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

The rate of change formula is one of the most important mathematical formulas across academics, particularly in chemistry, physics and accounting.

It’s most often applied when talking about velocity, though it has many uses across many industries. Due to its usefulness, this formula is something that learners should grasp.

This article will share the rate of change formula and how you should solve them.

Average Rate of Change Formula

In mathematics, the average rate of change formula denotes the variation of one figure in relation to another. In practical terms, it's employed to identify the average speed of a variation over a certain 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 change of x.

The variation within the numerator and denominator is portrayed by the greek letter Δ, expressed as delta y and delta x. It is also denoted as the variation within 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 portrayed as:

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

Average Rate of Change = Slope

Plotting out these figures in a Cartesian plane, is useful when working with differences in value A when compared to value B.

The straight line that connects 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 between two figures is equivalent to 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. 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 know the slope formula and what the figures mean, finding the average rate of change of the function is feasible.

To make studying this principle less complex, here are the steps you need to keep in mind to find the average rate of change.

Step 1: Find Your Values

In these sort of equations, mathematical problems 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 find 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 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 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 in place, all that is left is to simplify the equation by subtracting all the values. Thus, our equation becomes something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As shown, 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 before, the rate of change is relevant to numerous diverse scenarios. The previous examples were more relevant to the rate of change of a linear equation, but this formula can also be used in functions.

The rate of change of function observes an identical principle but with a distinct formula because of the unique values that functions have. This formula is:

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

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

Negative Slope

If you can remember, the average rate of change of any two values can be graphed. The R-value, 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 Cartesian plane.

This means that the rate of change is decreasing in value. For example, velocity can be negative, which means a declining position.

Positive Slope

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

Examples of Average Rate of Change

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

Example 1

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

In the given example, all we need to do is a straightforward substitution because the delta values are already provided.

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 provided, the average rate of change is the same as 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 calculating 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 substitute the values on the equation using the values given 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

Once we have 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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