How to find the average rate of change — step by step
The average rate of change tells you how much a function value changes on average per unit of x. It is the slope of the secant through two points and follows the formula (f(b)−f(a))/(b−a). Worked example: from (1, 3) to (4, 15) gives 4. This topic is the gateway into calculus, grade 10–11.
Quick answer
The average rate of change is the slope of the secant between two points: (f(b)−f(a))/(b−a). You divide the change in y by the change in x. Example: from (1, 3) to (4, 15) → (15−3)/(4−1) = 12/3 = 4.
At a glance
| Example | (1, 3) → (4, 15) |
|---|---|
| Method | Secant slope Δy/Δx |
| Steps | 3 |
| Result | 4 |
| Meaning | function rises 4 per x on average |
| Grade level | Grade 10–11 |
Worked example: (1, 3) → (4, 15)
We form the change in y and the change in x, then divide the first by the second.
The steps to find the average rate of change
These steps work for any point pair (a, f(a)) and (b, f(b)) with a ≠ b.
Step 1 · Start
(1, 3) → (4, 15)Write the two points with a=1, f(a)=3, b=4, f(b)=15.Step 2 · Formula
( f(b) − f(a) ) ÷ ( b − a )The average rate of change is the secant slope Δy/Δx.Step 3 · Substitute
(15 − 3) ÷ (4 − 1)Plug the four values into the formula.Step 4 · Simplify
12 ÷ 3Compute Δy = 12 and Δx = 3.Step 5 · Result
= 4The average rate of change is 4.
Why the secant slope is the rate of change
The secant is the straight line through the two points (a, f(a)) and (b, f(b)). Its slope is “rise over run”, that is Δy/Δx — exactly the average change per unit of x across the interval. A positive result means the function rises on average; a negative one means it falls. As b moves closer and closer to a, the secant turns into the tangent and the average rate becomes the instantaneous rate of change — the derivative.