# What is the rate of change?

Computes the average rate of change, Δy ÷ Δx, between two points or across a table of values, with the rate on each interval, whether the rate is constant, and a chart.

- Page: https://www.acalculator.org/math/rate-of-change-calculator
- JSON spec: https://www.acalculator.org/math/rate-of-change-calculator.json
- Version: 9805d832c984

## Default answer

Example with the default inputs (Points (x, y) [x 2,007, y 2.84; x 2,009, y 2.41]): The average rate of change is -0.215 (-0.43 ÷ 2).

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| points | Points (x, y) | The input x and the output y of each point, in order: two points, or a table of values. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| rate | Average rate of change | (last y − first y) ÷ (last x − first x). |
| fraction | As a fraction | The average rate of change as an exact fraction in lowest terms. |
| dy | Change in y (Δy) | Last y − first y. |
| dx | Change in x (Δx) | Last x − first x. |
| percent | Percent change in y | Δy ÷ \|first y\| × 100, when the first y is not 0. |
| constant | Is the rate constant? | Whether every interval has the same rate, so the points lie on one straight line. |
| intervals | Rate on each interval | The average rate of change between each pair of neighbouring points, as a fraction. |

## Method

Average rate of change = (y_last − y_first) ÷ (x_last − x_first); on each interval, (yₙ − yₙ₋₁) ÷ (xₙ − xₙ₋₁).

## Assumptions

- The points are taken in the order typed; the overall rate runs from the first point to the last.
- Each value is read as the exact decimal typed, so the rates are exact fractions.

## Worked examples

1. points = {"x":2007,"y":2.84} or {"x":2009,"y":2.41} gives rate = -0.215, fraction = -43/200, percent = -15.140845%. Source: OpenStax Algebra and Trigonometry 2e, section 3.3, Rates of Change and Behavior of Graphs, Example 1: ($2.41 − $2.84) ÷ (2009 − 2007) ≈ −$0.22 per year (https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-3-rates-of-change-and-behavior-of-graphs, retrieved 2026-10-01).
2. points = {"x":0,"y":0} or {"x":1,"y":1} gives rate = 3, constant = No: the rate changes from one interval to the next., intervals = (0, 0) to (1, 1): 1; (1, 1) to (2, 4): 3; (2, 4) to (3, 9): 5.
3. points = {"x":1,"y":5} or {"x":3,"y":11} gives rate = 3, dy = 9, dx = 3, percent = 180%, constant = Yes: every interval has the same rate, so the points lie on one line..

## FAQ

### How do I find the average rate of change?

Divide the change in the output by the change in the input: (y₂ − y₁) ÷ (x₂ − x₁). Gas cost $2.84 a gallon in 2007 and $2.41 in 2009, so the rate is (2.41 − 2.84) ÷ (2009 − 2007) = −0.215, a fall of about 22 cents a year.

### How do I find the rate of change from a table?

Take the first and last rows for the average over the whole table. For each pair of neighbouring rows, divide the change in y by the change in x to get the rate on that interval. The calculator shows both.

### What does a constant rate of change mean?

Every interval has the same rate, so the points lie on one straight line: the function is linear, and the rate is its slope. For (1, 5), (3, 11), (4, 14) every interval rises 3 per unit of x.

### Can the rate of change be negative?

Yes. A negative rate means the output falls as the input grows, as with the gas prices from 2007 to 2009.

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

Between two points, yes: it is the slope of the line through them (the secant line). For a curve, the rate changes from interval to interval; the rate at a single point is the instantaneous rate of change, the derivative.

### Why is there no answer when two x values are equal?

The change in x would be 0, and you cannot divide by 0. Two points with the same x lie on a vertical line, which has no rate of change.

## Sources

- OpenStax, Algebra and Trigonometry 2e, section 3.3, Rates of Change and Behavior of Graphs (average rate of change = Δy ÷ Δx = (y₂ − y₁) ÷ (x₂ − x₁); Example 1, gasoline prices 2007 to 2009), CC BY 4.0, retrieved 2026-10-01. https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-3-rates-of-change-and-behavior-of-graphs