Problem: One of the advantages of rational functions is that even rational functions with low-order polynomials can provide an excellent fit for complex experimental data. Linear-to-linear rational functions have been used to describe earthquake plates.
As another example, a linear-quadratic fit has been used to describe lung function after patients have been treated with x-rays; cubic/quadratic equations are used to model the stiffness of various materials.
To explore the versatility of rational functions, choose a second-order/third-order (e.g., x2/x3) and a third-order/second-order (e.g., x3/x2) rational function. Provide a graph for the second-order rational function (e.g., x2), choosing x values in the range from -10 through +10.
Then, provide at least three variations of the function plotted on the same graph. Include separate changes to a coefficient in the numerator, to a coefficient in the denominator, and to a constant. The changes should be increased or decreased by a factor of 2 in each case.
Repeat the procedure making a second graph for the third-order rational function (e.g., x3). For each of the two graphs, describe how changes in coefficients and constants change the behavior of the function.