Calculate and interpret the average rate of change of a function over a specified interval (LA A1: F-IF.B.6).

What this topic is asking

Standard A1: F-IF.B.6 asks you to compute and interpret the average rate of change of a function over an interval, from a graph, a table, or a formula. On LEAP 2025 these are Type I and Type II items. For a linear function the average rate of change is just the slope; for a nonlinear function it depends on the interval you pick.

The formula

This is the slope between the two endpoints. The numerator is the change in output; the denominator is the change in input.

From a table

With a table, pick the two rows at the interval's endpoints and divide the change in output by the change in input, exactly the slope between those rows.

Linear versus nonlinear

For a line, the average rate of change is the same over every interval, that constant is the slope, and it is why lines model constant-rate situations. For a curve, the average rate of change varies: a parabola is steeper far from its vertex and flatter near it, so the rate over $[0, 1]$ differs from the rate over $[3, 4]$. This is why you must name the interval.

How LEAP examines this topic

• Equation response. Compute the average rate of change over a stated interval.
• Type II reasoning. Interpret the rate in context, with units, and explain "on average."
• Multiple choice. Compare rates over different intervals, or read the rate from a table.

A clarifying idea: average rate of change is the same computation as slope ($\frac{\text{rise}}{\text{run}}$), just applied to two points on a function's graph, which is why this topic links directly to slope of a line.

Why average rate of change generalizes slope

Average rate of change extends the idea of slope from straight lines to every function, which is the conceptual reach of F-IF.B.6. A line has a single slope because it climbs at one steady rate, so $\frac{\text{rise}}{\text{run}}$ gives the same value no matter which two points you choose. A curve has no single slope, it bends, but you can still ask "between these two points, how fast did the output change on average?" The answer is the slope of the secant line joining those points, and that is precisely $\frac{f(b) - f(a)}{b - a}$. This is powerful because it lets you summarize change for relationships that are not constant: how fast a population grew over a decade, how fast a ball's height changed during the first second. The dependence on the interval is the honest part: because the curve's steepness varies, the average over a long interval can hide faster and slower stretches inside it, which is exactly why the standard requires you to specify the interval and to say "on average." This idea is the Algebra I doorway to instantaneous rate of change in calculus, where the interval shrinks to a single point.

Try this

Q1. For $f(x) = x^2$, find the average rate of change from $x = 2$ to $x = 5$. [2 points]

• Cue. $\frac{25 - 4}{5 - 2} = \frac{21}{3} = 7$.

Q2. A car's distance is $0$ mi at hour $0$ and $180$ mi at hour $3$. What is the average rate of change? [2 points]

• Cue. $\frac{180 - 0}{3 - 0} = 60$ miles per hour.