Quick answer
Average rate of change = (f(b) − f(a)) / (b − a). It is the slope of the secant line between (a, f(a)) and (b, f(b)).
Formula
- (f(b) − f(a)) / (b − a)
- Change in output ÷ change in input
Introduction
Students often meet this topic in algebra and again in calculus. The vocabulary changes, but the core idea stays stable: compare two states, measure how much the output changed, divide by how much the input changed. When you are ready to compute and visualize, use the Average Rate of Change Calculator on our homepage guide.
A common mistake is treating the phrase like a buzzword. It is a measurable summary of behavior across an interval. If you already know slope as rise over run, you are closer than you think; our average rate of change vs slope guide explains when the two names describe the same arithmetic.
Before numbers enter a worksheet, define the interval in words. Are you measuring revenue from Q1 to Q4, position from 0 s to 4 s, or a function value from x = 1 to x = 5? Naming the interval prevents mixing unrelated rows and keeps your subtraction aligned with the question.
This article stays at the concept level. After you can explain the definition in your own words, move to calculation steps and worked examples so the fraction becomes automatic.
Definition, meaning, and slope interpretation
Let f be a rule or data series that assigns an output to each input. The average rate of change from a to b is (f(b) − f(a)) divided by (b − a), provided b ≠ a.
Meaning: the result tells you the average change in f for each 1-unit increase in the input across that span. If the rate is negative, the output tends to decrease as the input increases over the interval.
Graphically, plot the two points (a, f(a)) and (b, f(b)). Draw the secant segment connecting them. The steepness of that segment equals the average rate of change.
Real applications include average velocity in physics labs, average revenue growth per quarter in business summaries, and secant slopes on polynomial graphs in precalculus homework. Tables with many rows still use endpoint values when the question asks for an interval rate.
Formula connection
- (f(b) − f(a)) / (b − a)
- Equivalent table form: (f(x_n) − f(x_1)) / (x_n − x_1)
The fraction is called a difference quotient in calculus courses. The numerator measures total change in output; the denominator measures total change in input.
Keep subtraction order consistent. If you define Δf = f(b) − f(a), then Δx must be b − a, not the reverse.
- State the interval in context. Write sentences like "from March to September" or "from x = 2 to x = 8" so readers know which endpoints you chose.
- Read f(a) and f(b). Evaluate the function rule or pull the correct table rows. Label units.
- Compute Δf and Δx. Subtract in the same order you will use in the fraction.
- Divide and interpret. Attach units such as meters per second or dollars per month.
- Optional graph check. Sketch points and confirm the secant slope matches your division.
Worked example
A delivery truck records distance 0 km at time 0 h and 60 km at time 2 h. Treat distance as f(t) with t as time.
Δf = 60 − 0 = 60 km. Δt = 2 − 0 = 2 h. Average rate = 60/2 = 30 km/h.
That value is an average over the whole two-hour window, not proof that the truck moved at 30 km/h every minute. For step-by-step calculation methods, see how to calculate average rate of change.