Average Rate of Change Calculator

average rate of change = Δy ÷ Δx = (y₂ − y₁) ÷ (x₂ − x₁)

x₁

The input at the first endpoint of the interval — the earlier value of whatever the function takes, time in the velocity case, position in the geometric one. Nothing constrains it, and it does not have to be smaller than x₂: the endpoints may be given in either order.

x₂

The input at the second endpoint. It may not equal x₁. If it does, the denominator of the ratio is zero, and there is nothing to average over: the function is being asked to change with no change in its input, which is a vertical segment rather than a rate.

y₁, y₂

The function's output at those two inputs. Whether the function is a formula, a table of readings or a curve on a screen, all the page needs is the two values — and the order matters for the two change columns even though it cancels out of the ratio.

Δy ÷ Δx

The average rate itself: the total change in the output divided by the length of the interval over which it happened. In delta notation the Greek letter stands for a difference, so Δy is the change in y and Δx is the change in x. Reading the expression as per-unit language is the point — the answer is how much y moved for each single unit of x.

(y₂ − y₁) ÷ (x₂ − x₁)

The same quantity written out. This is numerically identical to the slope of the line joining the two points, which is why the two calculators get the same answer to different questions: the slope page asks how steep a line is, and this page asks how fast a function moved over an interval. Only the second framing survives into calculus, where the interval is taken narrower and narrower.

1. The average rate of change of x² on the interval from 1 to 3

   1. The interval is two units long: Δx = 3 − 1 = 2
   2. The output changed by eight: Δy = 9 − 1 = 8
   3. Divide: 8 ÷ 2 = 4
   4. So the function gained an average of 4 units of output per unit of input

   For the squaring function the answer follows a rule worth knowing: on an interval from a to b its average rate is always a + b, here 1 + 3 = 4. That rule holds because b² minus a² factors as (b − a)(b + a), and the b − a cancels with the denominator. It is also why the reference table below is built from this one function.

2. An average speed over three seconds

   1. The input is time in seconds, from 0 to 3
   2. The output is position, which moved from 10 to 70
   3. The position changed by 60 over 3 seconds: 60 ÷ 3 = 20
   4. The average speed is 20 units of distance per second

   The reading that makes the quantity concrete: with position on the vertical axis and time on the horizontal one, the average rate of change is the average speed. It is an average in a real sense — the actual speed may have varied throughout, and this ratio says nothing about the middle of the interval, only about the two endpoints.

3. Swapping the two endpoints

   1. The same two points as the first example, listed in the other order
   2. Δx = 1 − 3 = −2 and Δy = 1 − 9 = −8
   3. The ratio is −8 ÷ −2 = 4, the same value as before
   4. Both change columns have flipped sign while the rate has not

   A reader who sees a negative Δx beside a positive rate has not caught an error. The ratio is unchanged when numerator and denominator both change sign, so the average rate does not depend on which endpoint is called first; the two change columns do, because they describe a direction of travel between the points. The main result answers how fast, and the change columns answer from where to where.

The two inputs must differ. When x₂ equals x₁ the ratio has a zero denominator, and the page refuses it rather than returning infinity: a vertical segment has no rate of change per unit of x, because x is not changing at all. The endpoints may be given in either order, and the average rate is the same either way, but the two change columns are signed and swap when the order does. The page reports a rate over an interval and never a rate at a point — the instantaneous rate is a limit this page does not take, and it requires knowing the function rather than two of its values. Every coordinate is limited to a magnitude of one trillion, beyond which the subtraction would lose precision and the six decimals shown would be misleading. The result is rounded to six decimals while the arithmetic runs at full precision. The page prints no angle: an angle is a property of a straight line, and a function over an interval does not have one, which is one of the ways this page differs from the slope calculator it links to.

What is the average rate of change?

It is the total change in a function's output over an interval, divided by the length of that interval. If position moves from 10 to 70 over three seconds, the average rate is 60 ÷ 3, or 20 units per second. The word average is doing real work: the quantity summarizes the whole interval with a single number and says nothing about how the function behaved inside it.

How is this different from slope?

The arithmetic is identical — both divide the change in the vertical quantity by the change in the horizontal one — but the question is different. Slope describes how steep a line is, a property of the line itself. Average rate of change describes how fast a function moved between two points, which is the quantity that keeps its meaning when the interval is narrowed toward zero. Only one of those ideas survives into calculus.

Why is the answer the same when I swap the two points?

Because the ratio does not change when numerator and denominator both change sign, and swapping the endpoints flips both at once. The two change columns do flip, since they record a direction of travel from the first point to the second. So the rate answers how fast, and the change columns answer from where to where — both are correct at the same time.

Can the average rate of change be negative?

Yes, and it means the output fell as the input rose across that interval. On the interval from −3 to −1 the squaring function drops from 9 to 1, so its average rate is −4. A negative value is not a sign that something was entered the wrong way round; it is the direction of the change, and it is why the two change columns are printed alongside the ratio.

What happens if the two inputs are equal?

The page refuses the calculation. With x₂ equal to x₁ the denominator is zero, so there is no interval to average over — the function would be asked to change with no change in its input. That is a vertical segment, and it has no rate per unit of x. The page reports the refusal rather than returning infinity, which is not a rate.

Is the average rate the same as the instantaneous rate?

Only when the function is a straight line, where the rate is constant everywhere and the two coincide. Otherwise the average describes the interval as a whole and the instantaneous rate describes a single input value. Taking the interval narrower and narrower is what turns the average into the instantaneous rate, and that limit is the derivative — a step this page does not take, because it works from two values rather than from the function's formula.

• Average Rate of Change — the definition over an interval, the secant line reading, and the passage to the instantaneous rate — Wolfram MathWorld (United States)
• Secant Line — the line through the two endpoints, whose slope is the number this page computes — Wolfram MathWorld (United States)
• Derivative — the limit of the average rate of change as the interval shrinks, which is where this quantity is heading — Wolfram MathWorld (United States)