#### Project Title

Cell Phones

#### Author

David Graser, Yavapai College, Prescott, AZ (David_Graser@yc.edu)

#### Abstract

This project involves modeling the revenue per cell phone subscriber based on data for the number of cell phone subscribers and the total revenue from those subscribers. There are many different ways you could model this data. For instance, you could divide the revenue by the subscribers for each year and model the resulting quotients. Unfortunately, modeling this quotient with a polynomial leads to a function that grows or decays as x approaches infinity. Realistically, you would expect the revenue per subscriber to level off. A strategy is outlined for the student to model the subscribers with a polynomial and the revenue with another polynomial. Then the revenue per subscriber is modeled by a rational function formed from the two polynomials.

#### Project Content

#### Scaffolding Resources

None

#### Notes

- My initial attempts with this project lead to students taking the easy way out by modeling the quotient with a polynomial. I added a false strategy to the letter to encourage students to a better strategy. They also have a tendency to model with polynomials with different degrees. A great aha! moment occurs when they realize that they have to use the same degree to have a nonzero horizontal asymptote.
- I have never tried it, but I imagine that you could also model the subscriber and revenue data using exponential functions.
- In College Algebra and Precalculus, this project is useful to demonstrate rational functions and horizontal asymptotes. In Calculus, it can be used to analyze limits of rational functions.