Calculate and interpret the average rate of change of a function over a specified interval from a graph, a table, or an equation (MA.912.F.1.4).

What this topic is asking

MA.912.F.1.4 asks for the average rate of change of a function across an interval, the same idea as slope, but applied to any function, not just a line. On the B.E.S.T. Algebra 1 EOC you compute it from a graph, a table, or an equation, and interpret it in context (speed, growth per year, dollars per item).

The formula

This is "rise over run" between the two endpoints. The numerator is the change in the function's value; the denominator is the change in the input. Both must be subtracted in the same order (top: $b$ first; bottom: $b$ first).

Reading it from a table or graph

From a table, pick the two rows at the interval's endpoints and divide the output difference by the input difference. From a graph, read the two endpoint coordinates and compute rise over run, the slope of the line connecting them (the secant line). You do not need the points in between.

How the B.E.S.T. EOC examines this topic

• Equation editor. Compute the average rate of change from a function on a given interval.
• Multiple choice. Find it from a table or graph, with "forgot to divide" distractors.
• Context items. Interpret the rate with correct units (speed, cost per unit, growth per year).

A clarifying idea: average rate of change is slope generalized. For a straight line the answer is the same on every interval, because the slope never changes. For a parabola or exponential it differs interval to interval, which is exactly what distinguishes nonlinear functions, and a frequent EOC comparison.

Why it is the slope of the secant line

The formula is identical to the slope formula, only the names change. Slope is $\frac{y_2 - y_1}{x_2 - x_1}$, and here $y_2 = f(b)$, $y_1 = f(a)$, with $x_2 = b$, $x_1 = a$. So computing the average rate of change is literally computing the slope of the straight line, the secant line, joining the two endpoints of the interval. This is why a linear function's average rate of change is constant: every secant line lies along the function itself, so they all share one slope. For a curve, different intervals give secant lines of different steepness, which is the picture behind "the rate is changing." Seeing the connection means you never need a separate formula; average rate of change is just slope between two chosen points.

Interpreting in context

The units come straight from the quantities: outputs per inputs. If $f(t)$ is distance in meters and $t$ is in minutes, the rate is meters per minute, an average speed. If $C(n)$ is cost in dollars and $n$ is items, the rate is dollars per item. Always attach units, because the EOC interpretation items award credit for the meaning, not just the number.

Try this

Q1. For $f(x) = 5x + 2$, find the average rate of change from $x = 1$ to $x = 6$. [1 point]

• Cue. It is linear, so the answer is the slope, $5$.

Q2. A balloon's height (m) is $20$ at $t = 1$ s and $44$ at $t = 4$ s. Find the average rate of change with units. [2 points]

• Cue. $\frac{44 - 20}{4 - 1} = \frac{24}{3} = 8$ meters per second.