Average Rate of Change Calculator
Enter two points, or a function’s value at two inputs, and get the average rate of change: how much y changes per unit of x between them.
- Runs in your browser
- Shows the formula filled in
- Slope of the secant line
- Free
Two points
Show the working
How to find the average rate of change
Change in y
Subtract the first y from the second: 12 − 3 = 9. If you have a function, evaluate it at both x values first.
Change in x
Subtract the first x from the second: 4 − 1 = 3.
Divide
Δy ÷ Δx = 9 ÷ 3 = 3. On average, y rises by 3 for every 1 that x rises between those points. It is the slope of the line through them.
What it means
Positive
y goes up as x goes up. Distance vs time with a positive rate is average speed. Average calculator →
Negative
y goes down as x goes up, like a cooling cup of coffee or a draining tank.
Zero
y ends where it started. The function may have moved in between, but on average it did not change.
Questions people ask
What is the formula for average rate of change?
Average rate of change = (y₂ − y₁) ÷ (x₂ − x₁). For a function f over the interval [a, b] it is (f(b) − f(a)) ÷ (b − a).
Is average rate of change the same as slope?
Between two points, yes. It is the slope of the straight line (the secant line) that joins them. For a curve, it is the average slope over that interval, not the slope at a single point.
How do I find the average rate of change of a function?
Plug the two x values into the function to get f(x₁) and f(x₂), then use (f(x₂) − f(x₁)) ÷ (x₂ − x₁). For f(x) = x² between 1 and 4: (16 − 1) ÷ (4 − 1) = 5.
Why does the calculator say undefined?
When x₁ equals x₂, the change in x is zero and you cannot divide by zero. Two points with the same x value lie on a vertical line, which has no defined slope.
What is the difference between average and instantaneous rate of change?
Average rate of change is measured over an interval, between two points. Instantaneous rate of change is at a single point and is what the derivative gives you in calculus.