Rate of Change of a Function
The Rate of Change of a function is how much the y-value changes as the x-value changes by some amount. For a linear function, we would call that "slope". On a position vs time graph, it measures change in position per change in time, which we call velocity. If we measure this between two distinct points (with two distinct x-values), we call it the Average Rate of Change (AROC). In calculus, we will use the AROC to find the Instantaneous Rate of Change (IROC) at a single point (single x-value).
The app begins with "Show f" and "Show Secant Line" both unchecked. There are two points in the plane, (a, f(a)) and (b, f(b)). Looking at just these two points, we can easily find the slope of a line beween them using the familiar slope formula (with function notation instead of y notation). Check the "Show Secant Line" box to display the segment between the points.
If these two points existed in isolation, then all we would know is the change in x and the change in y between them, from which we calculate a slope. But if we think of these points as being points on a graph (check the "Show f" box), we now see that the slope calculation actually finds the AROC of the function between the two points.
Drag the x values (a and b) to various places, and observe the behavior with various functions. Notice that the AROC of a linear function is constant everywhere, equal to the slope of the line.