How do you calculate average rate of change?

Divide the change in output by the change in input. A is equal to f of b minus f of a over b minus a, which is also equal to y2 minus y1 over x2 minus x1. For example, for the quadratic function f of x equals x times x, if x varies from 1 to 3, then the function values are f of 1 equals 1 and f of 3 equals 9. So the average rate of change is equal to 9 minus 1 over 3 minus 1, which gives us 4.

Is average rate of change and slope the same thing?

For a straight line: yes. A straight line has only one slope everywhere so the average rate of change between any two points is equal to that slope. For a curve they are different. The slope changes depending on where you are while the average rate of change represents the average steepness over the entire interval and is equal to the slope of the secant connecting the two endpoints.

What does a negative average rate of change represent?

It shows that the output decreases as input increases. In this region, the end value is lower than the starting value, resulting in an average decrease. When the rate of change is zero, the output at the endpoint exactly matches the starting value, and there's no net change even if it went up along the way and then back down again.

What is the difference between average rate of change and instantaneous rate of change?

The average rate of change uses a secant line which connects two points and represents the speed of change over an interval. The instantaneous rate of change uses one point and its tangent at that point to represent the speed of change at a specific moment in time. As the two points get closer together, the average rate of change approaches the instantaneous rate of change, with the latter being the derivative.

Does order matter for subtraction?

The order must be consistent. Both the numerator and denominator are subtracted from the second value to the first value. If the output is in one order and the input is in the opposite order when subtracting, two wrong signs can cancel each other out and result in a correct answer. However, if only one of the orders is reversed, the sign of the rate of change will be changed, which could cause an increase to be interpreted as a decrease.

Average Rate of Change Calculator

Enter your data

How do you have your data?

Pick how your two points are described. The math is the same either way.

Two points (x, y)

A function on an interval

Your two points, (x₁, y₁) and (x₂, y₂).

Average rate of change A

Project a value at this rate

Assume the average rate continues and read off the value at any x on the secant line.

Average rate of change

Average rate of change
A = 4

Over the interval the output changed by Δy = 8 across a run of Δx = 2, and 8 ÷ 2 = 4.

The rate is positive, so on average the output rises as the input increases — about 4 unit(s) of output for every 1 unit of input.

Secant line through your points: y = 4x - 3. Its slope is the average rate of change.

Average rate of change (A)

Change in output (Δy)
Change in input (Δx)
Secant y-intercept (b)

The secant line joining (1, 1) and (3, 9) is y = 4x - 3. Its slope, 4, is the average rate of change.

Graph and steps

Show the secant-line graph

Plot the secant line whose slope is the average rate of change.

Show the step-by-step working

Walk through the change in y, the change in x, and the division.

From your two points to the average rate of change
Step	Working
Identify the two points	(x1, y1) = (1, 1) and (x2, y2) = (3, 9)
Change in output Δy = y2 - y1	= 9 - (1) = 8
Change in input Δx = x2 - x1	= 3 - (1) = 2
Divide A = Δy / Δx	= 8 / 2 = 4
The secant line y = A x + b	y = 4x - 3

Average rate of change indicates how much one quantity is changing with respect to another quantity on average. On a graph it corresponds to the slope of the line joining two points, this line being called the secant. This calculator takes in values of a function at two points or at the end points of an interval and gives back the average rate of change, the change in each direction, the secant and an easy to understand explanation in English on what these numbers mean.

If you leave one of the five values free, then this value is calculated by the calculator. Normally you let the rate of change be free, but it is also possible to first set a target rate of change and then calculate the missing coordinates backwards.

What is the average rate of change?

For a function f, the average rate of change from a to b is the quotient of the change in output value over the change in input value.

A = \frac{f(b) - f(a)}{b - a} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}

The Greek letter delta simply means change, so the change in y is just a subtraction and not a new variable. If you are reading data from a table or graph, use the second form since you know both points. If you are given an equation, calculate the values of f at the endpoints of the interval and use the first form. Both forms represent the same slope.

This slope is the secant, that is, the line going through both points. The average rate of change is closely related to the concept of slope, as it gives exactly how steep this line is.

Instructions for using this calculator:

Choose the method that fits your data. In two point mode, you will enter the coordinates of two points. In function mode, you will enter the endpoints a and b of the interval as well as the values f(a) and f(b) of the function at those two points. Regardless of which method you choose, the calculator will give you the same rate of change.

The rate of change answer box is initially blank and will be filled in by the calculator. If you already know a target rate of change, enter that value and leave the desired coordinates field blank. The calculator will then calculate the corresponding value. By enabling the projection feature, you can see where the trend will reach at any given x-value. Opening the graph allows you to verify the tangent, and expanding the steps lets you follow along with the calculation process.

A complete example:

Consider the function $f(x) = x^2$ on the interval from 1 to 3. The values at the endpoints are f(1) = 1 and f(3) = 9, so the two points are (1, 1) and (3, 9). The change in output is 9 - 1 while the change in input is 3 - 1.

A = \frac{f(3) - f(1)}{3 - 1} = \frac{9 - 1}{3 - 1} = \frac{8}{2} = 4

So the output is increasing by an average of 4 units for every 1 unit increase in input over this interval. The curve is not a straight line, but the average rate of change summarizes the entire interval with the slope of a single line.

The rate of change can be negative as well. If a price falls from 2.84 to 2.41 and two years pass, the average rate of change is the difference between 2.41 and 2.84 divided by 2, or a decrease of 0.22 per year. The price on average decreases about 22 cents each year.

Difference between average rate of change and slope:

For a straight line the two concepts are exactly the same. Since a straight line has the same slope everywhere, the average rate of change between any two points will always be that constant slope. The average rate of change for the line y = 2x is 2 on every interval.

For a curve the two concepts are different. Since a curve is curved, its slope changes from point to point. The average rate of change over some portion of the curve is an average of all those changes and is represented by a secant line. If you pick another portion of the curve, you will usually get a different average value.

Average rate of change and instantaneous rate of change:

The average rate of change considers two points and the secant that connects them. The instantaneous rate of change considers one point and the tangent which touches the curve exactly at that point. As you move the second point closer to the first, the secant gradually approaches the tangent, and the average rate of change gradually approaches the instantaneous rate of change. This limit is what we call the derivative in differential calculus.

So the average rate of change is kind of a stepping stone to derivatives. It can be calculated exactly and easily with two points. This is the right tool when you don't want to know the speed at one particular moment, but rather how fast it's changing over an entire period of time.

What can be derived from symbols and units:

A positive rate of change means that the output increases as the input increases. A negative rate of change means that the output decreases as the input increases, also known as negative growth. If the rate of change is zero, then the output at the end point will be exactly the same as the starting point, so there is no net change despite possible fluctuations along the way.

A rate of change always has a unit. It is the result of dividing the units of output by the units of input. For example, if you divide distance by time, you get speed, such as miles per hour. If you divide dollars by years, you get price trend in dollars per year. If you divide population by years, you get rate of change in people per year. A single number 4 is not meaningful, but a notebook for $4 or 4 meters per second contains much information.

Common mistakes to avoid:

Do not confuse the function value with the rate of change. Just because f(3) = 9 does not mean that the rate of change is 9. The rate of change requires two points, not just one.

The order of subtraction for the numerator and denominator must be the same. Both must subtract the second value from the first. If only one changes its order, it will change the sign and what looks like an increase can suddenly look like a decrease.

Remember that the actual intervals given in the problem are used and that the average hides details. With only two points, it is not possible to tell if the output is increasing steadily or whether it increases rapidly at first and then decreases again. The average just gives the net effect between the two endpoints.

Quantity	Formula	What it answers
Average rate of change	$\frac{f(b)-f(a)}{b-a}$	Overall pace between two points (secant slope)
Net change	$f(b)-f(a)$	Total change in output across the interval
Instantaneous rate	$f'(x)$	Pace at a single point (tangent slope, the derivative)

Frequently asked questions

How do you calculate average rate of change?: Divide the change in output by the change in input. A is equal to f of b minus f of a over b minus a, which is also equal to y2 minus y1 over x2 minus x1. For example, for the quadratic function f of x equals x times x, if x varies from 1 to 3, then the function values are f of 1 equals 1 and f of 3 equals 9. So the average rate of change is equal to 9 minus 1 over 3 minus 1, which gives us 4.
Is average rate of change and slope the same thing?: For a straight line: yes. A straight line has only one slope everywhere so the average rate of change between any two points is equal to that slope. For a curve they are different. The slope changes depending on where you are while the average rate of change represents the average steepness over the entire interval and is equal to the slope of the secant connecting the two endpoints.
What does a negative average rate of change represent?: It shows that the output decreases as input increases. In this region, the end value is lower than the starting value, resulting in an average decrease. When the rate of change is zero, the output at the endpoint exactly matches the starting value, and there's no net change even if it went up along the way and then back down again.
What is the difference between average rate of change and instantaneous rate of change?: The average rate of change uses a secant line which connects two points and represents the speed of change over an interval. The instantaneous rate of change uses one point and its tangent at that point to represent the speed of change at a specific moment in time. As the two points get closer together, the average rate of change approaches the instantaneous rate of change, with the latter being the derivative.
Does order matter for subtraction?: The order must be consistent. Both the numerator and denominator are subtracted from the second value to the first value. If the output is in one order and the input is in the opposite order when subtracting, two wrong signs can cancel each other out and result in a correct answer. However, if only one of the orders is reversed, the sign of the rate of change will be changed, which could cause an increase to be interpreted as a decrease.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

OpenStax Calculus Volume 1: Derivatives as Rates of Change
Average versus instantaneous rate of change, with worked examples.
Lumen Learning / OpenStax: Rates of Change and Behavior of Graphs
Textbook treatment of average rate of change from tables, graphs, and formulas.
Khan Academy: Average rate of change
Lessons and practice on average rate of change and secant lines.