Calculate and interpret the average rate of change of a function over a specified interval from an equation, table, or graph, and connect it to slope (Ohio F-IF.6).

What this topic is asking

Ohio standard F-IF.6 asks you to calculate and interpret the average rate of change of a function over an interval, from an equation, a table, or a graph. The average rate of change is the slope of the line connecting the two endpoints, change in output over change in input. It is a key Functions skill that ties slope to function language, and it appears on both parts.

The formula

The average rate of change is a difference quotient over the chosen interval.

This is exactly the slope formula applied to the two endpoints, which is why a constant rate of change is the signature of a line.

Computing from a table

A table gives the outputs directly; pick the two rows at the interval ends.

Computing from a graph

On a graph, the average rate of change is the slope of the secant line through the two endpoints: count the rise (vertical change) over the run (horizontal change) between $(a, f(a))$ and $(b, f(b))$. A steeper secant means a larger average rate.

Linear versus nonlinear

For a linear function, the average rate of change is the same over every interval, that constant is the slope $m$. For a nonlinear function (a parabola, an exponential), the average rate of change varies: comparing it over equal-width intervals reveals the curvature. A rate that grows over successive equal intervals is a hallmark of acceleration, like an upward parabola or exponential growth.

How Ohio examines this topic

• Numeric response. Compute the average rate of change over a given interval.
• Multiple choice and multiple-select. Compare the average rate over two intervals, or match a rate to its meaning.
• Tables and graphs. Read endpoints from a table or secant from a graph and compute.

Why it equals slope

The average rate of change formula, $\frac{f(b) - f(a)}{b - a}$, is the slope formula with the endpoints written in function notation: $f(a)$ and $f(b)$ are just the $y$-coordinates at $x = a$ and $x = b$. So the average rate of change is the slope of the straight line (the secant) connecting those two points on the graph, regardless of what the curve does between them. This is why a line, whose secant always lies on the line itself, has one constant rate, while a curve has different secant slopes for different intervals. Seeing the rate as a slope lets you compute it the same way every time and interpret it as "how fast the output changes per unit of input."

Why a changing rate signals a nonlinear function

If you compute the average rate of change over several equal-width intervals and the values differ, the function cannot be linear, because a line has a single constant slope everywhere. A table whose outputs increase by a constant amount for equal input steps is linear; one whose increases grow (or shrink) is nonlinear. This is a quick diagnostic the test rewards: equal first differences mean linear, growing first differences suggest a quadratic, and a constant ratio between outputs points to exponential. Average rate of change is therefore not just a number but a tool for classifying how a function behaves.

Try this

Q1. $g(0) = 2$ and $g(4) = 14$. Find the average rate of change over $[0, 4]$. [2 points]

• Cue. $\frac{14 - 2}{4 - 0} = \frac{12}{4} = 3$.

Q2. A line has slope $5$. What is its average rate of change over any interval? [1 point]

• Cue. Always $5$, the slope is constant.