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

The rate of change formula is one of the most important mathematical concepts throughout academics, most notably in physics, chemistry and finance.

It’s most often applied when discussing velocity, however it has numerous uses across different industries. Because of its value, this formula is a specific concept that students should grasp.

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 shows the variation of one figure in relation to another. In practice, it's used to identify the average speed of a variation over a specific period of time.

Simply put, the rate of change formula is expressed 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 represented by the greek letter Δ, read as delta y and delta x. It is additionally expressed as the variation between 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 described as:

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

Average Rate of Change = Slope

Plotting out these figures in a Cartesian plane, is helpful when reviewing dissimilarities in value A in comparison with value B.

The straight line that connects 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 why the 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. 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 know the slope formula and what the values mean, finding the average rate of change of the function is achievable.

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

Step 1: Understand Your Values

In these types of equations, mathematical scenarios typically give you two sets of values, from which you solve to find x and y values.

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

In this situation, next you have to locate the values along the x and y-axis. Coordinates are usually 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 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 numbers. So, our equation will look something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

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

Average Rate of Change of a Function

As we’ve shared earlier, the rate of change is applicable to many diverse scenarios. The previous examples focused on the rate of change of a linear equation, but this formula can also be used in functions.

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

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

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

Negative Slope

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

Occasionally, the equation concludes in a slope that is negative. This means that the line is descending from left to right in the X Y axis.

This translates to the rate of change is decreasing in value. For example, rate of change can be negative, which results in a declining position.

Positive Slope

On the other hand, a positive slope shows that the object’s rate of change is positive. This tells us that the object is increasing in value, and the secant line is trending upward from left to right. In relation to our last example, if an object has positive velocity and its position is increasing.

Examples of Average Rate of Change

Now, we will review the average rate of change formula through some examples.

Example 1

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

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

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 joining 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 finding the rate of change of a function with the formula:

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

When calculating the rate of change of a function, calculate the values of the functions in the equation. In this situation, we simply substitute 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 need 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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