The rate of change between two points is the change in y divided by the change in x, or rise over run. Learn the formula, see it worked out for two points, and understand how rate of change becomes the constant slope of a straight line.
What rate of change means
The rate of change between two points measures how much the output (y) changes for every unit the input (x) changes. It answers "how fast is this changing?" and is calculated as rise over run: the change in y divided by the change in x.
Calculating it from two points
Given two points (x₁, y₁) and (x₂, y₂), the rate of change is (y₂ − y₁) ÷ (x₂ − x₁). For the points (2, 3) and (8, 15): the rise is 15 − 3 = 12, and the run is 8 − 2 = 6, so the rate of change is 12 ÷ 6 = 2.
Rate of change = rise over run between two points on a line.
Rate of change and slope
For a straight line, the rate of change is the same between any two points on it — that constant value is what we call the slope. This is why you can compute it once from any two known points and use it to read linear relation graphs or set up equations for real-world linear applications.
Example: cost per unit
If a phone plan costs $30 for 100 minutes and $45 for 250 minutes, the rate of change is (45 − 30) ÷ (250 − 100) = 15 ÷ 150 = $0.10 per minute — the cost per additional minute.