AFM Exam Review pages 5,6: Calculator Modeling

Multiple Choice

Identify the choice that best completes the statement or answers the question.

____1.Find an exponential function to model the data.

x / y
1 / 7
2 / 16
3 / 30
4 / 61
5 / 124
6 / 271
7 / 522
a. / f(x) = 116.4 - 42.8 lnx / c. / f(x) = 3.56(2.04)x
b. / f(x) = 2.04(3.56)x / d. / f(x) = -42.8 + 116.4 lnx

____2.Find a logarithmic function to model the data.

x / y
1 / 60
2 / 54
3 / 51
4 / 50
5 / 46
6 / 45
7 / 44
a. / f(x) = 60.73(0.95)x / c. / f(x) = 60.04 - 8.25 lnx
b. / f(x) = 0.93(60.73)x / d. / f(x) = 8.25 - 60.04 lnx

____3.Find a logistic function to model the data.

x / y
0 / 98
2 / 110
4 / 117
6 / 121
8 / 123
10 / 124
12 / 125
14 / 126
a. / f(x) = / c. / 105.28 + 8.15 lnx
b. / f(x) = / d. / f(x) = 105.6(1.0155)x

____4.As automobiles age, the average miles traveled per gallon decreases.Determine the regression equation that best models the data.

Age (years) / MPG
1 / 35
3 / 34
5 / 33
7 / 31
9 / 28
11 / 26
13 / 23
15 / 18
a. / power / c. / quadratic
b. / logarithmic / d. / exponential

____5.Find the linear regression equation for the data according to the given model.

power

x / y
1 / 50
2 / 140
3 / 260
4 / 400
5 / 560
6 / 750
7 / 925
8 / 1130
a. / 49.79x1.50 / c. / 156.13x - 175.71
b. / 5.48x0.32 / d. / 1.5x + 3.91

Short Answer

6.Use a graphing calculator to write a polynomial function to model the data.

7.When rabbits were introduced to the continent of Australia they quickly multiplied and spread across the continent since there were only primitive marsupial competitors and predators to interfere with the exponential growth of their population. The data in the following table can be used to create a model of rabbit population growth.

Time (months) / 0 / 3 / 6 / 9 / 12
No. of Rabbits / 6 / 32 / 107 / 309 / 770

a. Find the regression equation for the rabbit population as a function of time x.

b. Write the regression equation in terms of base e.

c. Use part b’s equation to estimate the time for the rabbits to exceed 10,000.

8.The normal monthly temperatures (F) for Omaha, Nebraska, are recorded below.

Month / Jan / Feb / Mar / Apr / May / Jun / Jul / Aug / Sep / Oct / Nov / Dec
t / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 12
Temp. / 21 / 27 / 39 / 52 / 62 / 72 / 77 / 74 / 65 / 53 / 39 / 25

a. Write a sinusoidal function that models Omaha’s monthly temperature variation.

b. Use the model to estimate the normal temperature during the month of April.

9.Create a scatter plot of the data, and determine a power function to model the data. Then, calculate the value of the model at x = 10.

x / y
1 / 6.69
2 / 142.22
3 / 850.22
4 / 3,023.5
5 / 8,088.8
6 / 18,074.78
7 / 35,670.44
8 / 64,276.61

10.Use a graphing calculator to write a polynomial function to model the data.

11.Use a graphing calculator to write a polynomial function to model the data.

x / -2.5 / -2 / -1.5 / -1 / -0.5 / 0 / 0.5 / 1 / 1.5
f(x) / 23 / 11 / 7 / 6 / 6 / 5 / 3 / 2 / 4

12.Use a graphing calculator to write a polynomial function to model the data.

x / 30 / 35 / 40 / 45 / 50 / 55 / 60 / 65 / 70 / 75
f(x) / 52 / 41 / 32 / 44 / 61 / 88 / 72 / 59 / 66 / 93

13.Use a graphing calculator to write a polynomial function to model the data.

x / -3 / -2 / -1 / 0 / 1 / 2 / 3
f(x) / 8.75 / 7.5 / 6.25 / 5 / 3.75 / 2.5 / 1.25

14.Use a graphing calculator to write a polynomial function to model the data.

x / 5 / 7 / 8 / 10 / 11 / 12 / 15 / 16
f(x) / 2 / 5 / 6 / 4 / -1 / -3 / 5 / 9

15.Astronomers classify stars according to their brightness by assigning them a stellar “magnitude.” The higher the magnitude the dimmer the star. The dimmest stars visible to the naked eye have stellar magnitudes of 6. The table below shows the relative brightness of different stellar magnitudes.

Stellar Magnitude / 1 / 2 / 3 / 4 / 5 / 6
Relative Brightness / 100 / 40 / 16 / 6.3 / 2.5 / 1

a. Find an equation that gives the relative brightness in terms of stellar magnitude.

b. Use this equation to find the relative brightness of a star with magnitude 9.

16.Find an exponential function to model the data and find the value of the model at x = 2.25.

x / y
1 / 7.20
2 / 3.20
3 / 1.45
4 / 0.66
5 / 0.30
6 / 0.13
7 / 0.06

17.Find a logarithmic function to model the data and find the value of the model at x = 15.

x / y
1 / 21.0
2 / 25.16
3 / 27.60
4 / 29.32
5 / 30.66
6 / 31.75
7 / 32.68

18.Find a logistic function to model the data and find the value of the model at x = 16.5.

x / y
5 / 121
10 / 125
15 / 128
20 / 131
35 / 133
30 / 134
35 / 135
40 / 135

Linearize the data according to the given model and find the linear regression equation. Then use the linear model to find a model for the original data.

19.quadratic

x / y
0 / 0
1 / 1
2 / 10
3 / 30
4 / 50
5 / 80
6 / 115
7 / 155

20.The hourly temperature at Portland, Oregon, on a particular day is recorded below.

1 A.M. / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 12 Noon
46 / 44 / 43 / 41 / 40 / 40 / 41 / 43 / 46 / 52 / 65 / 69
1 P.M. / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 12 Midnight
72 / 74 / 75 / 75 / 77 / 75 / 74 / 70 / 62 / 55 / 51 / 48

a. Find the amplitude of a sinusoidal function that models this temperature variation.

b. Find the vertical shift of a sinusoidal function that models this temperature variation.

c. What is the period of a sinusoidal function that models this temperature variation?

d. Use t = 0 at 5 P.M. to write a sinusoidal function that models this temp. variation.