Comprehending the “average rate of change” of a function.
by Rabbit
Firstly, this material was written by a pre-calculus student. Therefore, it will probably be most beneficial towards pre-calculus inquiries. This is not a tutorial. I do not explain every concept used in this essay. I will move quickly and assume you have asked the same questions I have asked to arrive at this point.
If you find yourself being asked to “find the average rate of change of function” and that function happens to be non-linear (e.g. x^2), you’re probably being asked to find the average rate of change of the secant line of that same function (x^2). I say this because it simply doesn’t make sense to find the average rate of change for a function that does not change at a constant rate.
Worth your consideration when dealing with average rates of change for any function in two variables: if you are given two x-values, you actually have two points. Place each x-value into the function and solve for y to get these points.
With two points in hand, your goal is to write the slope-intercept form of the secant line of the given function between the the two points you calculated above. (Recall how to do this? Given points P1 and P2, find the slope. Given P1 and the slope, write the equation of the line in point-slop format, then evaluate this equation to produce the slope-intercept form.)
The secant line is just like any other line, except the points that form a secant line lie on the graph of some function. In pre-calculus, the secant line isn’t of particular interest, as the average rate of change of a secant line can be manipulated to read zero for a graph that clearly is never constant (e.g. x^2 with P1 = (4,4) and P2 = (-4,4)). However, I believe that doing such a thing is actually an obvious misuse of the idea behind a secant line. To the best of my knowledge, the secant line is intended to tell you the instantaneous rate of change at one point along a curve.
It may confuse you to consider the purpose of the secant line when dealing with average rate of change of a non-linear function.
As an aside, consider this: the secant line of a linear function is the graph of the function itself.
Here’s a completely worked out example of finding the secant line of a non-linear function given two x-values and a function f.
Ponder this: what would be the average rate of change of f(x) = x^2 if you are given the following two x-values: -4, 4?
