Average Rates of Change

Below is a rudimentary graph that could represent the price of a stock over the course of a year. One might be interested in knowing how much the price of that stock changed, on average, from month to month, when you compare the price it started at and the price it finished at.

The line connecting the starting point of the graph to the ending point is called a Secant line and it represents the straight-line path from the start point to the end point. In other words, the secant line between two points on a graph essentially gives you the “gist” of what happened to the original graph between the two points “without all (possibly irrelevant) ups and downs along the way.”

The slope of the secant line joining two points gives you the average rate of change of the original graph from the starting point to the ending point. In the case of the graph above, this represents the stocks average increase in price from month to month.

Average rates of change are useful for summarizing “noisy” information in graphs to determine by how much, on average, a given quantity changes from one time period to the next. They are also useful for predicting where a data trend may go next.

Finding Average Rates of Change

This \(m\) is the slope of the line joining the points \((a,f(a))\) and \((b,f(b))\). It represents how much the graph changes in \(y\) (on average) given a certain amount of change in \(x\).

NOTE: Finding the average rate of change of a function over an interval is exactly the same as finding the slope of the line between two points. The two points, in this case, are given by the starting \(x\)-value and its corresponding \(y\)-value on the graph, and the ending \(x\)-value along with its \(y\)-value on the graph.