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

The rate of change formula is one of the most widely used math concepts throughout academics, specifically in chemistry, physics and accounting.

It’s most frequently applied when discussing velocity, though it has multiple uses across many industries. Because of its usefulness, this formula is something that students should grasp.

This article will go over the rate of change formula and how you can work with them.

Average Rate of Change Formula

In math, the average rate of change formula shows the change of one value when compared to another. In every day terms, it's utilized to evaluate the average speed of a change over a specific period of time.

To put it simply, the rate of change formula is expressed as:

R = Δy / Δx

This computes the change of y in comparison to the variation of x.

The change within the numerator and denominator is shown by the greek letter Δ, read as delta y and delta x. It is additionally expressed as the difference within the first point and the second point of the value, or:

Δy = y2 - y1

Δx = x2 - x1

As a result, the average rate of change equation can also be shown as:

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

Average Rate of Change = Slope

Plotting out these values in a Cartesian plane, is useful when working with differences in value A versus value B.

The straight line that links these two points is called the 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 short, in a linear function, the average rate of change between two values is equal to the slope of the function.

This is the reason why the average rate of change of a function is the slope of the secant line passing through two random endpoints on the graph of the function. Simultaneously, 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 know the slope formula and what the values mean, finding the average rate of change of the function is achievable.

To make understanding this concept simpler, here are the steps you should follow to find the average rate of change.

Step 1: Determine Your Values

In these sort of equations, mathematical problems typically give you two sets of values, from which you solve to find x and y values.

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

In this instance, then you have to locate the values on the x and y-axis. Coordinates are typically given in an (x, y) format, as in this example:

x1 = 1

x2 = 3

y1 = 2

y2 = 4

Step 2: Subtract The Values

Calculate the Δx and Δy values. As you may recall, 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 values in place, all that we have to do is to simplify the equation by deducting all the numbers. So, our equation becomes something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As stated, just by replacing all our values and simplifying the equation, we obtain the average rate of change for the two coordinates that we were given.

Average Rate of Change of a Function

As we’ve mentioned before, the rate of change is pertinent to numerous different situations. The previous examples focused on the rate of change of a linear equation, but this formula can also be relevant for functions.

The rate of change of function obeys an identical rule but with a unique formula due to the unique values that functions have. This formula is:

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

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

Negative Slope

As you might remember, the average rate of change of any two values can be plotted. The R-value, then is, identical to its slope.

Sometimes, the equation concludes in a slope that is negative. This means that the line is trending downward from left to right in the X Y axis.

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

Positive Slope

In contrast, a positive slope denotes that the object’s rate of change is positive. This shows us 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 ascending.

Examples of Average Rate of Change

In this section, we will run through the average rate of change formula via 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 plain substitution because the delta values are already given.

R = Δy / Δx

R = 10 / 2

R = 5

Example 2

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

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

R = y2 - y1 / x2 - x1

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

R = 8 / 2

R = 4

As provided, the average rate of change is equal to the slope of the line connecting two points.

Example 3

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

The third example will be extracting the rate of change of a function with the formula:

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

When extracting the rate of change of a function, solve for the values of the functions in the equation. In this situation, 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

With all our values, all we have to 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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