Average Rate of Change Calculator
This average rate of change calculator works for a function ax² + bx + c on an interval or for two points. Example: f(x) = x² between x = 1 and x = 4 has an average rate of change of (16 − 1) / 3 = 5.
Source: Wikipedia – Differenzenquotient. Updated: .
How it is calculated
The average rate of change tells you how much a quantity changes per unit on average, and it equals the slope of the secant line through two points of the graph. It is also called the difference quotient and is the forerunner of the derivative.
Formula and steps
For two points (x₁, y₁) and (x₂, y₂), m = Δy / Δx = (y₂ − y₁) / (x₂ − x₁). For a function, first compute f(x₁) and f(x₂) and use them as the y-values. The calculator also gives the secant equation y = m·x + n. The order of the two points does not matter as long as x and y are swapped together.
Examples
For f(x) = x² on [1, 4], f(1) = 1 and f(4) = 16, so m = 15 / 3 = 5 and the secant is y = 5x − 4. The points (1, 3) and (4, 12) give m = 9 / 3 = 3 and y = 3x. An everyday case: a car covering 240 km in 3 hours has an average rate of change of distance of 80 km/h.
- A positive m means growth on average, a negative one a decline.
- The smaller the interval, the closer m comes to the instantaneous rate, that is, the derivative.
Frequently asked questions
What is the difference between average and instantaneous rate of change?
The average rate applies to a whole interval and is the secant slope. The instantaneous rate applies at a single point and is the tangent slope, that is, the derivative.
How do I find the average rate of change of a function?
Compute f(x₂) − f(x₁) and divide by x₂ − x₁. For f(x) = x² between 1 and 4 that gives (16 − 1) / 3 = 5.
Can the average rate of change be negative?
Yes, when the values fall over the interval. The secant then slopes downward and m is below 0.
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