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

The rate of change formula is one of the most important math formulas throughout academics, especially in chemistry, physics and accounting.

It’s most frequently utilized when talking about momentum, although it has many uses across different industries. Because of its usefulness, this formula is something that learners should learn.

This article will share 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 figure when compared to another. In practical terms, it's utilized to define the average speed of a variation over a specific period of time.

At its simplest, the rate of change formula is expressed as:

R = Δy / Δx

This computes the variation of y compared to the variation of x.

The change within the numerator and denominator is shown by the greek letter Δ, expressed as delta y and delta x. It is also expressed as the variation within 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 shown as:

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

Average Rate of Change = Slope

Plotting out these numbers in a X Y graph, is helpful when working with dissimilarities in value A in comparison with value B.

The straight line that joins 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

To summarize, in a linear function, the average rate of change between two figures is equal to the slope of the function.

This is mainly why 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. 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 understand the slope formula and what the values mean, finding the average rate of change of the function is possible.

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

Step 1: Determine Your Values

In these equations, mathematical problems usually offer you two sets of values, from which you extract x and y values.

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

In this instance, next you have to search for the values on the x and y-axis. Coordinates are usually provided 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 found all the values of x and y, we can add the values as follows.

R = 4 - 2 / 3 - 1

Step 3: Simplify

With all of our figures in place, all that is left is to simplify the equation by deducting all the values. So, our equation then becomes the following.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As stated, by plugging in 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 mentioned previously, the rate of change is pertinent to many different scenarios. The aforementioned examples focused on the rate of change of a linear equation, but this formula can also be used in functions.

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

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

In this situation, the values given will have one f(x) equation and one Cartesian plane value.

Negative Slope

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

Occasionally, the equation results in a slope that is negative. This denotes that the line is trending downward from left to right in the X Y graph.

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

Positive Slope

In contrast, a positive slope indicates that the object’s rate of change is positive. This means that the object is gaining value, and the secant line is trending upward from left to right. With regards to our aforementioned example, if an object has positive velocity and its position is ascending.

Examples of Average Rate of Change

In this section, we will review the average rate of change formula with some examples.

Example 1

Calculate 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 provided.

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 Cartesian plane.

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 you can see, the average rate of change is equivalent to the slope of the line joining 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 calculating the rate of change of a function, solve for the values of the functions in the equation. In this situation, we simply replace the values on the equation with the values provided 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

Now that we have 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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