What is the rate of change?
Type two points, or a whole table of x and y values. The rate of change calculator divides the change in y by the change in x, shows the rate on every interval, and tells you whether the rate is constant.
- Average rate of change
- -0.215
The average rate of change is -0.215 (-0.43 ÷ 2).
- As a fraction
- -43/200
- Change in y (Δy)
- -0.43
- Change in x (Δx)
- 2
- Percent change in y
- -15.14%
- Is the rate constant?
- Two points: one interval.
- Rate on each interval
- (2007, 2.84) to (2009, 2.41): -43/200
Average rate of change: -0.215. The average rate of change is -0.215 (-0.43 ÷ 2).
How does y change with x?
What is the rate on each interval?
How to calculate
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.
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).
Method: Average rate of change = (y_last − y_first) ÷ (x_last − x_first); on each interval, (yₙ − yₙ₋₁) ÷ (xₙ − xₙ₋₁).
- 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
Each example is checked against the calculator on every build.
- Points (x, y) 2007 2.84; 2009 2.41 gives Average rate of change -0.215, As a fraction -43/200, Percent change in y -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)
- Points (x, y) 0 0; 1 1; 2 4; 3 9 gives Average rate of change 3, Is the rate constant? No: the rate changes from one interval to the next., Rate on each interval (0, 0) to (1, 1): 1; (1, 1) to (2, 4): 3; (2, 4) to (3, 9): 5.
- Points (x, y) 1 5; 3 11; 4 14 gives Average rate of change 3, Change in y (Δy) 9, Change in x (Δx) 3, Percent change in y 180%, Is the rate constant? Yes: every interval has the same rate, so the points lie on one line..
How it works
Type the points in order, each with its input x and output y; two to twenty rows. With (x₁, y₁) the first point and (xₙ, yₙ) the last:
- Average rate of change = (yₙ − y₁) ÷ (xₙ − x₁), shown to 6 significant figures (the exact value is in the fraction).
- As a fraction: the same rate as an exact fraction in lowest terms, written as a mixed number when its size is above 1 (7/2 shows as 3 1/2).
- Change in y = yₙ − y₁ and change in x = xₙ − x₁, to 12 significant figures.
- Percent change in y = (yₙ − y₁) ÷ |y₁| × 100, to 2 decimals; left out when y₁ = 0.
- Rate on each interval: (yₖ − yₖ₋₁) ÷ (xₖ − xₖ₋₁) between neighbouring rows, as a fraction, one step per interval, written "(x, y) to (x, y): rate" with each coordinate to at most 10 significant figures, rounded half up from the value typed.
- Is the rate constant? With two points: "Two points: one interval." With more: "Yes: every interval has the same rate, so the points lie on one line." when every interval rate equals the average rate exactly, otherwise "No: the rate changes from one interval to the next."
Each value is read as the exact decimal you typed, so the rates are exact fractions; decimals are rounded half up for display only. The chart draws straight lines between the points, sorted by x.
Rules
- Two neighbouring rows with the same x have no answer: "Points k and k + 1 have the same x, so the rate between them is undefined."
- A first and last row with the same x have no answer: "The first and last points have the same x, so the rate of change is undefined."
- The x values do not have to increase; the rows are used in the order typed.
Worked examples by hand
Gas prices. (2.41 − 2.84) ÷ (2009 − 2007) = −0.43 ÷ 2 = −0.215 = −43/200 dollars a year. Percent change = −0.43 ÷ 2.84 × 100 = −15.14%.
y = x² at x = 0, 1, 2, 3. The average rate is (9 − 0) ÷ (3 − 0) = 3. The intervals give (1 − 0) ÷ 1 = 1, (4 − 1) ÷ 1 = 3 and (9 − 4) ÷ 1 = 5, so the rate is not constant.
(1, 5), (3, 11), (4, 14). The average rate is (14 − 5) ÷ (4 − 1) = 9 ÷ 3 = 3. The intervals give 6 ÷ 2 = 3 and 3 ÷ 1 = 3, so the rate is constant. Percent change = 9 ÷ 5 × 100 = 180%.
Other questions people ask
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.