Activity 5.2

Major League Salaries: Rates of Change and Concavity

10 points

Due at the beginning of class, Friday, February 20, 2009

In this activity you will investigate major league baseball salary data and consider connections between rates of change and the graph of a function and what these tell you about the data.

1. Simply download the file, EA5.2 Major League Salaries.xls. Note that the table in this file gives the average yearly salary of major league baseball players from 1980 to 2007.

2. Calculate (using a calculator or Excel) the average rate of change in average yearly salary from 1980 to 1981 and from 1981 to 1982.

3. Now use the following Excel instructions to calculate the average rate of change in salary per year from one year to the next. When you are done, paste the entire table, including the new column of rates of change, into your Word document.

Instructions to Use Excel to Calculate Average Rate of Change
3. a. To enter the formula for rate of change of average salary, click cell C3, type =(B3 - B2) / (A3 - A2), and press Enter. (Does the number in C3 agree with one of the two in #2?)

3. b. Click C3 and autofill to the end of your data.

3. c. Enter an appropriate title for column C in cell C1.

3. d. You can use Audit mode (Ctrl and ` together) to check that your formula for rate of change autofilled correctly. (Press Ctrl and ` together again to exit Audit mode.)

4. What do the numbers in Column C represent?

5. Use the data now in your table (do not use Excel to graph the data yet) to answer the following:

5. a. Give the interval(s) of time when the average yearly salary was increasing.

5. b. Give the interval(s) of time when the average yearly salary was decreasing.

5. c. Within the years 1986-1997, find the interval(s) of time when the average rate of change per year in average salary is increasing. (Think about why this question is different than the question in part a.)

5. d. Should the graph of the function that gives the average yearly salary of major-league players be concave-up or concave-down over the intervals indicated in part c? Why?

5. e. Within the years 1986-1997, give the intervals(s) of time when the average rate of change per year in average salary is decreasing.

5. f. Should the graph of the function that gives the average yearly salary of major league players be concave downward or concave upward over the intervals of time indicated in part e? Why?

6. Use the information obtained in question 5 a - f to draw a rough sketch (on a separate sheet of paper, which you do not need to turn in) of the function from 1986 to 1997. Your graph should have an appropriate scale on the vertical axis.

7. Use Excel to draw a graph of the average major-league salary function over time. (On your Excel graph, include all data given, starting in 1980.) How does this graph compare with your sketch?

8. What happened to the average salary of major league players over the interval of time when the graph of the function is increasing and concave downward?

9. What happened to the average salary of major league players over the interval of time when the graph of the function is increasing and concave upward?

10. What portion of the graph could be approximated by a line? Explain why you chose that portion.

11. Write an equation of a line that approximates the values of the given function from 1980 to 1986. To simplify the numbers, take the year 1980 as time t = 0, 1981 as time t = 1, and so on. Explain how you obtained your line.

Summary
In this activity, you used Excel to find average rates of change of salaries for major league baseball players. You used these to draw a rough graph of the function that gives the players' average yearly salary and then used Excel to create the graph. You also approximated a portion of the graph using a line and then found an equation of the line.