Average Rate of Change

The average rate of change measures how much a function's output changes, on average, for each unit of change in the input over a specific interval. If you look at the graph of a function, the average rate of change between two points is exactly the slope of the straight line (called the secant line) connecting those two points.

The Formula

For a function $f(x)$ over the interval $[a, b]$, the average rate of change is calculated as the change in the $y$-values divided by the change in the $x$-values:

$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$

Notice that this is just the familiar slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ applied to the points $(a, f(a))$ and $(b, f(b))$.

Example: Calculating Average Rate of Change

Problem: Find the average rate of change of the function $f(x) = x^2$ from $x = 1$ to $x = 4$.

Solution:

1. Identify the interval endpoints: $a = 1$ and $b = 4$.
2. Find the function values at these endpoints:
   • $f(1) = 1^2 = 1$
   • $f(4) = 4^2 = 16$
3. Plug these into the formula:

$\frac{f(4) - f(1)}{4 - 1} = \frac{16 - 1}{4 - 1} = \frac{15}{3} = 5$

The average rate of change is $5$. This means that, on average, the function's value increases by $5$ units for every $1$ unit increase in $x$ between $x = 1$ and $x = 4$.

Increasing, Decreasing, and Constant Intervals

By looking at a function's graph from left to right, we can describe its behavior over different intervals based on its rate of change.

• Increasing: A function is increasing on an interval if the $y$-values go up as the $x$-values go up. The graph slopes upward. The average rate of change between any two points in this interval is positive.
• Decreasing: A function is decreasing if the $y$-values go down as the $x$-values go up. The graph slopes downward. The average rate of change here is negative.
• Constant: A function is constant if the $y$-values stay the exact same as the $x$-values change. The graph is a flat, horizontal line. The average rate of change is zero.

When identifying these intervals from a graph, always state the intervals using the $x$-values. For example, if a graph goes upwards from $x = -2$ to $x = 3$, we say the function is increasing on the interval $[-2, 3]$.