Rate of Change Formula - What Is the Rate of Change Formula? Examples

The rate of change formula is one of the most used math formulas across academics, especially in chemistry, physics and finance.

It’s most often utilized when discussing velocity, however it has many applications throughout different industries. Because of its usefulness, this formula is a specific concept that students should grasp.

This article will share the rate of change formula and how you can solve it.

Average Rate of Change Formula

In mathematics, the average rate of change formula denotes the change of one value in relation to another. In practice, it's employed to define the average speed of a change over a specific period of time.

Simply put, the rate of change formula is written as:

R = Δy / Δx

This measures the variation of y compared to the change of x.

The variation through the numerator and denominator is shown by the greek letter Δ, expressed as delta y and delta x. It is further expressed as the difference between the first point and the second point of the value, or:

Δy = y2 - y1

Δx = x2 - x1

Because of this, the average rate of change equation can also be portrayed as:

R = (y2 - y1) / (x2 - x1)

Average Rate of Change = Slope

Plotting out these numbers in a X Y graph, is beneficial when working with dissimilarities in value A when compared to value B.

The straight line that connects these two points is also known as secant line, and the slope of this line is the average rate of change.

Here’s the formula for the slope of a line:

y = 2x + 1

In summation, in a linear function, the average rate of change among two values is equal to the slope of the function.

This is why the average rate of change of a function is the slope of the secant line passing through two arbitrary endpoints on the graph of the function. Meanwhile, the instantaneous rate of change is the slope of the tangent line at any point on the graph.

How to Find Average Rate of Change

Now that we have discussed the slope formula and what the figures mean, finding the average rate of change of the function is possible.

To make grasping this principle easier, here are the steps you must keep in mind to find the average rate of change.

Step 1: Determine Your Values

In these equations, mathematical scenarios usually provide you with two sets of values, from which you will get x and y values.

For example, let’s take the values (1, 2) and (3, 4).

In this instance, next you have to locate the values via the x and y-axis. Coordinates are generally provided in an (x, y) format, as you see in the example below:

x1 = 1

x2 = 3

y1 = 2

y2 = 4

Step 2: Subtract The Values

Calculate the Δx and Δy values. As you may remember, the formula for the rate of change is:

R = Δy / Δx

Which then translates to:

R = y2 - y1 / x2 - x1

Now that we have obtained all the values of x and y, we can input the values as follows.

R = 4 - 2 / 3 - 1

Step 3: Simplify

With all of our numbers in place, all that is left is to simplify the equation by subtracting all the numbers. So, our equation will look something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As we can see, by replacing all our values and simplifying the equation, we obtain the average rate of change for the two coordinates that we were provided.

Average Rate of Change of a Function

As we’ve shared earlier, the rate of change is applicable to multiple different scenarios. The previous examples were more relevant to the rate of change of a linear equation, but this formula can also be relevant for functions.

The rate of change of function follows the same principle but with a different formula because of the distinct values that functions have. This formula is:

R = (f(b) - f(a)) / b - a

In this scenario, the values given will have one f(x) equation and one X Y axis value.

Negative Slope

If you can recollect, the average rate of change of any two values can be plotted on a graph. The R-value, therefore is, equal to its slope.

Every so often, the equation concludes in a slope that is negative. This denotes that the line is descending from left to right in the Cartesian plane.

This means that the rate of change is decreasing in value. For example, rate of change can be negative, which results in a declining position.

Positive Slope

On the other hand, a positive slope denotes that the object’s rate of change is positive. This means that the object is increasing in value, and the secant line is trending upward from left to right. In terms of our previous example, if an object has positive velocity and its position is increasing.

Examples of Average Rate of Change

Now, we will review the average rate of change formula with some examples.

Example 1

Extract the rate of change of the values where Δy = 10 and Δx = 2.

In the given example, all we need to do is a straightforward substitution because the delta values are already given.

R = Δy / Δx

R = 10 / 2

R = 5

Example 2

Extract the rate of change of the values in points (1,6) and (3,14) of the X Y axis.

For this example, we still have to find the Δy and Δx values by employing the average rate of change formula.

R = y2 - y1 / x2 - x1

R = (14 - 6) / (3 - 1)

R = 8 / 2

R = 4

As you can see, the average rate of change is equal to the slope of the line linking two points.

Example 3

Find the rate of change of function f(x) = x2 + 5x - 3 on the interval [3, 5].

The final example will be calculating the rate of change of a function with the formula:

R = (f(b) - f(a)) / b - a

When calculating the rate of change of a function, calculate the values of the functions in the equation. In this case, we simply substitute the values on the equation using the values specified in the problem.

The interval given is [3, 5], which means that a = 3 and b = 5.

The function parts will be solved by inputting the values to the equation given, such as.

f(a) = (3)2 +5(3) - 3

f(a) = 9 + 15 - 3

f(a) = 24 - 3

f(a) = 21

f(b) = (5)2 +5(5) - 3

f(b) = 25 + 10 - 3

f(b) = 35 - 3

f(b) = 32

Once we have all our values, all we must do is plug in them into our rate of change equation, as follows.

R = (f(b) - f(a)) / b - a

R = 32 - 21 / 5 - 3

R = 11 / 2

R = 11/2 or 5.5

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