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Average Rate of Change Calculator

Find the average rate of change step by step — the secant slope (f(b)−f(a))/(b−a) between two points, with the full working and worked examples.

Quick answer
How do you find the average rate of change?
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.
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Comma or dot as decimal separator, negative values allowed.
Step-by-step
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HowTo

Average rate of change — 3 steps

Using (1, 3) and (4, 15)
  1. 1
    Step 1 of 3

    Note the two points

    Write the x-values a, b and the function values f(a), f(b), e.g. a=1, f(a)=3, b=4, f(b)=15.

  2. 2
    Step 2 of 3

    Form the differences

    Δy = f(b)−f(a) = 15−3 = 12 and Δx = b−a = 4−1 = 3.

  3. 3
    Step 3 of 3

    Divide Δy by Δx

    Average rate of change = Δy/Δx = 12/3 = 4.

Examples

Average rate of change — worked examples

Typical point pairs with the working
(1, 3) → (4, 15)
(15−3)/(4−1)
12/3
4
(0, 2) → (5, 2)
(2−2)/(5−0)
0/5
0
(2, 10) → (6, 2)
(2−10)/(6−2)
−8/4
−2
(−1, 4) → (3, 4)
(4−4)/(3−(−1))
0/4
0
(1, 1) → (4, 16)
(16−1)/(4−1)
15/3
5
Theory

Secant slope and rate of change

The average rate of change of a function over an interval [a, b] describes how much the function value changes on average per unit of x. Geometrically it is the slope of the secant — the line through the two points (a, f(a)) and (b, f(b)) on the graph: m = (f(b)−f(a))/(b−a). The numerator is the change in the y-direction (Δy), the denominator the change in the x-direction (Δx). A positive result means the function rises on average, a negative one means it falls; zero means the start and end values are equal. Important: a and b must differ, otherwise you divide by zero. As b approaches a, the average rate of change becomes the instantaneous rate of change — the derivative. That makes it the gateway into differential calculus and a standard topic in upper secondary (grade 10–11).

Pitfalls

Common mistakes

Swapping numerator and denominator

It is Δy/Δx, not Δx/Δy. The y-difference on top, divided by the x-difference.

Dropping the sign

If the value falls, the rate is negative. (2−10)/(6−2) = −2, not 2.

Inconsistent subtraction order

Keep the order consistent: f(b)−f(a) on top, b−a below — both from the same endpoint.

Using a = b

If the two x-values are equal, Δx = 0 and the rate is undefined (division by zero).
FAQ

Frequently asked questions

Glossary

Glossary — key terms explained simply

Average rate of change
Average change per unit of x over an interval.
Secant
A line through two points of a function’s graph.
Δy (delta y)
Change in the y-values: f(b)−f(a).
Δx (delta x)
Change in the x-values: b−a.
Slope
The ratio Δy/Δx, the measure of steepness.
Derivative
Limit of the average rate as b approaches a.