Type a function and an interval [a, b], two points, or a table of values to find the average rate of change. You also get f(a), f(b), the secant line equation, the percent change and a graph showing the secant line on the curve.
Last updated:
Input type
Use x, + − * / ^, parentheses, sqrt, sin, cos, tan, ln, log, exp, abs, pi, e. Example: 2x^3 - 5x or sqrt(x+1). Trig uses radians.
Leave blank to use the first row.
Leave blank to use the last row.
Enter your values to see the result.
Advertisement
Advertisement
How to use the Average Rate of Change Calculator
Choose the input type: a function, two points or a data table.
Enter the function and the interval, for example x^2 + 3x − 4 from a = 1 to b = 4.
Or type the points or table rows as x, y pairs. Add units if you want a rate like miles per hour.
Press Calculate rate to get the average rate of change, f(a), f(b) and the secant line.
Check the graph and steps to see how the slope between A and B was found.
The average rate of change formula
For a function f on the interval [a, b]:
Average rate of change = (f(b) − f(a)) ÷ (b − a) = Δy ÷ Δx
Geometrically, it is the slope of the secant line through A(a, f(a)) and B(b, f(b)). The secant line equation is y = m(x − a) + f(a), where m is the average rate of change. With units, the rate reads as "units of y per unit of x", such as miles per hour, dollars per year or degrees per minute.
Examples
Function or data
Interval
f(a), f(b)
Average rate of change
f(x) = x² + 3x − 4
[1, 4]
0, 24
24 ÷ 3 = 8
f(x) = x³
[−1, 2]
−1, 8
9 ÷ 3 = 3
f(x) = √x
[4, 9]
2, 3
1 ÷ 5 = 0.2
f(x) = 2ˣ
[0, 3]
1, 8
7 ÷ 3 ≈ 2.333
f(x) = 1/x
[1, 2]
1, 0.5
−0.5 ÷ 1 = −0.5
Car trip: 0 mi at 0 h, 120 mi at 2.5 h
[0, 2.5]
0, 120
48 miles per hour
Tips
Keep the order consistent: if you subtract f(a) from f(b) on top, subtract a from b on the bottom. Swapping both gives the same answer.
The average rate only depends on the two endpoints. Use the subinterval table to see how the rate changes inside the interval.
The difference quotient [f(x + h) − f(x)] ÷ h is the same formula with a = x and b = x + h. Letting h shrink to 0 gives the derivative.
For trigonometric functions, angles are in radians, so sin(x) on [0, π/2] has an average rate of 1 ÷ 1.5708 ≈ 0.6366.
Frequently asked questions
What is the average rate of change formula?
Average rate of change = [f(b) − f(a)] ÷ (b − a). It is the change in the output divided by the change in the input over the interval [a, b], which is the slope of the straight line joining the points (a, f(a)) and (b, f(b)).
How do you find the average rate of change of a function?
Evaluate the function at both endpoints, subtract to get the change in y, subtract the x-values to get the change in x, and divide. For f(x) = x² + 3x − 4 on [1, 4]: f(4) = 24 and f(1) = 0, so the rate is (24 − 0) ÷ (4 − 1) = 8.
Is average rate of change the same as slope?
Yes. It equals the slope of the secant line through the two endpoints. For a straight line the average rate of change is the same on every interval; for a curve it changes depending on the interval you choose.
What is the difference between average and instantaneous rate of change?
The average rate uses two points and describes the overall change across an interval. The instantaneous rate is the derivative at a single point, the slope of the tangent line. As the interval shrinks toward one point, the average rate approaches the instantaneous rate.
How do I find the average rate of change from a table?
Pick the two rows for the start and end of the interval, then divide the difference in y-values by the difference in x-values. In table mode the calculator does this for your chosen rows and for every pair of consecutive rows.
What does a negative average rate of change mean?
It means the output decreased overall from a to b. For example, a temperature falling from 20° to 8° over 4 hours has an average rate of change of −3 degrees per hour.
Can the average rate of change be zero when the function changes?
Yes. If f(a) and f(b) are equal, the average rate is 0 even if the function goes up and down in between. For f(x) = x² on [−2, 2], both ends equal 4, so the average rate of change is 0.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.