# What is the average rate of change?

Finds the average rate of change of a function f(x) on an interval [a, b], (f(b) − f(a)) ÷ (b − a), with f(a), f(b), the secant line and a graph, exactly in fractions when it can.

- Page: https://www.acalculator.org/math/average-rate-of-change-calculator
- JSON spec: https://www.acalculator.org/math/average-rate-of-change-calculator.json
- Version: ccd6d6f0a1c2

## Default answer

Example with the default inputs (Function f(x) x^2 - 1/x, From x = a 2, To x = b 4): The average rate of change of f(x) = x^2 - 1/x from x = 2 to x = 4 is 6.125.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| f | Function f(x) | The function, typed like x^2 - 1/x, 2/x^2 or sqrt(x). |
| a | From x = a | The start of the interval. |
| b | To x = b | The end of the interval. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| rate | Average rate of change | (f(b) − f(a)) ÷ (b − a), the slope of the secant line. |
| fraction | As a fraction | The rate as an exact fraction, when f uses only numbers, x, + − × ÷ and whole powers. |
| fa | f(a) | The value of f at the start of the interval. |
| fb | f(b) | The value of f at the end of the interval. |
| dy | Change in f (Δy) | The change in f: f(b) − f(a). |
| dx | Change in x (Δx) | The width of the interval: b − a. |
| secant | Secant line | The line through (a, f(a)) and (b, f(b)). |

## Method

Average rate of change = (f(b) − f(a)) ÷ (b − a), the slope of the secant line through (a, f(a)) and (b, f(b)).

## Assumptions

- The variable is x; angles are in radians; ln and log are the natural logarithm, log10 is base 10.
- A formula of numbers, x, + − × ÷ and whole-number powers is worked out exactly in fractions from the typed decimals; any other formula in double precision.
- f must have a real value at both a and b; a and b must differ. The order of a and b does not change the answer.

## Worked examples

1. f = x^2 - 1/x, a = 2, b = 4 gives rate = 6.125, fraction = 49/8, fa = 3.5, fb = 15.75, dy = 12.25, dx = 2. Source: OpenStax, Algebra and Trigonometry 2e, §3.3 Rates of Change and Behavior of Graphs. https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-3-rates-of-change-and-behavior-of-graphs, retrieved 2026-10-02, Example 4: f(x) = x² − 1/x on [2, 4] has an average rate of change of 49/8.
2. f = 2/x^2, a = 2, b = 6 gives rate = -0.111111, fraction = −1/9, fa = 0.5, fb = 0.055556. Source: OpenStax, Algebra and Trigonometry 2e, §3.3 Rates of Change and Behavior of Graphs. https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-3-rates-of-change-and-behavior-of-graphs, retrieved 2026-10-02, Example 5: F(d) = 2/d² on [2, 6] changes at −1/9 per unit on average.
3. f = x^2 + 3x + 1, a = 0, b = 2 gives rate = 5, fraction = 5, fa = 1, fb = 11. Source: OpenStax, Algebra and Trigonometry 2e, §3.3 Rates of Change and Behavior of Graphs. https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-3-rates-of-change-and-behavior-of-graphs, retrieved 2026-10-02, Example 6: g(t) = t² + 3t + 1 on [0, a] has rate a + 3, so 5 for a = 2.

## FAQ

### What is the average rate of change?

How much a function's output changes per unit of input over an interval: (f(b) − f(a)) ÷ (b − a). It is the slope of the straight line, called the secant line, through the two points (a, f(a)) and (b, f(b)).

### How do I find the average rate of change of f(x) = x² − 1/x on [2, 4]?

f(2) = 4 − 1/2 = 7/2 and f(4) = 16 − 1/4 = 63/4. The change is 63/4 − 7/2 = 49/4, and Δx = 4 − 2 = 2, so the average rate of change is 49/4 ÷ 2 = 49/8 = 6.125.

### Is the average rate of change the same as the slope?

For a straight line, yes: the rate is the same on every interval. For a curve it depends on the interval, and it equals the slope of the secant line through the two end points, not of the curve itself.

### What is the difference from the instantaneous rate of change?

The instantaneous rate of change is the rate at one point, the derivative f′(a). It is the limit of the average rate of change on [a, a + h] as h shrinks to 0.

### What does a negative average rate of change mean?

The function is lower at b than at a, so on balance it falls over the interval. F(x) = 2/x² falls from 1/2 at x = 2 to 1/18 at x = 6, an average rate of −1/9 per unit.

### Can I use a table of values instead of a formula?

Yes. Use the rate of change calculator, which takes points (x, y) and also shows the rate on each interval between them.

## Sources

- OpenStax, Algebra and Trigonometry 2e, §3.3 Rates of Change and Behavior of Graphs (average rate of change = Δy/Δx = (f(x₂) − f(x₁))/(x₂ − x₁); Example 4: f(x) = x² − 1/x on [2, 4] gives 49/8; Example 5: F(d) = 2/d² on [2, 6] gives −1/9; Example 6: g(t) = t² + 3t + 1 on [0, a] gives a + 3). https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-3-rates-of-change-and-behavior-of-graphs (retrieved 2026-10-02)