Teacher Guide and Answer Key, Algebra II Activity

Based on AP* Statistics Problem 6, 1997

Algebra II TEKS addressed: (a)(3); (a)(5); (2A.1)(A); (2A.1)(B); (2A.11)(F)

This problem is a good Pre-AP* problem because it requires students to work with exponential and linear functions in a problem-solving situation. Students are expected to use their knowledge of these functions to write a mathematical model for the given data. The data can be plotted on a graphing calculator. A student can see a graph of his/her model on the calculator then adjust his/her model as necessary. This problem builds a greater understanding of transformations of functions. This problem also introduces a piecewise function. Although piecewise functions are not addressed in the TEKS until Precalculus, piecewise functions can be written and interpreted in an Algebra II Pre-AP class using pieces of the functions that are studied in the Algebra II TEKS.

Algebra II Worksheet, Figure 1

2. y = (1.17)x The asking price using this exponential function is too low for 5 = years = 7 and too high for 12 = years = 14.

3. The asking price for year 9 would be about $4108.40 if you use the exponential model.

Algebra II Worksheet, Figure 2

5. The asking price for year 9 would be about $4300 if you use the piecewise model.

6. Using the exponential model for the year 23, the asking price would be about $37,006.22. Using the piecewise model for the year 23, the asking price would be about $20,400. The piecewise model would be more appropriate because the exponential model expected asking price is too high for the years between 12 and 14 thus I think it would be too high for any year after the 14th.

7. I think the piecewise function better models the data because values of the data are closer to the piecewise function than to the exponential functions. Also, the rate of change in the exponential function continues to increase which does not seem to fit the context for the asking price of the automobiles. It appears from the scatter plot that the rate of change for the asking price of automobiles is a constant value and therefore linear.

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