Average Rate of Change

The rate of change of a nonlinear function is not constant — it may even change from point to point. To measure the change for this type of function, an average rate of change is defined by averaging the total change of the function over a specific interval. The following graph illustrates how the average rate of change depends on the interval considered.

The average rate of change can be thought of as an approximation for the rate of change of the nonlinear function. This approximation is done using the slope of a linear function passing through the endpoints of a specific interval.

The slope of the linear function can be calculated by using the points common to both functions, (x_1,y_1) and (x_2,y_2). slope = y_2-y_1/x_2-x_1 The y-values that correspond to x_1 and x_2 are also the nonlinear function's values f(x_1) and f(x_2), respectively. Then, the slope formula can be rewritten and identified as the average rate of change of the function f(x) over the interval [x_1, x_2]. Average Rate of Change = f(x_2)-f(x_1)/x_2-x_1