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 principles across academics, most notably in chemistry, physics and finance.

It’s most often applied when discussing momentum, although it has many uses across different industries. Because of its value, this formula is something that learners should understand.

This article will go over the rate of change formula and how you should solve them.

Average Rate of Change Formula

In mathematics, the average rate of change formula denotes the change of one value in relation to another. In practical terms, it's utilized to determine the average speed of a variation over a specific period of time.

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

R = Δy / Δx

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

The variation within the numerator and denominator is represented by the greek letter Δ, read as delta y and delta x. It is further denoted as the variation 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 described as:

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

Average Rate of Change = Slope

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

The straight line that connects 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 summation, in a linear function, the average rate of change among two figures is the same as the slope of the function.

This is why the average rate of change of a function is the slope of the secant line intersecting two arbitrary 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 feasible.

To make learning this principle easier, here are the steps you need to follow to find the average rate of change.

Step 1: Understand Your Values

In these sort of equations, math scenarios generally offer you two sets of values, from which you extract x and y values.

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

In this scenario, next you have to locate the values via 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 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. Thus, our equation will look something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As stated, by simply 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 relevant to numerous diverse scenarios. The previous examples were more relevant to the rate of change of a linear equation, but this formula can also be applied to functions.

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

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

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

Negative Slope

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

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

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

Positive Slope

In contrast, a positive slope indicates 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. With regards to our last example, if an object has positive velocity and its position is increasing.

Examples of Average Rate of Change

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

Example 1

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

In the given example, all we have to do is a simple substitution due to the fact that the delta values are already specified.

R = Δy / Δx

R = 10 / 2

R = 5

Example 2

Calculate 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 look for the Δy and Δx values by using 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 joining two points.

Example 3

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

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

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

When finding the rate of change of a function, determine the values of the functions in the equation. In this instance, we simply replace the values on the equation using 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

With all our values, all we have to do is substitute 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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