Calculate and interpret the average rate of change of a function over an interval from a graph or table (NC.M1.F-IF.6).

What this topic is asking

NC.M1.F-IF.6 asks you to calculate and interpret the average rate of change of a function over a specified interval, estimating it from a graph or table, for linear, quadratic, and exponential functions. The average rate of change is the slope of the line connecting the two endpoints of the interval.

The formula

The average rate of change generalizes slope to any function.

This is exactly the slope formula applied to the two endpoint points.

Computing from a table

When values come in a table, pick the two endpoints of the interval.

Why it changes for non-linear functions

For a line, the rate is the same everywhere. For a quadratic like $f(x) = x^2$, the average rate of change from $1$ to $2$ is $\frac{4 - 1}{1} = 3$, but from $2$ to $3$ it is $\frac{9 - 4}{1} = 5$, larger, because the parabola steepens. For an exponential, the rate grows even faster. This is why F-IF.6 specifies an interval: the answer depends on which interval you choose.

Interpreting in context with units

The average rate of change always carries units of output per input. If $f(t)$ is distance in miles and $t$ is hours, the rate is miles per hour (a speed). If $f(t)$ is a balance in dollars and $t$ is years, it is dollars per year. Stating the units is usually required for full points.

How the NC Math 1 EOC examines this topic

• Gridded response. Compute the average rate of change over a stated interval.
• Multiple choice. Choose the correct rate, or interpret it in context.
• Calculator-active. Often paired with a table or graph in the calculator-active section.

This idea connects slope (constant rate for lines) to comparing function families, where linear, quadratic, and exponential functions differ precisely in how their rate of change behaves.

Why average rate of change unifies slope and growth

For lines, "rate of change" and "slope" are the same single number, but real situations often curve. Average rate of change extends the slope idea to any function by asking: over this stretch, how much did the output change per unit of input on average? That single question describes a car's average speed, a population's average growth, or a balance's average gain, regardless of the shape of the underlying graph. It also reveals the defining difference between function families: a constant average rate signals linear, a rising average rate signals quadratic or exponential. So computing it is not just arithmetic; it is a diagnostic for what kind of function you are looking at, which is exactly why F-IF.6 sits at the center of the Functions strand.

Try this

Q1. For $f(x) = 2x + 1$, find the average rate of change from $x = 1$ to $x = 5$. [1 point]

• Cue. $\frac{f(5) - f(1)}{5 - 1} = \frac{11 - 3}{4} = 2$ (the slope, constant for a line).

Q2. For $g(x) = x^2$, find the average rate of change from $x = 2$ to $x = 5$. [2 points]

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