8 − 17x + 109x2

D (x) ,

2x + y = 4

2x − 3y = 6

5x − 3y = 15





In Exercises 20–23, calculate the slope of the line passing through the given points. If the slope is undefined, so state. Then indicate whether the line rises, falls, ishorizontal, or is vertical.

20. and

21. and

22. and

23. and

In Exercises 24–27, graph each linear function using the slope and y-intercept.

24.

25.

26.

27.

In Exercises 28–30, a. Write the equation in slope-intercept form; b. Identify the slope and the y-intercept; c. Use the slope and y-intercept to graph the line.

28.

29.

30.

In Exercises 31–33, graph each horizontal or vertical line.

31.

32.

33.

34. Shown, again, is the scatter plot that indicates a relationship between the percentage of adult females in a country who are literate and the mortality of childrenunder five. Also shown is a line that passes through or near the points.

Source: United Nations

a. According to the graph, what is the y-intercept of the line? Describe what this represents in this situation.

b. Use the coordinates of the two points shown to compute the slope of the line. Describe what this means in terms of the rate of change.

c. Use the y-intercept from part (a) and the slope from part (b) to write a linear function that models child mortality, f(x), per thousand, for children under five in acountry where x% of adult women are literate.

d. Use the function from part (c) to predict the mortality rate of children under five in a country where 50% of adult females are literate.

7.3

In Exercises 35–37, solve each system by graphing. Check the coordinates of the intersection point in both equations.

35.

36.



(3, 2) (5, 1)

(−1, 2) (−3, −4)

(−3, 4) (6, 4)

(5, 3) (5, −3)

y = 2x − 4

y = − x + 523

f (x) = x − 234

y = x + 012

2x + y = 0

3y = 5x

3x + 2y = 4

x = 3

y = −4

x + 2 = 0

{ x + y = 53x − y = 3

{2x − y = −1 x + y = −5

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37.

In Exercises 38–40, solve each system by the substitution method.

38.

39.

40.

In Exercises 41–43, solve each system by the addition method.

41.

42.

43.

In Exercises 44–46, solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express theirsolution sets.

44.

45.

46.

47. A company is planning to manufacture computer desks. The fixed cost will be $60,000 and it will cost $200 to produce each desk. Each desk will be sold for $450.

a. Write the cost function, C, of producing x desks.

b. Write the revenue function, R, from the sale of x desks.

c. Determine the break-even point. Describe what this means.

48. The graph shows the number of guns in private hands in the United States and the country's population, both expressed in millions, from 1995 through 2020, withprojections from 2012 onward.

d

Source: Mother Jones, November/December 2012

a. Use the graphs to estimate the point of intersection. In what year will there be a gun for every man, woman, and child in the United States? What will be thepopulation and the number of firearms in that year?

b. In 1995, there were an estimated 200 million firearms in private hands. This has increased at an average rate of 6 million firearms per year. Write a function thatmodels the number of civilian firearms in the United States, y, in millions, x years after 1995.

c. The function models the U.S. population, y, in millions, x years after 1995. Use this model and the model you obtained in part (b) to determinethe year in which there will be a gun for every U.S. citizen. According to the models, what will be the population and the number of firearms in that year?

d. How well do the models in parts (b) and (c) describe the point of intersection of the graphs that you estimated in part (a)?

{y = −x + 52x − y = 4

{2x + 3y = 2x = 3y + 10

{y = 4x + 13x + 2y = 13

{x + 4y = 142x − y = 1

{x + 2y = −3x − y = −12

{2x − y = 22x + 2y = 11

{5x + 3y = 13x + 4y = −6

{y = −x + 43x + 3y = −6

{3x + y = 82x − 5y = 11

{3x − 2y = 66x − 4y = 12

y − 3x = 263




7.4

In Exercises 49–55, graph each linear inequality.

49.

50.

51.

52.

53.

54.

55.

In Exercises 56–61, graph the solution set of each system of linear inequalities.

56.

57.

58.

59.

60.

61.

7.5

62. Find the value of the objective function at each corner of the graphed region shown. What is the maximum value of the objective function? What isthe minimum value of the objective function?

63. Consider the objective function and the following constraints:

a. Graph the system of inequalities representing the constraints.

b. Find the value of the objective function at each corner of the graphed region.

c. Use the values in part (b) to determine the maximum and minimum values of the objective function and the values of x and y for which they occur.

64. A paper manufacturing company converts wood pulp to writing paper and newsprint. The profit on a unit of writing paper is $500 and the profit on a unit ofnewsprint is $350.



x − 3y ≤ 6

2x + 3y ≥ 12

2x − 7y > 14

y > x35

y ≤ − x + 212

x ≤ 2

y > −3

{3x − y ≤ 6x + y ≥ 2

{x + y < 4x − y < 4

{x ≤ 3y > −2

{4x + 6y ≤ 24y > 2

{x + y ≤ 6y ≥ 2x − 3

{y < −x + 4y > x − 4

z = 2x + 3y

z = 2x + 3y

x ≤ 6, y ≤ 5, x + y ≥ 2, .x ≥ 0, y ≥ 0

Quadrant I and its

boundary

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a. Let x represent the number of units of writing paper produced daily. Let y represent the number of units of newsprint produced daily. Write the objective functionthat models total daily profit.

b. The manufacturer is bound by the following constraints:

• Equipment in the factory allows for making at most 200 units of paper (writing paper and newsprint) in a day.

• Regular customers require at least 10 units of writing paper and at least 80 units of newsprint daily.

Write a system of inequalities that models these constraints.

c. Graph the inequalities in part (b). Use only the first quadrant and its boundary, because x and y must both be nonnegative. (Suggestion: Let each unit along thex- and y-axes represent 20.)

d. Evaluate the objective profit function at each of the three vertices of the graphed region.

e. Complete the missing portions of this statement: The company will make the greatest profit by producing _______ units of writing paper and _______ units ofnewsprint each day. The maximum daily profit is $ _______.

7.6

In Exercises 65–66, use a table of coordinates to graph each exponential function. Begin by selecting 0, 1, and 2 for x.

65.

66.

67. Graph by rewriting the equation in exponential form. Use a table of coordinates and select 0, 1, and 2 for y.

In Exercises 68–69,

a. Determine if the parabola whose equation is given opens upward or downward.

b. Find the vertex.

c. Find any x-intercepts.

d. Find the y-intercept.

e. Use (a)–(d) to graph the quadratic function.

68.

69.

In Exercises 70–72,

a. Create a scatter plot for the data in each table.

b. Use the shape of the scatter plot to determine if the data are best modeled by a linear function, an exponential function, a logarithmic function, or a quadraticfunction.

70.

AGE OF U.S. DRIVERS AND FATAL CRASHESAge Fatal Crashes per 100 Million Miles Driven20 6.225 4.135 2.845 2.455 3.065 3.875 8.0

Source: Insurance Institute for Highway Safety

71.

NUMBER OF PAGES INTHE IRS 1040 TAX

INSTRUCTIONBOOKLET

Year Number of Pages

Year Number of Pages

1945 41955 161965 171975 391985 521995 842005 1422010 179

Source: National Taxpayers Union

−2, −1,

f (x) = 2x

y = 2x+1

y = xlog2 −2, −1,

y = − 6x − 7x2

f (x) = − − 2x + 3x2



7 Algebra: Graphs, Functions and Linear Systems > Chapter 7 Test

72.

INTENSITY AND LOUDNESS LEVEL OF VARIOUS SOUNDSLoudness Level (decibels)

0.1 (loud thunder) 1101 (rock concert, 2 yd from speakers) 12010 (jackhammer) 130100 (jet takeoff, 40 yd away) 140

73. Just browsing? Take your time. Researchers know, to the dollar, the average amount the typical consumer spends at the shopping mall. And the longer you stay,the more you spend.

d

Source: International Council of Shopping Centers Research

The data in the bar graph can be modeled by the functions

where and model the average amount spent, in dollars, at a shopping mall after x hours.

a. What is the slope of the linear model? What does this mean in terms of the average amount spent at a shopping mall?

b. Which function, the linear or the exponential, is a better model for the average amount spent after 3.5 hours of browsing?

74. The bar graph shows that people with lower incomes are more likely to report that their health is fair or poor.

d

Source: William Kornblum and Joseph Julian, Social Problems, 12th Edition, Prentice Hall, 2007.

The data can be modeled by and where f(x) and g(x) model the percentage of Americans reporting fair or poorhealth in terms of annual income, x, in thousands of dollars. Which function, the linear or the logarithmic, is a better model for an annual income of $60 thousand?

Chapter 7 Test1. Graph Let 0, 1, 2, and 3.

2. If find

In Exercises 3–4, use the vertical line test to identify graphs in which y is a function of x.



Intensity( )watts per meter2

f (x) = 47x + 22 and g (x) = 42.2 ,(1.56)x

f (x) g (x)

f (x) = −0.4x + 25.4 g (x) = 54.8 − 12.3 ln x,

y = |x| − 2. x = −3, −2, −1,

f (x) = 3 − 7x − 5,x2 f (−2) .

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3. d

4. d

5. The graph shows the height, in meters, of an eagle in terms of its time, in seconds, in flight.

d

a. Is the eagle's height a function of time? Use the graph to explain why or why not.

b. Find Describe what this means in practical terms.

c. What is a reasonable estimate of the eagle's maximum height?

d. During which period of time was the eagle descending?


f (15) .





6. Use the x- and y-intercepts to graph

7. Find the slope of the line passing through and

In Exercises 8–9, graph each linear function using the slope and y-intercept.

8.

9.

10. In a 2010 survey of more than 200,000 freshmen at 279 colleges, only 52% rated their emotional health high or above average, a drop from 64% in 1985.

d

Source: UCLA Higher Education Research Institute

a. Find the slope of the line passing through the two points shown by the voice balloons.

b. Use your answer from part (a) to complete this statement:

For each year from 1985 through 2010, the percentage of U.S. college freshmen rating their emotional health high or above average decreased by _______. Therate of change was _______ per _______.

11. Studies show that texting while driving is as risky as driving with a 0.08 blood alcohol level, the standard for drunk driving. The bar graph shows the number offatalities in the United States involving distracted driving from 2005 through 2008. Although the distracted category involves such activities as talking on cellphones,conversing with passengers, and eating, experts at the National Highway Traffic Safety Administration claim that texting while driving is the clearest menace becauseit requires looking away from the road. Shown to the right of the bar graph is a scatter plot with a line passing through two of the data points.

d

Source: National Highway Traffic Safety Administration

a. According to the scatter plot shown on the right, what is the y-intercept? Describe what this represents in this situation.

b. Use the coordinates of the two points shown in the scatter plot to compute the slope. What does this represent in terms of the rate of change in the number ofhighway fatalities involving distracted driving?

c. Use the y-intercept shown in the scatter plot and the slope from part (b) to write a linear function that models the number of highway fatalities involving distracteddriving, in the United States x years after 2005.

d. In 2010, surveys showed overwhelming public support to ban texting while driving, although at that time only 19 states and Washington, D.C., outlawed thepractice. Without additional laws that penalize texting drivers, use the linear function you obtained from part (c) to project the number of fatalities in the United



4x − 2y = −8.

(−3, 4) (−5, −2) .

y = x − 123

f (x) = −2x + 3

f (x) ,




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States in 2015 involving distracted driving.

12. Solve by graphing:

13. Solve by substitution:

14. Solve by addition:

15. A company is planning to produce and sell a new line of computers. The fixed cost will be $360,000 and it will cost $850 to produce each computer. Eachcomputer will be sold for $1150.

a. Write the cost function, C, of producing x computers.

b. Write the revenue function, R, from the sale of x computers.

c. Determine the break-even point. Describe what this means.

Graph each linear inequality in Exercises 16–18.

16.

17.

18.

19. Graph the system of linear inequalities:

20. Find the value of the objective function at each corner of the graphed region shown. What is the maximum value of the objective function? What isthe minimum value of the objective function?

d

21. Find the maximum value of the objective function subject to the following constraints:


{ x + y = 64x − y = 4.

{ x = y + 43x + 7y = −18.

{5x + 4y = 103x + 5y = −7.

3x − 2y < 6

y ≤ x − 112

y > −1

{2x − y ≤ 42x − y > −1.

z = 3x + 2y

z = 3x + 5y x ≥ 0, y ≥ 0, x + y ≤ 6, x ≥ 2.





22. A manufacturer makes two types of jet skis, regular and deluxe. The profit on a regular jet ski is $200 and the profit on the deluxe model is $250. To meetcustomer demand, the company must manufacture at least 50 regular jet skis per week and at least 75 deluxe models. To maintain high quality, the total number ofboth models of jet skis manufactured by the company should not exceed 150 per week. How many jet skis of each type should be manufactured per week to obtainmaximum profit? What is the maximum weekly profit?

23. Graph Use 0, 1, and 2 for x and find the corresponding values for y.

24. Graph by rewriting the equation in exponential form. Use a table of coordinates and select 0, 1, and 2 for y.

25. Use the vertex and intercepts to graph the quadratic function

In Exercises 26–29, determine whether the values in each table belong to an exponential function, a logarithmic function, a linear function, or a quadratic function.

26.

x y0 31 1234

27.

x y

1 03 19 227 3

28.

x y0 11 52 253 1254 625

29.

x y0 121 32 03 34 12

30. The bar graph and the scatter plot show what it cost the United States Mint to make a penny for five selected years from 1982 through 2012. The data can bemodeled by the functions

where and model what it cost to make a penny x years after 1982.



f (x) = .3x −2, −1,

y = xlog3 −2, −1,

f (x) = − 2x − 8.x2

−1−3−5

13

−1

f (x) = 0.03x + 0.63 and g (x) = 0.72 ,(1.03)x

f (x) g (x)




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d

Source: U.S. Mint

a. What is the slope of the linear model? What does this mean in terms of the change in the cost of making a penny?

b. Based on the shape of the scatter plot, which function, the linear or the exponential, is a better model for the data? Explain your answer.

c. Use each function to find the cost of making a penny in 2012. Where necessary, round to two decimal places. Is either function a particularly good model for the2012 data? Which function serves as a better model? Is this consistent with your answer in part (b)?



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Moreland MATH - HOME · 56. Each group member should consult an almanac, newspaper, magazine, or the Internet to find data that can be modeled by linear, exponential, logarithmic,