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

The rate of change formula is one of the most widely used mathematical concepts across academics, specifically in chemistry, physics and finance.

It’s most often utilized when talking about velocity, however it has numerous applications across many industries. Due to its utility, this formula is a specific concept that students should understand.

This article will discuss 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 denotes the variation of one figure in relation to another. In practical terms, it's used to define the average speed of a variation over a specific period of time.

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

R = Δy / Δx

This computes the variation 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 denoted 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 X Y graph, is helpful when reviewing dissimilarities in value A versus value B.

The straight line that links 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 the same as 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 intersecting two random endpoints on the graph of the function. At the same time, 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 achievable.

To make understanding this topic easier, here are the steps you must obey to find the average rate of change.

Step 1: Understand Your Values

In these sort of equations, mathematical questions usually offer you 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 situation, then you have to locate the values along 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

Find the Δx and Δy values. As you can recollect, 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 plug-in the values as follows.

R = 4 - 2 / 3 - 1

Step 3: Simplify

With all of our values inputted, all that we have to do is to simplify the equation by subtracting all the values. Therefore, 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 achieve the average rate of change for the two coordinates that we were provided.

Average Rate of Change of a Function

As we’ve stated earlier, the rate of change is pertinent to numerous different situations. The aforementioned examples were applicable to the rate of change of a linear equation, but this formula can also be used in functions.

The rate of change of function follows the same rule but with a different formula because of the unique 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

Previously if you recall, the average rate of change of any two values can be graphed. The R-value, therefore is, equivalent 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 axis.

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 tells us 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 run through 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 have to do is a plain substitution since 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 search 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 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 calculating 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, calculate the values of the functions in the equation. In this case, 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 replace 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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