Stability or lack of it is important in analyzing these functions. The trigonometry problem which requires the student to find the angles for tunneling from both sides of a mountain is an example of a problems that can be used to explore stability. The data can be analyzed by fitting a linear curve to the data. Writing an equation or function to fit the data is one part of the mathematical modeling process. The goal is to find a function which accurately reflects the data and to use that function to answer questions about the dynamical system.
Residuals are a measure of the closeness of the fit.
The sum of the absolute values of the squares of the residuals can also be used, and will provide the student with some background for the least-squares technique of curve fitting. Advanced students will see that iteration of a linear function leads to exponential growth. Geometric analysis of linear dynamics can be done with Web Diagrams. These diagrams will provide a strong visual representation of the seed, the orbit, repelling fixed points, and attracting fixed points. They can be used to explore the effect of changing the slope at the fixed point.