Math
Average Rate of Change Calculator
Type a polynomial such as x^2 - 3x + 2 and two x-values, or enter two points, and mathcheck computes the average rate of change [f(b) - f(a)] / (b - a).
You get the exact fraction and decimal, the change in y and x, the equation of the secant line through the two points, and whether the function is increasing or decreasing on average over the interval.
Polynomial in x. Use ^ for powers, for example 2x^3 - x + 5.
Try an example
Result
Average rate of change
—
- Decimal
- —
- Change in y: f(b) - f(a)
- —
- Change in x: b - a
- —
- Secant line
- —
- Point at a
- —
- Point at b
- —
More details (1 more)
- Trend on the interval
- —
Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What average rate of change means
The average rate of change of a function over an interval [a, b] is how much the output changes per unit of input, on average, between x = a and x = b. It is the How steep a line is: how much y changes for each step right in x. of the secant line through the points (a, f(a)) and (b, f(b)). For a straight line it equals the slope everywhere; for a curve it is a single summary number for the whole interval.
f(x) = x^2 - 3x + 2 from x = 1 to x = 4
f(1) = 0 and f(4) = 6. Average rate = (6 - 0) / (4 - 1) = 6/3 = 2. The secant line through (1, 0) and (4, 6) is y = 2x - 2.
Average versus instantaneous rate
The average rate uses two points and describes the whole interval. The instantaneous rate (the derivative) describes a single point and is the limit of the average rate as the interval shrinks. Average rate of change is the algebra-class version; it needs no calculus.
| Result | Meaning on the interval |
|---|---|
| Positive | f(b) > f(a): the function ends higher than it started |
| Negative | f(b) < f(a): the function ends lower than it started |
| Zero | f(b) = f(a): no net change, even if the function moved in between |
How to use this calculator
- Choose whether you have a formula for f(x) or just two points.
- For a formula, type the polynomial and the two x-values a and b.
- For points, enter (x1, y1) and (x2, y2).
- Read the average rate of change as an exact fraction with its decimal, and the secant line equation.
- Open Show the work to see f(a), f(b), and the division.
How to read the answer
A rate of 2 means that, on average, y rises by 2 for each 1-unit increase in x across the interval, even if the function rises faster in some places and slower in others. The sign tells you the overall direction, and the secant line is the straight line that connects the two endpoints of the interval on the graph.
Common mistakes and edge cases
- Subtracting in mismatched order: f(b) - f(a) must go over b - a in the same order.
- Plugging the x-values into the formula as if they were y-values.
- Using the same x-value twice, which makes the The bottom number of a fraction; it says how many equal parts make one whole. 0.
- Reading a zero average rate as a constant function; it only means the endpoints have the same height.
- Confusing average rate of change with the derivative at a point.
Worked examples
Quadratic over [1, 4]
f(x) = x^2 - 3x + 2
Average rate of change
2
Decreasing on the interval
f(x) = -x^2 + 4 from 0 to 3
Average rate of change
-3
Fraction answer
f(x) = x^3 from 1 to 3
Average rate of change
13
Zero net change
f(x) = x^2 - 4x from 0 to 4
Average rate of change
0
From two points
(2, 5) and (6, 8)
Average rate of change
3/4 (0.75)
Decimal x-values
f(x) = 2x^2 from 0.5 to 1.5
Average rate of change
4
Same x-value twice
Average rate of change
—
Unreadable function
Average rate of change
—
Frequently asked questions
What is the formula for average rate of change?+
[f(b) - f(a)] / (b - a): the change in the output divided by the change in the input over the interval from a to b.
Is average rate of change the same as slope?+
It is the slope of the secant line through the two endpoints of the interval. For a linear function that is the line's slope; for a curve it is the slope of the chord.
Can the average rate of change be negative?+
Yes. A negative value means the function's output is lower at the end of the interval than at the start.
What if I only have a table of values?+
Use Two points mode and enter the two rows you want as (x1, y1) and (x2, y2). The calculator computes the rate directly from them.
How is this different from the derivative?+
The derivative is the instantaneous rate at one point, found by letting the interval shrink to zero width. The average rate of change uses a whole interval and needs only algebra.
What functions can I type?+
Polynomials in x with integer or decimal coefficients, such as 2x^3 - 0.5x + 7. For other functions, evaluate the two points yourself and use Two points mode.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
Related calculators
Slope Calculator
Find slope from two points, solve a missing coordinate from a known slope, or turn an angle or percentage grade into a slope. Includes equations, intercepts, perpendicular slope, graph, and step-by-step work.
Slope Intercept Form Calculator
Write the equation of a line in slope-intercept form y = mx + b from two points, a point and a slope, or standard form. Exact fractions, intercepts, a graph, and every other form of the line.
Point Slope Form Calculator
Write the equation of a line in point-slope form y - y1 = m(x - x1) from a point and a slope or from two points, then convert it to slope-intercept and standard form with a graph.
Quadratic Formula Calculator
Solve ax^2 + bx + c = 0 with exact simplified radicals, reduced fractions, or complex roots, plus discriminant, vertex, factored and vertex forms, a graph, and every step.
Parabola Calculator
Find the vertex, focus, directrix, axis of symmetry, focal length, latus rectum, and intercepts of a parabola from standard form, vertex form, or focus and directrix, with a graph.
Percentage Change Calculator
Find percentage increase or decrease between an original value and a new value with signed change, absolute change, and steps.
Last updated: September 4, 2026