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 concepts throughout academics, most notably in physics, chemistry and accounting.

It’s most frequently used when discussing momentum, although it has multiple uses throughout many industries. Due to its usefulness, this formula is a specific concept that learners 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 shows the variation of one value when compared to another. In every day terms, it's used to evaluate the average speed of a change over a specified period of time.

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

R = Δy / Δx

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

The variation through the numerator and denominator is shown by the greek letter Δ, read as delta y and delta x. It is also expressed as the difference within the first point and the second point of the value, or:

Δy = y2 - y1

Δx = x2 - x1

Because of this, the average rate of change equation can also be shown 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 when compared to value B.

The straight line that joins 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

In summation, in a linear function, the average rate of change among two figures 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 going 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 understand the slope formula and what the figures mean, finding the average rate of change of the function is feasible.

To make learning this topic simpler, here are the steps you need to obey to find the average rate of change.

Step 1: Determine Your Values

In these sort of equations, math scenarios typically 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 instance, next you have to locate the values on the x and y-axis. Coordinates are typically provided 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 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 add the values as follows.

R = 4 - 2 / 3 - 1

Step 3: Simplify

With all of our figures in place, all that remains is to simplify the equation by deducting all the numbers. Thus, our equation then becomes the following.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As we can see, 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 stated previously, the rate of change is relevant to numerous different 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 follows the same rule but with a distinct formula due to the different values that functions have. This formula is:

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

In this instance, the values given will have one f(x) equation and one Cartesian plane value.

Negative Slope

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

Sometimes, the equation results in a slope that is negative. This means that the line is descending from left to right in the X Y graph.

This means that the rate of change is diminishing 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 indicates 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. In terms of our last example, if an object has positive velocity and its position is increasing.

Examples of Average Rate of Change

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

Example 1

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

In the given example, all we need to do is a simple 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 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 identical 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 last 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, determine 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

Now that we have all our values, all we must 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

Grade Potential Can Help You Improve Your Math Skills

Math can be a challenging topic to learn, but it doesn’t have to be.

With Grade Potential, you can get paired with an expert tutor that will give you customized teaching depending on your abilities. With the quality of our tutoring services, working with equations is as simple as one-two-three.

Connect with us now!