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 physics, chemistry and finance.

It’s most often utilized when talking about momentum, though it has numerous uses throughout many industries. Because of its value, this formula is a specific concept that students should learn.

This article will go over 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 in relation to another. In practical terms, it's employed to define the average speed of a variation over a specified period of time.

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

R = Δy / Δx

This measures the change of y compared to the variation of x.

The change within the numerator and denominator is represented by the greek letter Δ, read as delta y and delta x. It is also portrayed as the variation between 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 figures in a X Y graph, is beneficial when talking about dissimilarities in value A when compared to value B.

The straight line that links 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 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 going through two arbitrary 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 have discussed the slope formula and what the figures mean, finding the average rate of change of the function is achievable.

To make understanding this principle easier, here are the steps you should keep in mind to find the average rate of change.

Step 1: Understand Your Values

In these sort of equations, mathematical problems generally provide you with 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 case, next you have to locate the values on the x and y-axis. Coordinates are typically 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

Calculate 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 values in place, all that remains is to simplify the equation by deducting all the numbers. Thus, our equation will look something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As shown, by plugging in all our values and simplifying the equation, we achieve the average rate of change for the two coordinates that we were given.

Average Rate of Change of a Function

As we’ve shared before, the rate of change is relevant to multiple different situations. The previous examples were applicable to the rate of change of a linear equation, but this formula can also be applied to functions.

The rate of change of function observes 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 case, the values given will have one f(x) equation and one X Y graph value.

Negative Slope

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

Occasionally, the equation concludes in a slope that is negative. This indicates that the line is descending 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 results in a declining position.

Positive Slope

In contrast, a positive slope shows 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 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 the given example, all we have to do is a plain substitution due to the fact that the delta values are already given.

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 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 equivalent to the slope of the line joining two points.

Example 3

Find 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 extracting the rate of change of a function, calculate 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

With 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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