About the Average Rate of Change Calculator
This average rate of change calculator measures how much a function changes, on average, per unit of x between two points. Type a function such as x^2 + 3x, 50*1.04^x or sin(x), choose the start and end of the interval, and it returns f(a), f(b), the change in output, the change in input and their ratio — the slope of the secant line joining the two points on the graph.
It is useful for algebra and pre-calculus homework, for checking the difference quotient before taking a limit in calculus, and for real-world questions like average speed over a trip, average growth per year of an investment or population, or average cost per extra unit produced.
The graph draws your function together with the secant line so you can see the rate visually. Trigonometric functions use radians, ln(x) is the natural logarithm and log10(x) the common logarithm. If you only have measurements rather than a formula, switch to two-point mode and type the two (x, y) readings directly.
With the default inputs, the average rate of change is 8. Change any value above to recalculate instantly.
How to use the average rate of change calculator
- 1Choose whether to enter a function or two data points.
- 2Type the function using x as the variable, e.g. x^2 + 3x.
- 3Enter the start a and end b of the interval.
- 4Read the average rate of change, Δy and Δx.
- 5Use the secant line equation and graph to see the rate as a slope.
Formula and method
The average rate of change of f on [a, b] is the change in output divided by the change in input. Geometrically it is the slope of the secant line through (a, f(a)) and (b, f(b)); the secant line is y = f(a) + m(x − a), where m is the rate.
Writing b = a + h gives the difference quotient [f(a + h) − f(a)] ÷ h. As h shrinks toward 0 the average rate approaches the instantaneous rate of change, which is the derivative f′(a). For a linear function the average rate is the same on every interval and equals the slope.
- a, b
- Start and end of the interval
- f(a), f(b)
- Function values at the endpoints
- Δy
- Change in output, f(b) − f(a)
- Δx
- Change in input, b − a
Worked examples
f(x) = x² + 3x on [1, 4]
f(1) = 1 + 3 = 4 and f(4) = 16 + 12 = 28, so Δy = 24 over Δx = 3. The average rate of change is 24 ÷ 3 = 8, and the secant line through (1, 4) with slope 8 is y = 8x − 4.
Investment growing 5% a year, years 0 to 10
$1,000 growing at 5% reaches 1000 × 1.05¹⁰ ≈ $1,628.89 after 10 years. The average rate of change is 628.89 ÷ 10 ≈ $62.89 per year, even though growth is slower early and faster later.
Average speed from two readings
A car has travelled 120 miles after 2 hours and 315 miles after 5 hours. Over those 3 hours it covered 195 miles, an average speed of 195 ÷ 3 = 65 mph.
sin(x) from 0 to π/2
sin(0) = 0 and sin(π/2) = 1, so the average rate is 1 ÷ (π/2) = 2/π ≈ 0.6366 — less than the slope of 1 at the origin because the curve flattens near π/2.
Frequently asked questions
What is the average rate of change?+
It is how much a quantity changes per unit of another quantity over an interval: [f(b) − f(a)] ÷ (b − a). It is the slope of the straight line (secant) joining the two endpoints of the graph.
How is average rate of change different from the derivative?+
The average rate covers a whole interval, while the derivative is the instantaneous rate at a single point. The derivative is the limit of the average rate of change as the interval width b − a shrinks to zero.
What is the difference quotient?+
The difference quotient [f(x + h) − f(x)] ÷ h is the average rate of change over an interval of width h starting at x. Simplifying it and letting h → 0 is how derivatives are defined from first principles.
Can the average rate of change be negative?+
Yes. A negative value means the function decreased overall between a and b, for example a car slowing down or a balance being paid off. A value of zero means f(a) = f(b), even if f moved in between.
What units does the average rate of change have?+
Output units per input unit — miles per hour if f is distance in miles and x is time in hours, dollars per year for a balance over time, or degrees per minute for temperature readings.