Calculate and interpret the average rate of change of a function over a specified interval from a graph or table (TN A1.F.IF.C.6).

What this topic is asking

Standard A1.F.IF.C.6 asks you to compute and interpret the average rate of change of a function over an interval $[a, b]$. It is the change in output divided by the change in input, the same idea as slope, applied between two points on any function (linear or not). For a linear function it is constant; for a curve it depends on the interval.

The formula and how to use it

Compute the two outputs, then divide their difference by the difference of inputs.

From tables and graphs

From a table, pick the two rows at the interval's endpoints and apply the formula. From a graph, read the two points and compute the slope of the segment between them. The average rate of change is exactly that segment's slope, which is why "rate of change" and "slope" are interchangeable for lines.

How TNReady examines this topic

• Numeric response. Compute the average rate of change from a function, table, or graph.
• Multiple choice. Choose the rate or its interpretation (average speed, cost per item), with "forgot to divide" distractors.
• Inline choice. Describe whether a function increases faster on one interval than another.

A clarifying idea is that for a linear function the average rate of change is the same on every interval (it is the constant slope), while for a quadratic it differs interval to interval, faster where the curve is steeper. Comparing rates across intervals is how the test probes the difference between linear and nonlinear growth.

Why order and units matter

Two small things decide whether the answer is right. First, consistent order: the numerator must be $f(b) - f(a)$ and the denominator $b - a$, both starting from the same endpoint. Reversing one but not the other flips the sign, turning an increase into a decrease. A safe habit is to write the later point first in both differences. Second, units carry the meaning: distance over time is a speed, dollars over items is a unit price, population over years is a growth rate. The units come directly from the axes, so a rate of change on a distance-time graph is automatically a speed. Stating the units is often part of the interpretive credit, and it doubles as a check, if the units come out wrong, the setup is wrong. This is the same reasoning used in units and quantities.

Comparing rates across intervals

A favorite TNReady item gives a curve and asks where it changes fastest, which is just comparing average rates of change on different intervals. For $f(x) = x^2$, the rate over $[0, 1]$ is $\frac{1 - 0}{1 - 0} = 1$, but over $[3, 4]$ it is $\frac{16 - 9}{4 - 3} = 7$. The same function is increasing much faster on the later interval, which is the visual fact that a parabola gets steeper as you move away from the vertex. This is impossible for a line, whose rate is identical everywhere, and it is exactly how the test distinguishes linear from nonlinear behavior using rate-of-change language.

Try this

Q1. For $f(x) = 3x + 2$, find the average rate of change from $x = 1$ to $x = 5$. [1 point]

• Cue. $\frac{f(5) - f(1)}{5 - 1} = \frac{17 - 5}{4} = 3$ (the slope).

Q2. A table: at $x = 0$, $y = 4$; at $x = 2$, $y = 16$. Find the average rate of change. [1 point]

• Cue. $\frac{16 - 4}{2 - 0} = 6$.