What is the average rate of change?
Type a function and the two ends of an interval. The calculator finds how fast the function changes on average between them, exactly as a fraction when it can.
- Average rate of change
- 6.125
The average rate of change of f(x) = x^2 - 1/x from x = 2 to x = 4 is 6.125.
- As a fraction
- 49/8
- f(a)
- 3.5
- f(b)
- 15.75
- Change in f (Δy)
- 12.25
- Change in x (Δx)
- 2
- Secant line
- y − 7/2 = (49/8)(x − 2)
Average rate of change: 6.125. The average rate of change of f(x) = x^2 - 1/x from x = 2 to x = 4 is 6.125.
The curve y = f(x) from a to b
How to calculate
Finds the average rate of change of a function f(x) on an interval [a, b], (f(b) − f(a)) ÷ (b − a), with f(a), f(b), the secant line and a graph, exactly in fractions when it can.
Example with the default inputs (Function f(x) x^2 - 1/x, From x = a 2, To x = b 4): The average rate of change of f(x) = x^2 - 1/x from x = 2 to x = 4 is 6.125.
Method: Average rate of change = (f(b) − f(a)) ÷ (b − a), the slope of the secant line through (a, f(a)) and (b, f(b)).
- The variable is x; angles are in radians; ln and log are the natural logarithm, log10 is base 10.
- A formula of numbers, x, + − × ÷ and whole-number powers is worked out exactly in fractions from the typed decimals; any other formula in double precision.
- f must have a real value at both a and b; a and b must differ. The order of a and b does not change the answer.
Worked examples
Each example is checked against the calculator on every build.
- Function f(x) x^2 - 1/x, From x = a 2, To x = b 4 gives Average rate of change 6.125, As a fraction 49/8, f(a) 3.5, f(b) 15.75, Change in f (Δy) 12.25, Change in x (Δx) 2.Source: OpenStax, Algebra and Trigonometry 2e, §3.3 Rates of Change and Behavior of Graphs. https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-3-rates-of-change-and-behavior-of-graphs, retrieved 2026-10-02, Example 4: f(x) = x² − 1/x on [2, 4] has an average rate of change of 49/8
- Function f(x) 2/x^2, From x = a 2, To x = b 6 gives Average rate of change -0.111111, As a fraction −1/9, f(a) 0.5, f(b) 0.055556.Source: OpenStax, Algebra and Trigonometry 2e, §3.3 Rates of Change and Behavior of Graphs. https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-3-rates-of-change-and-behavior-of-graphs, retrieved 2026-10-02, Example 5: F(d) = 2/d² on [2, 6] changes at −1/9 per unit on average
- Function f(x) x^2 + 3x + 1, From x = a 0, To x = b 2 gives Average rate of change 5, As a fraction 5, f(a) 1, f(b) 11.Source: OpenStax, Algebra and Trigonometry 2e, §3.3 Rates of Change and Behavior of Graphs. https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-3-rates-of-change-and-behavior-of-graphs, retrieved 2026-10-02, Example 6: g(t) = t² + 3t + 1 on [0, a] has rate a + 3, so 5 for a = 2
How it works
The average rate of change of f on the interval from x = a to x = b is
(f(b) − f(a)) ÷ (b − a) = Δy ÷ Δx
It is the slope of the secant line through (a, f(a)) and (b, f(b)). The page also shows f(a), f(b), Δy = f(b) − f(a), Δx = b − a, and the secant line in point-slope form, y − f(a) = rate × (x − a).
Exact or decimal. When f is built only from numbers, x, + − × ÷ and whole-number powers (such as x^2 − 1/x), every value is worked out exactly in fractions from the typed decimals (0.1 is 1/10), and the rate is also shown as a fraction in lowest terms. Any other formula (pi, e, sqrt, ln, sin, a fraction power) is worked out in double-precision arithmetic.
Rules
- Type f with x as the variable: + − * / ^, brackets, pi, e, sqrt, ln (and log, both the natural logarithm), log10, sin, cos, tan (radians).
- a and b must be different. Swapping them gives the same rate.
- f must have a real value at a and at b (2/x² at x = 0 or sqrt(x) at x = −1 gives no answer). Values over about 1.8 × 10³⁰⁸ give no answer.
- The fraction is shown when the exact value is a fraction of at most 40 characters.
Output format. Decimals are shown to 10 significant figures. In the secant line, a typed a is written as typed, an exact value as a fraction in lowest terms (7/2, −1/18) of at most 40 characters, and other values to 10 significant figures, all with the minus sign −. A fraction or negative slope is put in brackets: y − 7/2 = (49/8)(x − 2).
Worked examples by hand
f(x) = x² − 1/x on [2, 4]. f(2) = 4 − 1/2 = 7/2. f(4) = 16 − 1/4 = 63/4. Δy = 63/4 − 7/2 = 49/4. Δx = 2. Rate = 49/4 ÷ 2 = 49/8 = 6.125.
f(x) = 2/x² on [2, 6]. f(2) = 2/4 = 1/2. f(6) = 2/36 = 1/18. Δy = 1/18 − 9/18 = −8/18 = −4/9. Δx = 4. Rate = −4/9 ÷ 4 = −1/9.
f(x) = x² + 3x + 1 on [0, 2]. f(0) = 1, f(2) = 4 + 6 + 1 = 11. Rate = (11 − 1) ÷ 2 = 5, which is a + 3 with a = 2.
f(x) = √x on [1, 4]. (2 − 1) ÷ 3 = 1/3 ≈ 0.3333333333 (a root, so the page works it out in double precision).
f(x) = 0.1x² on [0.1, 0.3]. f(0.1) = 0.001, f(0.3) = 0.009. Rate = 0.008 ÷ 0.2 = 0.04 = 1/25 exactly.
Other questions people ask
What is the average rate of change?
How much a function's output changes per unit of input over an interval: (f(b) − f(a)) ÷ (b − a). It is the slope of the straight line, called the secant line, through the two points (a, f(a)) and (b, f(b)).
How do I find the average rate of change of f(x) = x² − 1/x on [2, 4]?
f(2) = 4 − 1/2 = 7/2 and f(4) = 16 − 1/4 = 63/4. The change is 63/4 − 7/2 = 49/4, and Δx = 4 − 2 = 2, so the average rate of change is 49/4 ÷ 2 = 49/8 = 6.125.
Is the average rate of change the same as the slope?
For a straight line, yes: the rate is the same on every interval. For a curve it depends on the interval, and it equals the slope of the secant line through the two end points, not of the curve itself.
What is the difference from the instantaneous rate of change?
The instantaneous rate of change is the rate at one point, the derivative f′(a). It is the limit of the average rate of change on [a, a + h] as h shrinks to 0.
What does a negative average rate of change mean?
The function is lower at b than at a, so on balance it falls over the interval. F(x) = 2/x² falls from 1/2 at x = 2 to 1/18 at x = 6, an average rate of −1/9 per unit.
Can I use a table of values instead of a formula?
Yes. Use the rate of change calculator, which takes points (x, y) and also shows the rate on each interval between them.