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7 Algebra: Graphs, Functions and Linear Systems > 7.6 Modeling Data: Exponential, Logarithmic, and Quadratic Functions

Writing in Mathematics19. What kinds of problems are solved using the linear programming method?

20. What is an objective function in a linear programming problem?

21. What is a constraint in a linear programming problem? How is a constraint represented?

22. In your own words, describe how to solve a linear programming problem.

23. Describe a situation in your life in which you would like to maximize something, but you are limited by at least two constraints. Can linear programming be used inthis situation? Explain your answer.

Critical Thinking ExercisesMake Sense? In Exercises 24–27, determine whether each statement makes sense or does not make sense, and explain your reasoning.

24. In order to solve a linear programming problem, I use the graph representing the constraints and the graph of the objective function.

25. I use the coordinates of each vertex from my graph representing the constraints to find the values that maximize or minimize an objective function.

26. I need to be able to graph systems of linear inequalities in order to solve linear programming problems.

27. An important application of linear programming for businesses involves maximizing profit.

28. Suppose that you inherit $10,000. The will states how you must invest the money. Some (or all) of the money must be invested in stocks and bonds. Therequirements are that at least $3000 be invested in bonds, with expected returns of $0.08 per dollar, and at least $2000 be invested in stocks, with expected returns of$0.12 per dollar. Because the stocks are medium risk, the final stipulation requires that the investment in bonds should never be less than the investment in stocks.How should the money be invested so as to maximize your expected returns?

Group Exercises29. Group members should choose a particular field of interest. Research how linear programming is used to solve problems in that field. If possible, investigate thesolution of a specific practical problem. Present a report on your findings, including the contributions of George Dantzig, Narendra Karmarkar, and L. G. Khachion tolinear programming.

30. Members of the group should interview a business executive who is in charge of deciding the product mix for a business. How are production policy decisionsmade? Are other methods used in conjunction with linear programming? What are these methods? What sort of academic background, particularly in mathematics,does this executive have? Present a group report addressing these questions, emphasizing the role of linear programming for the business.

7.6 Modeling Data: Exponential, Logarithmic, and Quadratic Functions

What am I Supposed to Learn?After you have read this section, you should be able to:

1 Graph exponential functions.

2 Use exponential models.

3 Graph logarithmic functions.

4 Use logarithmic models.

5 Graph quadratic functions.

6 Use quadratic models.

7 Determine an appropriate function for modeling data.

IS THERE A RELATIONSHIP BETWEEN LITERACY AND child mortality? As the percentage of adult females who are literate increases, does the mortality of childrenunder five decrease? Figure 7.41, based on data from the United Nations, indicates that this is, indeed, the case. Each point in the figure represents one country.

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Page 462

dFIGURE 7.41 Source: United Nations

Data presented in a visual form as a set of points are called a scatter plot. Also shown in Figure 7.41 is a line that passes through or near the points. The line that bestfits the data points in a scatter plot is called a regression line. We can use the line's slope and y-intercept to obtain a linear model for under-five mortality, y, perthousand, as a function of the percentage of literate adult females, x. The model is given at the top of the next page.

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7 Algebra: Graphs, Functions and Linear Systems > 7.6 Modeling Data: Exponential, Logarithmic, and Quadratic Functions >Modeling with Exponential Functions

Using this model, we can make predictions about child mortality based on the percentage of literate adult females in a country.

In Figure 7.41, the data fall on or near a line. However, scatter plots are often curved in a way that indicates that the data do not fall near a line. In this section, we willuse functions that are not linear functions to model such data and make predictions.

Modeling with Exponential Functions

1 Graph exponential functions.

The scatter plot in Figure 7.42 has a shape that indicates the data are increasing more and more rapidly. Exponential functions can be used to model this explosivegrowth, typically associated with populations, epidemics, and interest-bearing bank accounts.

FIGURE 7.42

Definition of the Exponential FunctionThe exponential function with base b is defined by

where b is a positive constant other than 1 and and x is any real number.

Example 1 Graphing an Exponential FunctionGraph:

SOLUTION

We start by selecting numbers for x and finding the corresponding values for

x (x, y)

0



y = − x + 2552.3▲

For each percent increase in adult female literacy,

under-five mortality decreases by 2.3 per thousand.

y = or f (x) = ,bx bx

(b > 0 b ≠ 1)

f (x) = .2x

f (x) .

f (x) = 2x

−3 f (−3) = =2−3 1

8(−3, )1

8

−2 f (−2) = =2−2 1

4(−2, )1

4

−1 f (−1) = =2−1 1

2(−1, )1

2

f (0) = = 120 (0, 1)








63

x (x, y)

1

2

3

We plot these points, connecting them with a smooth curve. Figure 7.43 shows the graph of

FIGURE 7.43 The graph of

All exponential functions of the form or where b is a number greater than 1, have the shape of the graph shown in Figure 7.43. The graphapproaches, but never touches, the negative portion of the x-axis.

Check Point 1Graph:


f (1) = = 221 (1, 2)

f (2) = = 422 (2, 4)

f (3) = = 823 (3, 8)

f (x) = .2x

f (x) = 2x

y = ,bx f (x) = ,bx

f (x) = .3x









Blitzer BonusExponential Growth: The Year Humans Become Immortal

In 2011, Jeopardy! aired a three-night match between a personable computer named Watson and the show's two most successful players. The winner: Watson. In thetime it took each human contestant to respond to one trivia question, Watson was able to scan the content of one million books. It was also trained to understand thepuns and twists of phrases unique to Jeopardy! clues.

Watson's remarkable accomplishments can be thought of as a single data point on an exponential curve that models growth in computing power. According toinventor, author, and computer scientist Ray Kurzweil (1948–), computer technology is progressing exponentially, doubling in power each year. What does this meanin terms of the accelerating pace of the graph of that starts slowly and then rockets skyward toward infinity? According to Kurzweil, by 2023, a supercomputerwill surpass the brainpower of a human. As progress accelerates exponentially and every hour brings a century's worth of scientific breakthroughs, by 2045,computers will surpass the brainpower equivalent to that of all human brains combined. Here's where it gets exponentially weird: In that year (says Kurzweil), we willbe able to scan our consciousness into computers and enter a virtual existence, or swap our bodies for immortal robots. Indefinite life extension will become a realityand people will die only if they choose to.

2 Use exponential models.

Figure 7.44(a) shows world population, in billions, for seven selected years from 1950 through 2010. A scatter plot of the data is shown in Figure 7.44(b).



y = 2x








Page 464

dSource: U.S. Census Bureau, International Database

Because the data in the scatter plot appear to increase more and more rapidly, the shape suggests that an exponential function might be a good choice for modeling thedata. Furthermore, we can probably draw a line that passes through or near the seven points. Thus, a linear function would also be a good choice for a model.










Example 2 Comparing Linear and Exponential ModelsThe data for world population are shown in Table 7.3. Using a graphing utility's linear regression feature and exponential regression feature, we enter the data andobtain the models shown in Figure 7.45.

TABLE 7.3

▶x, Number of Years after 1949 y, World Population (billions)

1 (1950) 2.611 (1960) 3.021 (1970) 3.731 (1980) 4.541 (1990) 5.351 (2000) 6.161 (2010) 6.9

dFIGURE 7.45 A linear model and an exponential model for the data in Table 7.3

a. Use Figure 7.45 to express each model in function notation, with numbers rounded to three decimal places.

b. How well do the functions model world population in 2000?

c. By one projection, world population is expected to reach 8 billion in the year 2026. Which function serves as a better model for this prediction?

SOLUTION

a. Using Figure 7.45 and rounding to three decimal places, the functions

model world population, in billions, x years after 1949. We named the linear function f and the exponential function g, although any letters can be used.

b. Table 7.3 shows that world population in 2000 was 6.1 billion. The year 2000 is 51 years after 1949. Thus, we substitute 51 for x in each function's equation andthen evaluate the resulting expressions with a calculator to see how well the functions describe world population in 2000.

Because 6.1 billion was the actual world population in 2000, both functions model world population in 2000 extremely well.



Although the domain of y = a is the set of all real numbers,bx

some graphing calculators only accept positive values for x. That's why we

assigned x to represent the number of years after 1949.

f (x) = 0.074x + 2.294 and g (x) = 2.577(1.017)x

f (x)

f (51)

g (x)

g (51)

= 0.074x + 2.294

= 0.074 (51) + 2.294

≈ 6.1

= 2.577(1.017)x

= 2.577(1.017)51

≈ 6.1

This is the linear model.

Substitute 51 for x.

Use a calculator:

This is the exponential model.


Use a calculator:

2.577 1.017 (or )51 .× yx ⋀ =








Page 465









7 Algebra: Graphs, Functions and Linear Systems > 7.6 Modeling Data: Exponential, Logarithmic, and Quadratic Functions > TheRole of e in Applied Exponential Functions

c. Let's see which model comes closer to projecting a world population of 8 billion in the year 2026. Because 2026 is 77 years after 1949 we substitute 77 for x in each function's equation.

The linear function serves as a better model for a projected world population of 8 billion in 2026.

Blitzer BonusGlobal Population Increase

Exponential functions of the form model growth in which quantities increase at a rate proportional to their size. Populations that are growingexponentially grow extremely rapidly as they get larger because there are more adults to have offspring. Here's a way to put this idea into perspective:

By the time you finish reading Example 2 and working Check Point 2, more than 1000 people will have been added to our planet. By this time tomorrow, worldpopulation will have increased by more than 220,000.

Check Point 2Use the models and to solve this problem.

a. World population in 1970 was 3.7 billion. Which function serves as a better model for this year?

b. By one projection, world population is expected to reach 9.3 billion by 2050. Which function serves as a better model for this projection?

The Role of e in Applied Exponential FunctionsAn irrational number, symbolized by the letter e, appears as the base in many applied exponential functions. This irrational number is approximately equal to 2.72. Moreaccurately,

The number e is called the natural base. The function is called the natural exponential function.

Use a scientific or graphing calculator with an key to evaluate e to various powers. For example, to find press the following keys on most calculators:

The calculator display for should be approximately 7.389.

The number e lies between 2 and 3. Because and it makes sense that approximately 7.389, lies between 4 and 9.

Because the graph of lies between the graphs of and shown in Figure 7.46.



(2026 − 1949 = 77) ,

f (x)

f (77)

g (x)

g (77)

= 0.074x + 2.294

= 0.074 (77) + 2.294

≈ 8.0

= 2.577(1.017)x

= 2.577(1.017)77

≈ 9.4

This is the linear model.


Use a calculator:

This is the exponential model.


Use a calculator:

2.577 1.017 (or ) 77 .× yx ⋀ =

f (x) = 0.074x + 2.294

y = a , b > 1,bx

f (x) = 0.074x + 2.294 g (x) = 2.577(1.017)x

e ≈ 2.71828 … .

f (x) = ex

ex ,e2

Scientific calculator:

Graphing calculator:

2 ex

2 .ex ENTER

e2

≈ 7.389e2

= 422 = 9,32 ,e2

2 < e < 3, y = ex y = 2x y = ,3x








Page 466

dFIGURE 7.46 Graphs of three exponential functions

Example 3 Alcohol and Risk of a Car AccidentMedical research indicates that the risk of having a car accident increases exponentially as the concentration of alcohol in the blood increases. The risk is modeled by


R = 6 ,e12.77x








7 Algebra: Graphs, Functions and Linear Systems > 7.6 Modeling Data: Exponential, Logarithmic, and Quadratic Functions >Modeling with Logarithmic Functions

where x is the blood-alcohol concentration and R, given as a percent, is the risk of having a car accident. In every state, it is illegal to drive with a blood-alcoholconcentration of 0.08 or greater. What is the risk of a car accident with a blood-alcohol concentration of 0.08? How is this shown on the graph of R in Figure 7.47?

dFIGURE 7.47

SOLUTION

For a blood-alcohol concentration of 0.08, we substitute 0.08 for x in the exponential model's equation. Then we use a calculator to evaluate the resulting expression.

Perform this computation on your calculator.

The display should be approximately 16.665813. Rounding to one decimal place, the risk of a car accident is approximately 16.7% with a blood-alcohol concentrationof 0.08. This can be visualized as the point on the graph of R in Figure 7.47. Take a moment to locate this point on the curve.

Check Point 3Use the model in Example 3 to solve this problem. In many states, it is illegal for drivers under 21 years old to drive with a blood-alcohol concentration of 0.01 orgreater. What is the risk of a car accident with a blood-alcohol concentration of 0.01? Round to one decimal place.

Modeling with Logarithmic FunctionsThe scatter plot in Figure 7.48 starts with rapid growth and then the growth begins to level off. This type of behavior can be modeled by logarithmic functions.

FIGURE 7.48

Definition of the Logarithmic FunctionFor and



R

R

=

=

6e12.77x

6e12.77(0.08)

This is the given exponential model.

Substitute 0.08 for x.

Scientific calculator : 6 12.77 .08× ( × ) ex =

Graphing calculator : 6 12.77 .08× ex ( × ) ENTER

(0.08, 16.7)

x > 0 b > 0, b ≠ 1,








Page 467

The function is the logarithmic function with base b .

The equations

are different ways of expressing the same thing. The first equation is in logarithmic form, and the second equivalent equation is in exponential form.

Notice that a logarithm, y, is an exponent. You should learn the location of the base and exponent in each form.

Location of Base and Exponent in Exponential and Logarithmic Forms


y = x is equivalent to = x.logb by

f (x) = xlogb

y = x and = xlogb by

Logarithmic Form:

Exponent

▼

y = xlogb

▲

Base

Exponential Form:

Exponent

▼

= xby

▲

Base









3 Graph logarithmic functions.

Great Question!I know that means But what's the relationship between and

The coordinates for the logarithmic function with base 2 are the reverse of those for the exponential function with base 2. In general, reverses the x-and y-coordinates of

Example 4 Graphing a Logarithmic FunctionGraph:

SOLUTION

Because means we will use the exponential form of the equation to obtain the function's graph. Using we start by selecting numbersfor y and then we find the corresponding values for x.

(x, y)

0 (1, 0)1 (2, 1)2 (4, 2)

3 (8, 3)

We plot the six ordered pairs in the table, connecting the points with a smooth curve. Figure 7.49 shows the graph of

FIGURE 7.49 The graph of the logarithmic function with base 2

All logarithmic functions of the form or where have the shape of the graph shown in Figure 7.49. The graph approaches, butnever touches, the negative portion of the y-axis. Observe that the graph is increasing from left to right. However, the rate of increase is slowing down as the graphmoves to the right. This is why logarithmic functions are often used to model growing phenomena with growth that is leveling off.

Check Point 4Rewrite in exponential form. Then use the exponential form of the equation to obtain the function's graph. Select integers from to 2, inclusive, for y.

Scientific and graphing calculators contain keys that can be used to evaluate the logarithmic function with base 10 and the logarithmic function with base e.



y = xlog2 = x.2y y = xlog2 y = ?2x

y = xlogb

y = .bx

y = x.log2

y = xlog2 = x,2y = x,2y

x = 2y

Start with vaules for y.

▼

y

=2−2 14

−2 ( , −2)14

=2−1 12

−1 ( , −1)12

= 120

= 221

= 422

= 823

▲

Compute x using

x = .2y

y = x.log2

y = x,logb f (x) = x,logb b > 1,

y = xlog3 −2








Page 468

Key Function the Key Is Used to EvaluateKey Function the Key Is Used to Evaluate

4 Use logarithmic models.

Example 5 Dangerous Heat: Temperature in an Enclosed VehicleWhen the outside air temperature is anywhere from to Fahrenheit, the temperature in an enclosed vehicle climbs by in the first hour.


LOG y = x ◀log10

This is called the common logarithmic

function, usually expressed as y = log x.

LN y = x ◀loge

This is called the natural logarithmic

function, usually expressed as y = ln x.

72° 96° 43°









The bar graph in Figure 7.50(a) shows the temperature increase throughout the hour. A scatter plot of the data is shown in Figure 7.50(b).

dSource: Professor Jan Null, San Francisco State University

Because the data in the scatter plot increase rapidly at first and then begin to level off a bit, the shape suggests that a logarithmic function is a good choice for amodel. After entering the data, a graphing calculator displays the logarithmic model, shown in Figure 7.51.

FIGURE 7.51 Data (10, 19), (20, 29), (30, 34), (40, 38), (50, 41), (60, 43)

a. Express the model in function notation, with numbers rounded to one decimal place.

b. Use the function to find the temperature increase, to the nearest degree, after 50 minutes. How well does the function model the actual increase shown in Figure7.50(a)?

SOLUTION

a. Using Figure 7.51 and rounding to one decimal place, the function

models the temperature increase, in degrees Fahrenheit, after x minutes.

b. We find the temperature increase after 50 minutes by substituting 50 for x and evaluating the function at 50.

Perform this computation on your calculator.

The display should be approximately 40.821108. Rounding to the nearest degree, the logarithmic model indicates that the temperature will have increased byapproximately after 50 minutes. Because the increase shown in Figure 7.50(a) is the function models the actual increase extremely well.

Check Point 5



y = a + b ln x,

f (x) = −11.6 + 13.4 ln x

f (x) ,

f (x)

f (50)

=

=

−11.6 + 13.4 ln x

−11.6 + 13.4 ln 50

This is the logarithmic model from part (a).


Scientific calculator : 11.6 13.4 50+ −/ + × LN =

Graphing calculator : 11.6 13.4 50(−) + LN ENTER

41° 41°,








Page 469

Use the model obtained in Example 5(a) to find the temperature increase, to the nearest degree, after 30 minutes. How well does the function model the actualincrease shown in Figure 7.50(a)?

Great Question!How can I use a graphing calculator to see how well my models describe the data?

Once you have obtained one or more models for the data, you can use a graphing calculator's feature to numerically see how well each model describesthe data. Enter the models as and so on. Create a table, scroll through the table, and compare the table values given by the models to the actual data.


TABLE, ,y1 y2








7 Algebra: Graphs, Functions and Linear Systems > 7.6 Modeling Data: Exponential, Logarithmic, and Quadratic Functions >Modeling with Quadratic Functions

Modeling with Quadratic FunctionsThe scatter plot in Figure 7.52 has a shape that indicates the data are first decreasing and then increasing. This type of behavior can be modeled by a quadraticfunction.

FIGURE 7.52

Definition of the Quadratic FunctionA quadratic function is any function of the form

where a, b, and c are real numbers, with

The graph of any quadratic function is called a parabola. Parabolas are shaped like bowls or inverted bowls, as shown in Figure 7.53. If the coefficient of (the valueof a in ) is positive, the parabola opens upward. If the coefficient of is negative, the graph opens downward. The vertex (or turning point) of theparabola is the lowest point on the graph when it opens upward and the highest point on the graph when it opens downward.

dFIGURE 7.53 Characteristics of graphs of quadratic functions

Look at the unusual image of the word mirror shown below. The artist, Scott Kim, has created the image so that the two halves of the whole are mirror images of eachother. A parabola shares this kind of symmetry, in which a line through the vertex divides the figure in half. Parabolas are symmetric with respect to this line, called theaxis of symmetry. If a parabola is folded along its axis of symmetry, the two halves match exactly.



y = a + bx + c or f (x) = a + bx + c,x2 x2

a ≠ 0.

x2

a + bx + cx2 x2








Page 470

When graphing quadratic functions or using them as models, it is frequently helpful to determine where the vertex, or turning point, occurs.

The Vertex of a ParabolaThe vertex of a parabola whose equation is occurs where


y = a + bx + cx2

x = .−b

2a









5 Graph quadratic functions.

Several points are helpful when graphing a quadratic function. These points are the x-intercepts (although not every parabola has x-intercepts), the y-intercept, and thevertex.

Graphing Quadratic FunctionsThe graph of or called a parabola, can be graphed using the following steps:

1. Determine whether the parabola opens upward or downward. If it opens upward. If it opens downward.

2. Determine the vertex of the parabola. The x-coordinate is

The y-coordinate is found by substituting the x-coordinate into the parabola's equation and evaluating.

3. Find any x-intercepts by replacing y or with 0. Solve the resulting quadratic equation for x.

4. Find the y-intercept by replacing x with 0. Because (the constant term in the function's equation), the y-intercept is c and the parabola passes through(0, c).

5. Plot the intercepts and the vertex.

6. Connect these points with a smooth curve.

Example 6 Graphing a ParabolaGraph the quadratic function:

SOLUTION

We can graph this function by following the steps in the box.

Step 1 Determine how the parabola opens. Note that a, the coefficient of is 1. Thus, this positive value tells us that the parabola opens upward.

Step 2 Find the vertex. We know that the x-coordinate of the vertex is Let's identify the numbers a, b, and c in the given equation, which is in the form

Now we substitute the values of a and b into the expression for the x-coordinate:

The x-coordinate of the vertex is 1. We substitute 1 for x in the equation to find the y-coordinate:

The vertex is shown in Figure 7.54

d



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

a > 0, a < 0,

. −b

2a

f (x)

f (0) = c

y = − 2x − 3.x2

,x2 a > 0;

.−b

2a

y = a + bx + c.x2

y

=

x2

▲

a = 1

−

2x

▲

b = −2

−

3

▲

c = −3

x-coordinate of vertex = = = = 1.−b

2a

− (−2)

2 (1)

2

2

y = − 2x − 3x2

y-coordinate of vertex = − 2 ⋅ 1 − 3 = 1 − 2 − 3 = −4.12

(1, −4),








Page 471

FIGURE 7.54 The graph of

Step 3 Find the x-intercepts. Replace y with 0 in We obtain or We can solve this equation byfactoring.

The x-intercepts are 3 and The parabola passes through (3, 0) and shown in Figure 7.54.


y = − 2x − 3x2

y = − 2x − 3.x2 0 = − 2x − 3x2 − 2x − 3 = 0.x2

− 2x − 3x2

(x − 3) (x + 1)

x − 3 = 0 or x + 1

x = 3 x

=

=

=

=

0

0

0

−1

−1. (−1, 0),









Step 4 Find the y-intercept. Replace x with 0 in

The y-intercept is The parabola passes through shown in Figure 7.54.

FIGURE 7.54 (repeated) The graph of

Steps 5 and 6 Plot the intercepts and the vertex. Connect these points with a smooth curve. The intercepts and the vertex are shown as the four labeledpoints in Figure 7.54. Also shown is the graph of the quadratic function, obtained by connecting the points with a smooth curve.

Check Point 6Graph the quadratic function:

6 Use quadratic models.

Example 7 Modeling the Parabolic Path of a Punted FootballFigure 7.55 shows that when a football was kicked, the nearest defensive player was 6 feet from the point of impact with the kicker's foot. Table 7.4 shows fivemeasurements indicating the football's height at various horizontal distances from its point of impact. A scatter plot of the data is shown in Figure 7.56.

dFIGURE 7.55



y = − 2x − 3:x2

y = − 2 ⋅ 0 − 3 = 0 − 0 − 3 = −3.02

−3. (0, −3) ,

y = − 2x − 3x2

y = + 6x + 5.x2








Page 472

FIGURE 7.56 A scatter plot for the data in Table 7.4

TABLE 7.4x, Football's Horizontal Distance (feet) y, Football's Height (feet)0 230 28.460 36.890 27.2110 10.8

Because the data in the scatter plot first increase and then decrease, the shape suggests that a quadratic function is a good choice for a model. Using the data inTable 7.4, a graphing calculator displays the quadratic function, shown in Figure 7.57.

FIGURE 7.57 Data (0, 2), (30, 28.4), (60, 36.8), (90, 27.2), (110, 10.8)

a. Express the model in function notation.

b. How far would the nearest defensive player, who was 6 feet from the kicker's point of impact, have needed to reach to block the punt?

SOLUTION

a. Using Figure 7.57, the function

models the football's height, in feet, in terms of its horizontal distance, x, in feet.

b. Figure 7.55 shows that the defensive player was 6 feet from the point of impact. To block the punt, he needed to touch the football along its parabolic path. Thismeans that we must find the height of the ball 6 feet from the kicker. Replace x with 6 in the function,


y = a + bx + c,x2

f (x) = −0.01 + 1.18x + 2x2

f (x) ,

f (x) = −0.01 + 1.18x + 2.x2








Page 473


The defensive player would have needed to reach 8.72 feet above the ground to block the punt.

Assuming that the football was not blocked by the defensive player, the graph of the function that models the football's parabolic path is shown in Figure 7.58. The graphis shown only for indicating horizontal distances that begin at the football's impact with the kicker's foot and end with the ball hitting the ground. Notice how thegraph provides a visual story of the punted football's parabolic path.

dFIGURE 7.58 The parabolic path of a punted football

Check Point 7Use the model obtained in Example 7(a) to answer this question: If the defensive player had been 8 feet from the kicker's point of impact, how far would he haveneeded to reach to block the punt? Does this seem realistic? Identify the solution as a point on the graph in Figure 7.58.

7 Determine an appropriate function for modeling data.

Table 7.5 contains a description of the scatter plots we have encountered in this section, as well as the type of function that can serve as an appropriate model for eachdescription.

TABLE 7.5 Modeling DataDescription of Data Points in a Scatter Plot Model

Lie on or near a line Linear Function: or

Increasing more and more rapidly Exponential Function: or

Increasing, although rate of increase is slowing down

Logarithmic Function: or

Decreasing and then increasing

Quadratic Function: or

The vertex, is a minimum point on the parabola.

Increasing and then decreasing

Quadratic Function: or

The vertex, is a maximum point on the parabola.



f (6) = −0.01 + 1.18 (6) + 2 = −0.36 + 7.08 + 2 = 8.72(6)2

x ≥ 0,

y = mx + b f (x) = mx + b

y = bx f (x) = , b > 1bx

y = xlogb f (x) = x, b > 1 logb

(y = x means = x. )logb by

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

( , f ( )) ,−b

2a

−b

2a

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

( , f ( )) ,−b

2a

−b

2a










7 Algebra: Graphs, Functions and Linear Systems > 7.6 Modeling Data: Exponential, Logarithmic, and Quadratic Functions >Concept and Vocabulary Check

Once the type of model has been determined, the data can be entered into a graphing calculator. The calculator's regression feature will display the specific function ofthe type requested that best fits the data. That, in short, is how your author obtained the algebraic models you have encountered throughout this book. In this era oftechnology, the process of determining models that approximate real-world situations is based on a knowledge of functions and their graphs, and has nothing to do withlong and tedious computations.

Concept and Vocabulary CheckFill in each blank so that the resulting statement is true.

1. Data presented in a visual form as a set of points are called a/an ________. The line that best fits the data points is called a/an ________ line.

For each set of points in Exercises 2–5, determine whether an exponential function, a logarithmic function, a linear function, or a quadratic function is the best choicefor modeling the data.

2.

________

3.

________

4.

________

5.

________

6. The exponential function with base b is defined by

7. The irrational number e is approximately equal to ________. The function is called the ________ exponential function.

8. is equivalent to the exponential form ________,

9. is usually expressed as ______ and is called the ________ logarithmic function.

10. is usually expressed as _____ and is called the ________ logarithmic function.

11. The function or is called a/an ________ function. The graph of this function is called a/an ________. Thevertex, or turning point, occurs where ____.

Exercise Set 7.6Practice ExercisesIn Exercises 1–6, use a table of coordinates to graph each exponential function. Begin by selecting and 2 for x.

1.



y =__, b > 0 and b ≠ 1.

y = or f (x) =ex ex

y = xlogb x > 0, b > 0, b ≠ 1.

y = xlog10 y =

y = xloge y =

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

x =

−2, −1, 0, 1,

f (x) = 4x








Page 474

2.

3.

4.

5.

6.

In Exercises 7–8,

a. Rewrite each equation in exponential form.

b. Use a table of coordinates and the exponential form from part (a) to graph each logarithmic function. Begin by selecting 0, 1, and 2 for y.

7.

8.

In Exercises 9–14,

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

b. Find the vertex.

c. Find the x-intercepts.

d. Find the y-intercept.

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

9.

10.

11.

12.

13.

14.

In Exercises 15–22,

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.

15.

x y0 09 116 1.219 1.325 1.4

16.

x y0 0.38 115 1.218 1.324 1.4


f (x) = 5x

y = 2x+1

y = 2x−1

f (x) = 3x−1

f (x) = 3x+1

−2, −1,

y = xlog4

y = xlog5

y = + 8x + 7x2

y = + 10x + 9x2

f (x) = − 2x − 8x2

f (x) = + 4x − 5x2

y = − + 4x − 3x2

y = − + 2x + 3x2









17.

x y01 22 73 124 17

18.

x y0 51 32 134

19.

x y0 41 12 03 14 4

20.

x y012 034

21.

x y012 03 44 12

22.

x y0 41 52 73 114 19

Practice PlusIn Exercises 23–24, use a table of coordinates to graph each exponential function. Begin by selecting 0, 1, and 2 for x. Based on your graph, describe theshape of a scatter plot that can be modeled by

23.

24.

In Exercises 25–26, use the directions for Exercises 7–8 to graph each logarithmic function. Based on your graph, describe the shape of a scatter plot that can bemodeled by

25.

26.



−3

−1−3

−4−1

−1−4

−3−2

−2, −1,f (x) = , 0 < b < 1.bx

f (x) = (Equivalently, y = )( )12

x2−x

f (x) = (Equivalently, y = )( )13

x

3−x

f (x) = x, 0 < b < 1.logb

y = log x12

y = log x13








Page 475

In Exercises 27–28, use the directions for Exercises 9–14 to graph each quadratic function. Use the quadratic formula to find x-intercepts, rounded to the nearest tenth.

27.

28.

In Exercises 29–30, find the vertex for the parabola whose equation is given by writing the equation in the form

29.

30.

Application ExercisesIn 1900, Americans age 65 and over made up only 4.1% of the population. By 2010, that figure was 13%, or approximately 44.6 million people. Demographic projectionsindicate that by the year 2050, approximately 82 million Americans, or 20% of the U.S. population, will be at least 65. The bar graph shows the number of people in theUnited States age 65 and over, with projected figures for the year 2020 and beyond.

d

Source: U.S. Census Bureau

The graphing calculator screen displays an exponential function that models the U.S. population age 65 and over, y, in millions, x years after 1899. Use this informationto solve Exercises 31–32.

31.

a. Explain why an exponential function was used to model the population data.

b. Use the graphing calculator screen to express the model in function notation, with numbers rounded to three decimal places.

c. According to the model in part (b), how many Americans age 65 and over were there in 2010? Use a calculator with a key or a key, and round to onedecimal place. Does this rounded number overestimate or underestimate the 2010 population displayed by the bar graph? By how much?

d. According to the model in part (b), how many Americans age 65 and over will there be in 2020? Round to one decimal place. Does this rounded numberoverestimate or underestimate the 2020 population projection displayed by the bar graph? By how much?


f (x) = −2 + 4x + 5x2

f (x) = −3 + 6x − 2x2

y = a + bx + c.x2

y = + 2(x − 3)2

y = + 3(x − 4)2

yx ∧









32. Refer to the graph showing the U.S. population age 65 and over and the graphing calculator screen on the previous page.

a. Explain why an exponential function was used to model the population data.

b. Use the graphing calculator screen to express the model in function notation, with numbers rounded to three decimal places.

c. According to the model in part (b), how many Americans age 65 and over were there in 2000? Use a calculator with a key or a key, and round to onedecimal place. Does this rounded number overestimate or underestimate the 2000 population displayed by the bar graph? By how much?

d. According to the model in part (b), how many Americans age 65 and over will there be in 2030? Round to one decimal place. Does this rounded numberoverestimate or underestimate the 2030 population projection displayed by the bar graph? By how much?

Use a calculator with an key to solve Exercises 33–34.

Average annual premiums for employer-sponsored family health insurance policies more than doubled over 11 years. The bar graph shows the average cost of a familyhealth insurance plan in the United States for six selected years from 2000 through 2011.

d

Source: Kaiser Family Foundation

The data can be modeled by

in which f(x) and g(x) represent the average cost of a family health insurance plan x years after 2000. Use these functions to solve Exercises 33–34. Where necessary,round answers to the nearest whole dollar.

33.

a. According to the linear model, what was the average cost of a family health insurance plan in 2011?

b. According to the exponential model, what was the average cost of a family health insurance plan in 2011?

c. Which function is a better model for the data in 2011?

34.

a. According to the linear model, what was the average cost of a family health insurance plan in 2008?

b. According to the exponential model, what was the average cost of a family health insurance plan in 2008?

c. Which function is a better model for the data in 2008?

The data in the following table indicate that between the ages of 1 and 11, the human brain does not grow linearly, or steadily. A scatter plot for the data is shown to theright of the table.

GROWTH OF THE HUMAN BRAINAge Percentage of Adult Size Brain1 30%2 50%4 78%6 88%8 92%10 95%



yx ∧

ex

f (x) = 782x + 6564 and g (x) = 6875 ,e0.077x








Page 476

Age Percentage of Adult Size Brain11 99%

Source: Gerrig and Zimbardo, Psychology and Life, 18th Edition, Allyn and Bacon, 2008.

d

The graphing calculator screen displays the percentage of an adult size brain, y, for a child at age x, where Use this information to solve Exercises 35–36.

35.

a. Explain why a logarithmic function was used to model the data.

b. Use the graphing calculator screen to express the model in function notation, with numbers rounded to the nearest whole number.

c. According to the model in part (b), what percentage of an adult size brain does a child have at age 10? Use a calculator with an key and round to thenearest whole percent. Does this overestimate or underestimate the percent displayed by the table? By how much?

36.

a. Explain why a logarithmic function was used to model the data.

b. Use the graphing calculator screen to express the model in function notation, with numbers rounded to the nearest whole number.

c. According to the model in part (b), what percentage of an adult size brain does a child have at age 8? Use a calculator with an key and round to thenearest whole percent. How does this compare with the percent displayed by the table?


1 ≤ x ≤ 11.

LN

LN









The percentage of adult height attained by a girl who is x years old can be modeled by

where x represents the girl's age (from 5 to 15) and represents the percentage of her adult height. Use the function to solve Exercises 37–38.

37.

a. According to the model, what percentage of her adult height has a girl attained at age 13? Use a calculator with a key and round to the nearest tenth ofa percent.

b. Why was a logarithmic function used to model the percentage of adult height attained by a girl from ages 5 to 15, inclusive?

38.

a. According to the model, what percentage of her adult height has a girl attained at age ten? Use a calculator with a key and round to the nearest tenthof a percent.

b. Why was a logarithmic function used to model the percentage of adult height attained by a girl from ages 5 to 15, inclusive?

39. A ball is thrown upward and outward from a height of 6 feet. The table shows four measurements indicating the ball's height at various horizontal distances fromwhere it was thrown. The graphing calculator screen displays a quadratic function that models the ball's height, y, in feet, in terms of its horizontal distance, x, in feet.

x, Ball's Horizontal Distance (feet) y, Ball's Height (feet)0 61 7.63 64 2.8

a. Explain why a quadratic function was used to model the data. Why is the value of a negative?

b. Use the graphing calculator screen to express the model in function notation.

c. Use the model from part (b) to determine the x-coordinate of the quadratic function's vertex. Then complete this statement: The maximum height of the balloccurs ________ feet from where it was thrown and the maximum height is ________ feet.

40. A ball is thrown upward and outward from a height of 6 feet. The table shows four measurements indicating the ball's height at various horizontal distances fromwhere it was thrown. The graphing calculator screen displays a quadratic function that models the ball's height, y, in feet, in terms of its horizontal distance, x, in feet.

x, Ball's Horizontal Distance (feet) y, Ball's Height (feet)0 60.5 7.41.5 94 6

a. Explain why a quadratic function was used to model the data. Why is the value of a negative?

b. Use the graphing calculator screen to express the model in function notation.

c. Use the model from part (b) to determine the x-coordinate of the quadratic function's vertex. Then complete this statement: The maximum height of the balloccurs ________ feet from where it was thrown, and the maximum height is ________ feet.

Writing in Mathematics



f (x) = 62 + 35 log (x − 4) ,

f (x)

LOG

LOG








Page 477

41. What is a scatter plot?

42. What is an exponential function?

43. Describe the shape of a scatter plot that suggests modeling the data with an exponential function.

44. Describe the shape of a scatter plot that suggests modeling the data with a logarithmic function.

45. Would you prefer that your salary be modeled exponentially or logarithmically? Explain your answer.

46. Describe the shape of a scatter plot that suggests modeling the data with a quadratic function.

Critical Thinking ExercisesMake Sense? In Exercises 47–50, determine whether each statement makes sense or does not make sense, and explain your reasoning.

47. I'm looking at data that show the number of new college programs in green studies, and a linear function appears to be a better choice than an exponentialfunction for modeling the number of new college programs from 2005 through 2009.

d

Source: Association for the Advancement of Sustainability in Higher Education

48. I used two different functions to model the data in a scatter plot.

49. Drinking and driving is extremely dangerous because the risk of a car accident increases logarithmically as the concentration of alcohol in the blood increases.









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.415

7.2 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



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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1/9/2018 thinking mathematically, sixth edition€¦ · solution of a specific practical problem....

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