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Test 7, Chapter 3 Form A

Objective: Fit linear, quadratic, logarithmic, exponential, power, or logistic functions to data.

Part 1: No calculators allowed (1–11)

1. Write the general equation for

• A quadratic function: ________________________

• A power function: __________________________

2. Name the kind of function.

• f(x) = [pic]__________________________

• g(x) ’ 3 + 7 ln x ___________________________

3. Name the numerical pattern followed by regularly spaced points in

• An exponential function: ____________________

• A quadratic function: _______________________

4. Which type of function listed in this test’s objective has a graph like the one shown here?

[pic]

5. On the figure in Problem 4, sketch the graph of the inverse of the function.

6. The table shows x- and y-values measured from a real-world situation. The regression equation for the best-fitting linear function is

[pic] = –2x + 19.5

In the table, write the value of[pic], the residual, and the square of the residual for each point. Calculate SSres, the sum of the squares of the residuals.

x y [pic] Residual Residual2

1 16

3 16

5 9

7 5

SSres ’ _________________

7. If you found SSres for the data in Problem 6 using any line besides [pic] = –2x + 19.5, how would the value of SSres compare with the value you calculated in Problem 6?

8. The figure shows a scatter plot of the data in Problem 6 along with the regression line. The figure also shows a line plotted at y = [pic], where [pic] = 11.5 is the average of the y-values. For the point where x = 3, show the deviation and the residual.

[pic]

9. For the data in Problem 6, the sum of the squares of the deviations is SSdev = 89. Use this information to write the coefficient of determination, r2.

10. The correlation coefficient is the square root of the coefficient of determination. Which square root would you use for the data in Problem 6: the positive one or the negative one? Explain why.

11. The full name of “residual” is “residual deviation.” What is the meaning of “residual” used in this context?

(Hand in this page to get the rest of the test.)

Test 7, Chapter 3 continued Form A

Part 2: Graphing calculators allowed (12–28)

Exponential Function Problem (12–15): An exponential function contains these two points.

x y

3 17

6 34

12. Without finding the equation, show how to calculate quickly the value of y when x ’ 12.

13. Find algebraically the particular equation for the exponential function by substituting the two given points into the general equation and finding the two coefficients.

14. Show that the equation in Problem 13 gives the value of y you calculated in Problem 12 at x ’ 12.

15. Use the equation in Problem 13 to find y when

x ’ 4.7. Which does this calculation involve: interpolation or extrapolation? How can you tell?

Housing Problem (16–19): The table shows the number of houses in the new community of Scorpion Gulch at various numbers of years after the community was opened.

Year Houses

0 13

1 15

2 18

3 25

4 34

5 42

6 50

7 55

8 58

9 60

16. Enter the data in two lists in your grapher and make a scatter plot. Give a reason why a logistic function would be appropriate based on

• The pattern of the data

• The endpoint behavior

17. Run a logistic regression. Write the particular equation and plot it on your grapher. Use the result to predict the number of houses at year 12.

18. Based on the logistic model, about how many houses will Scorpion Gulch ultimately have?

19. Use the regression equation and the given data in an appropriate way to calculate SSres, the sum of the squares of the residuals.

Hot Water Problem (20–27): Calvin Butterball heats a pot of water. Here are its temperatures at various numbers of seconds since he started heating it.

Time (s) Temperature (˚C)

10 23.1

20 26.3

30 29.5

40 32.8

50 35.9

60 38.6

70 41.2

80 43.9

90 46.7

100 49.3

20. Find the best-fitting linear function for the data. Write the particular equation and the correlation coefficient. Store the equation as f1(x).

21. How can you tell from the correlation coefficient that a linear function fits the data quite well?

22. Find SSres for the data and the linear function.

23. Make a residual plot. What does the fact that the residuals follow a pattern tell you about how well a linear function fits the data?

24. Find the best-fitting quadratic function for the residuals. Write the equation. Store it as f2(x).

25. Let f3(x) ’ f1(x) + f2(x). Plot f3(x) and the scatter plot of the original data on the same screen. Sketch the result.

26. Find SSres for the function in f3(x). How do you interpret the fact that it is smaller than SSres in Problem 22 for the linear function?

27. Run a quadratic regression on the original data. Write the particular equation. Store it as f4(x). Plot f4(x) on the same screen as the plot in Problem 25. Use thick style. What do you notice about the resulting graph?

28. What did you learn as a result of taking this test that you did not know before?

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