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

It’s most often used when talking about velocity, although it has many applications across various industries. Because of its value, this formula is something that students should understand.

This article will discuss the rate of change formula and how you can solve it.

Average Rate of Change Formula

In math, the average rate of change formula denotes the change of one value in relation to another. In practical terms, it's used to identify the average speed of a variation over a certain period of time.

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

R = Δy / Δx

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

The change 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

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

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

Average Rate of Change = Slope

Plotting out these values in a X Y axis, is useful when reviewing dissimilarities in value A when compared to value B.

The straight line that joins these two points is 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 values is the same as the slope of the function.

This is mainly why average rate of change of a function is the slope of the secant line going 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 have discussed the slope formula and what the values mean, finding the average rate of change of the function is achievable.

To make learning this topic less complex, here are the steps you should keep in mind to find the average rate of change.

Step 1: Find Your Values

In these sort of equations, math questions typically 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 case, next you have to locate the values on the x and y-axis. Coordinates are generally given 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 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 numbers inputted, all that remains is to simplify the equation by deducting all the values. So, our equation will look something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As stated, just by plugging in 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 before, the rate of change is applicable to many diverse situations. The aforementioned 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 an identical principle but with a distinct formula due to 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 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, equivalent to its slope.

Occasionally, 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 graph.

This means that the rate of change is diminishing 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 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 increasing.

Examples of Average Rate of Change

In this section, we will run through 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 must do is a simple substitution because 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 X Y graph.

For this example, we still have to find 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 equivalent to the slope of the line linking two points.

Example 3

Calculate 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, solve for 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 must 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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