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 frequently used when discussing velocity, though it has many applications across many industries. Because of its utility, this formula is something that learners should grasp.

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

Average Rate of Change Formula

In math, the average rate of change formula describes the change of one figure when compared to another. In practical terms, it's employed to determine the average speed of a change over a specified period of time.

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

R = Δy / Δx

This measures the change of y in comparison 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 denoted as the variation within the first point and the second point of the value, or:

Δy = y2 - y1

Δx = x2 - x1

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

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

Average Rate of Change = Slope

Plotting out these values in a Cartesian plane, is beneficial when talking about differences in value A when compared to 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

To summarize, in a linear function, the average rate of change between two figures is equivalent 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 arbitrary endpoints on the graph of the function. In the meantime, 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 values mean, finding the average rate of change of the function is achievable.

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

Step 1: Find Your Values

In these types of equations, mathematical scenarios generally 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 case, 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

Find 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 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 plugged in, all that is left is to simplify the equation by deducting all the numbers. So, our equation will look something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As shown, just by replacing all our values and simplifying the equation, we get 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 many different scenarios. The previous examples were applicable to the rate of change of a linear equation, but this formula can also be relevant for functions.

The rate of change of function observes a similar principle but with a unique 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 Cartesian plane value.

Negative Slope

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

Sometimes, 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, velocity can be negative, which means a declining position.

Positive Slope

At the same time, 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 previous example, if an object has positive velocity and its position is increasing.

Examples of Average Rate of Change

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

Example 1

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

In this example, all we have to do is a straightforward substitution due to the fact that the delta values are already provided.

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 Cartesian plane.

For this example, we still have to look for 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 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 third example will be calculating 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 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

With all our values, all we must 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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