APPENDIX B

B.2

Mathematical Modeling

B11

Modeling Data with Quadratic Functions Classifying Scatter Plots • Fitting Quadratic Models to Data • Application

Classifying Scatter Plots In Appendix B.1, you saw how to fit linear models to data. In real life, many relationships between two variables are nonlinear. A scatter plot can be used to give you an idea of which type of model can be used to best fit a set of data.

EXAMPLE 1 Classifying Scatter Plots

Decide whether the data could be better modeled by a linear model y  ax  b or a quadratic model y  ax2  bx  c. (a) (0.2, 2.6), (0.4, 2.3), (0.6, (1.7, 2.5), (1.9, 2.8), (2.2, (3.2, 6.8), (3.4, 7.8) (b) (0.2, 4.4), (0.4, 4.9), (0.5, (1.7, 7.4), (1.9, 7.8), (2.0, (3.2, 10.4), (3.4, 10.8)

2.1), (0.8, 2.0), (1.1, 2.1), (1.3, 2.1), (1.4, 2.2), 3.3), (2.3, 3.8), (2.6, 4.4), (2.7, 5.1), (3.0, 5.9), 5.2), (0.7, 5.7), (1.1, 6.1), (1.2, 6.5), (1.5, 7.0), 8.2), (2.3, 8.7), (2.5, 9.0), (2.8, 9.5), (2.9, 9.9),

Solution

Begin by entering the data into a graphing utility. You should obtain the scatter plots shown in Figure B.9. 12

8

0

4 0

(a)

0

4 0

(b)

FIGURE B.9

From the scatter plots, it appears that the data in part (a) can be modeled by a quadratic function and the data in part (b) can be modeled by a linear function.

B12

APPENDIX B

Mathematical Modeling

Fitting Quadratic Models to Data Once you have used a scatter plot to determine the type of model to be fit to a set of data, there are several ways that you can actually find the model. Each method is best used with a computer or calculator, rather than with hand calculations.

EXAMPLE 2 Fitting a Model to Data

Fit the following data, from Example 1(a), with a linear model and a quadratic model. Which model do you think fits better? (0.2, 2.6), (0.4, 2.3), (0.6, 2.1), (0.8, 2.0), (1.1, 2.1), (1.3, 2.1), (1.4, 2.2), (1.7, 2.5), (1.9, 2.8), (2.2, 3.3), (2.3, 3.8), (2.6, 4.4), (2.7, 5.1), (3.0, 5.9), (3.2, 6.8), (3.4, 7.8) Solution

Begin by entering the data into a calculator or computer that has least squares regression programs. Then run the regression programs for linear and quadratic models. Linear:

y  ax  b

a  1.60

Quadratic: y  ax2  bx  c

b  0.7

a  0.983 b  1.93

c  2.9

From the graphs in Figure B.10, you can see that the quadratic model fits better. 8

8

0

4

0

4 0

0

Linear

Quadratic

FIGURE B.10

NOTE Deciding which model best fits a set of data is a problem that is studied in detail in statistics. In statistics, how well a model fits a set of data is measured by a number called the correlation coefficient, which is denoted by r. The closer r is to 1, the better the model fits the data. In Example 2, the correlation coefficient for the linear model is r  0.880, but the correlation coefficient for the quadratic model is r  0.998.



APPENDIX B

Mathematical Modeling

B13

EXAMPLE 3 Fitting a Model to Data

Fit the following data, from Example 1(b), with a linear model and a quadratic model. Which model do you think fits better? (0.2, 4.4), (0.4, 4.9), (0.5, 5.2), (0.7, 5.7), (1.1, 6.1), (1.2, 6.5), (1.5, 7.0), (1.7, 7.4), (1.9, 7.8), (2.0, 8.2), (2.3, 8.7), (2.5, 9.0), (2.8, 9.5), (2.9, 9.9), (3.2, 10.4), (3.4, 10.8) Solution

Begin by entering the data into a calculator or computer that has least squares regression programs. Then run the regression programs for linear and quadratic models. Linear:

y  ax  b

a  1.96 b  4.1

Quadratic: y  ax2  bx  c

a  0.001 b  1.96 c  4.1

From the scatter plots, notice that the data appears to follow a linear pattern. Although both models fit the data well, the linear model is better because it is simpler. The graphs of both models are shown in Figure B.11. 12

12

0

4 0

Linear FIGURE B.11

0

4 0

Quadratic

B14

APPENDIX B

Mathematical Modeling

Application EXAMPLE 4 Finding a Quadratic Model

The total number of pediatricians P (in thousands) in the United States in selected years from 1980 through 1998 are shown below. Decide whether a linear model or a quadratic model better fits the data. Then use the model to predict the number of pediatricians in the year 2005. In the list of data points t, P, t represents the year, with t  0 corresponding to 1980. (Source: American Medical Association) (0, 17.4), (5, 22.4), (10, 26.5), (12, 29.0), (13, 30.8), (14, 31.5), (15, 33.9), (16, 35.5), (17, 36.8), (18, 38.4) Solution

Begin by entering the data into a calculator or computer that has least squares regression programs. Then run the regression programs for linear and quadratic models. Linear:

y  ax  b

a  1.16

Quadratic: y  ax2  bx  c

b  16.3

a  0.030 b  0.61

c  17.7

From the graphs in Figure B.12, you can see that the quadratic model fits better. (The correlation coefficient for the quadratic model is r  0.996, which implies that the model is a good fit to the data.) From this model, you can predict the number of pediatricians in the year 2005 to be P  0.030252  0.6125  17.7  51.7 thousand which is more than two times the number of pediatricians in 1985. 40

0

40

18 0

Linear FIGURE B.12

0

18 0

Quadratic

APPENDIX B

Exercises

B.2

In Exercises 1–4, decide whether you would use the linear model y  ax  b or the quadratic model y  ax2  bx  c to model the data. y

1.

y

9.

3 12

(5, 2)

(4, 12)

y

2.

y

10. 2

8

(3, 6)

1

4

(−1, 1) −8

x y

3.

x y

4.

x

x

In Exercises 5–14, use a graphing utility to fit a linear model and a quadratic model to the points. Then plot the data and graph the models. Which model would you use? y

5.

y

6.

5

5

(2, 5)

4

4

3

(−2, 0)

(1, 2) (0, 1)

1

1

2

(−2, 0)

1

x

−3 −2 −1 −1

3

y

1

2

3

y

8. (−2, 6)

(0, 4)

(0, 7)

6

(1, 3)

3

(0, 2) (0, 1)

2 x

−3 −2 −1 −1

4

(2, 3)

3

(−1, 0) 2

7.

B15

Mathematical Modeling

(−4, 5)

4 2

(1, 1) (2, 0) 1

2

x 3

4

−4

−2

x

−1

x 4

(6, 2) (3, 1) (4, 1) (3, 0)

8

−1

4 5 6 7

(1, 0) (2, 0)

0, 0, 1, 1, 3, 4, 4, 2, 5, 5 2, 5, 1, 5, 0, 4, 1, 3, 2, 0 4, 5, 2, 6, 2, 6, 4, 5 0, 6, 4, 3, 5, 0, 8, 4, 10, 5 Stopping Distance The speeds (in miles per hour) and stopping distances (in feet) for an automobile braking system were recorded as follows. Speed (x)

30

40

50

60

70

Stopping distance (y)

25

55

105

188

300

(a) Use a graphing utility to fit a least squares regression quadratic to the data, then plot the data and graph the model in the same viewing window. (b) Use the model in part (a) to estimate the stopping distance when x  70. Compare this estimate with the actual value given in the table. 16. Wildlife Management A wildlife management team studied the reproduction rates of deer in five tracts of a wildlife preserve. In each tract, the number of females and the percent of females that had offspring were recorded as follows. Number x

80

100

120

140

160

Percent  y

80

75

68

55

30

(2, 6)

2 1

11. 12. 13. 14. 15.

−4

(2, 2)

(4, 2)

(4, 2) x 2

4

(a) Use a graphing utility to fit a least squares regression quadratic to the data. (b) Use the model in part (a) to predict the percent of females that would have offspring if x  170.

B16

APPENDIX B

Mathematical Modeling

17. Engine Performance After a new turbocharger for an automobile engine was developed, the following experimental data was obtained for speed in miles per hour at 2-second intervals. Time x

0

2

4

6

8

10

Speed  y

0

15

30

50

65

70

(a) Use a graphing utility to fit a least squares regression quadratic to the data. (b) Plot the data and graph the model in the same viewing window. 18. World Population The table shows the world population (in billions) for seven different years. (Source: U.S. Census Bureau) Year

1995 1996 1997 1998 1999 2000 2001

Population 5.7

5.8

5.8

5.9

6.0

6.1

6.2

Let x  5 represent 1995. (a) Use a graphing utility to fit a least squares regression line to the data. (b) Use a graphing utility to fit a least squares regression quadratic to the data. (c) Use both models in parts (a) and (b) to predict the world population for the year 2010. How do the two models differ as you extrapolate into the future? Which do you think is a better model? 19. Health Maintenance Organizations The table shows the number N (in millions) of people enrolled in HMOs in four selected years. (Source: The Interstudy Competitve Edge) Year

1990

1995

1998

1999

Enrollment

33.0

46.2

64.8

81.3

Let x  0 represent 1990. (a) Use a graphing utility to fit a least squares regression line to the data. (b) Use a graphing utility to fit a least squares regression quadratic to the data.

(c) Use a graphing utility to plot the data and graph the models in parts (a) and (b). (d) Use both models in part (c) to predict the enrollment in HMOs in the year 2010. How do the two models differ as you extrapolate into the future? 20. Cassette Singles The table shows the number N (in millions) of cassette singles sold for the years 1996 through 1999. (Source: Recording Industry Association of America) Year

1996

1997

1998

1999

Singles

59.9

42.2

26.4

14.2

Let x  6 represent 1996. (a) Use a graphing utility to fit a least squares regression line to the data. (b) Use a graphing utility to fit a least squares regression quadratic to the data. (c) Use a graphing utility to plot the data and graph the models in parts (a) and (b). Which do you think is a better model? 21. Path of a Ball After the path of a ball thrown by a baseball player is videotaped, it is analyzed on a television set with a grid covering the screen. The tape is paused three times and the positions of the ball are measured. The coordinates are approximately (0, 5.0), (15, 9.6), and (30, 12.4). The x-coordinate measures the horizontal distance from the player in feet and the y-coordinate measures the height in feet. (a) Use a graphing utility to fit a least squares regression quadratic to the data. (b) Use a graphing utility to plot the data and graph the model in part (a) in the same viewing window. How well does the model fit the data? Explain. (c) Graphically approximate the maximum height of the ball and the point at which it struck the ground. (d) Algebraically find the maximum height of the ball and the point at which it struck the ground.