First point coordinates
x₁
f(x₁)
Second point coordinates
x₂
f(x₂)
Result
Average rate of change

# Average Rate of Change Calculator

By Julia Żuławińska

The average rate of change calculator is here to help you understand the simple concept hidden behind a long, little bit confusing, name. What is the rate of change? Generally speaking, it shows the relationship between two factors. Look for a more precise average rate of change definition below. We will also demonstrate and explain the average rate of change formula with a couple of examples of how to use it.

## What is rate of change? - the average rate of change definition

Everything keeps moving. Change is inevitable. Starting with the acceleration of your bike or car, through to population growth, from the blood flow in your veins to symbiosis of your cells, the rate of change allows us to establish the value associated with those changes.

The average rate of change is a rate that describes how one number changes, on average, in relation to another. If you have a function, it is the slope of the line drawn between two points. But don't confuse it with slope, you can use the average rate of change for any given function, not only linear ones.

## Average rate of change formula

In the following picture, we marked two points to help you better understand how to find the average rate of change. The average rate of change formula is:

`A = [f(x2) - f(x1)] / [x2 - x1]`, where

• `(x1, f(x1))` are the coordinates of the first point.
• `(x2, f(x2))` are the coordinates of the second point.

If it's positive, it means that one coordinate increases as the other also increases. For example, the more you ride a bike, the more calories you burn.

It's equal to zero when one coordinate changes, but the other one does not. A good example might be not studying for your exams. As time starts running out, the amount of things to learn doesn't change.

The average rate of change is negative when one coordinate increases, while the other one decreases. Let's say you're going on a vacation. The more time you spend on your travel, the closer you are to your destination.

## How to find the average rate of change? - first example

Let's calculate the average rate of change of speed of a train going from Paris to Rome (1420,6 km). On the following chart you can see the change in distance over time: As you see, the speed wasn't constant. The train stopped two times, and in between stops, it went significantly slower. But for calculating the average speed, the only variables that matter are the change in distance and the change in time. So, if the coordinates of the first point are (0, 0), and the coordinates of the second point are the distance between two cities, and the time of the travel, (1420.6, 12.5), then:

`A = (1420.6 - 0) / (12.5 - 0) = 113.648 [km/h]`

On average, the train was going 113.648 kilometers per hour. Now, let's look at a more mathematical example.

## How to find the average rate of change? - second example

You have been given a function:

`f(x) = x2 + 5*x - 7`

Find the average rate of change over the interval [-4, 6].

1. Find values of your function for both points:
• `f(x1) = f(-4) = (-4)2 + 5 * (-4) - 7 = -11`
• `f(x2) = f(6) = 62 + 5 * 6 - 7 = 59`
1. Use the average rate of change equation:
• `A = [f(x2) - f(x1)] / [x2 - x1] = [f(6) - f(-4)] / [6 - (-4)] = [59 - (-11)] / [6 - (-4)] = 70 / 10 = 7`

We have a lot of maths calculators, just like this one! If you enjoyed the average rate of change calculator, feel free to check them out!

Julia Żuławińska

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