You can quantify how fast a function is changing on average over a specific interval by using the ratio = f(x_2)-f(x_1)x_2-x_1. Do you think this ratio will be the same all along the graph of the function?
See solution.
Practice makes perfect
To compare the growth rate of different functions we can quantify how the function is changing compared to the change in the corresponding x-values. Since this value is not constant for all functions, we can find an average rate of change (AROC) by calculating the slope of a line passing through two points on the graph of the function.
Rate of Change Formula [0.5em]
m = y_2-y_1/x_2-x_1 = f(x_2)-f(x_1)/x_2-x_1
In the equation above, (x_1,y_1) and (x_2,y_2) are two points on the graph of the function y=f(x). Since, the AROC is not constant — except for linear functions — for a better comparison we need to compare it between several pairs of points. This way, we can see how this rate is changing along the function's graph.
