Rate of Change

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.

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.

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.

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.

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