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, especially in physics, chemistry and accounting.

It’s most frequently applied when talking about velocity, however it has many uses throughout different industries. Due to its usefulness, this formula is a specific concept that students should understand.

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 describes the variation of one value when compared to another. In practical terms, it's used to define the average speed of a change over a specified period of time.

Simply put, the rate of change formula is expressed as:

R = Δy / Δx

This calculates the variation of y in comparison to the variation of x.

The change within the numerator and denominator is shown by the greek letter Δ, expressed as delta y and delta x. It is also expressed as the difference between 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 values in a X Y graph, is helpful when discussing dissimilarities 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

To summarize, in a linear function, the average rate of change between two values is equivalent to the slope of the function.

This is the reason why the average rate of change of a function is the slope of the secant line intersecting two random 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 understand the slope formula and what the figures mean, finding the average rate of change of the function is feasible.

To make studying this concept less complex, here are the steps you need to follow to find the average rate of change.

Step 1: Determine Your Values

In these types of equations, math problems typically give you two sets of values, from which you extract x and y values.

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

In this situation, then you have to find the values on the x and y-axis. Coordinates are typically 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 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 plugged in, all that remains is to simplify the equation by subtracting all the numbers. Thus, our equation will look something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As shown, just by plugging in 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 shared before, the rate of change is relevant to multiple diverse 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 follows the same principle but with a distinct formula due to the unique 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 Cartesian plane value.

Negative Slope

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

Occasionally, the equation concludes in a slope that is negative. This denotes that the line is trending downward 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

On the contrary, a positive slope denotes 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 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 run through the average rate of change formula with some examples.

Example 1

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

In this example, all we need to do is a plain substitution due to the fact that the delta values are already specified.

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 axis.

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 given, the average rate of change is equivalent to the slope of the line connecting two points.

Example 3

Extract 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 calculating the rate of change of a function, solve for the values of the functions in the equation. In this instance, we simply substitute the values on the equation with 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

Now that we have all our values, all we have 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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