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

The rate of change formula is one of the most important math principles across academics, especially in chemistry, physics and finance.

It’s most often utilized when discussing momentum, though it has many applications throughout different industries. Because of its utility, this formula is something that learners should understand.

This article will share 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 figure when compared to another. In practical terms, it's used to define the average speed of a change over a specific period of time.

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

R = Δy / Δx

This computes the change of y in comparison to the change 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 further denoted as the difference 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 described as:

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

Average Rate of Change = Slope

Plotting out these numbers in a X Y axis, is helpful when reviewing differences in value A versus value B.

The straight line that links 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 short, in a linear function, the average rate of change between two values 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. Simultaneously, 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 grasping this topic less complex, here are the steps you need to obey to find the average rate of change.

Step 1: Find Your Values

In these types of equations, mathematical problems usually give 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 situation, then you have to search for 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

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 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 values inputted, all that remains is to simplify the equation by deducting all the numbers. So, our equation will look 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 mentioned previously, the rate of change is pertinent to multiple different situations. The aforementioned examples were more relevant to the rate of change of a linear equation, but this formula can also be applied to functions.

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

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

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

Negative Slope

As you might recollect, the average rate of change of any two values can be plotted on a graph. The R-value, therefore is, equal to its slope.

Every so often, the equation results in a slope that is negative. This indicates that the line is trending downward from left to right in the X Y axis.

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

Positive Slope

On the contrary, a positive slope means 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 previous 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 via 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 plain substitution due to the fact that the delta values are already provided.

R = Δy / Δx

R = 10 / 2

R = 5

Example 2

Find 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 given, the average rate of change is equal to the slope of the line connecting 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 finding the rate of change of a function, determine the values of the functions in the equation. In this instance, we simply substitute the values on the equation with the values specified 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 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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