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

The rate of change formula is one of the most important math formulas throughout academics, particularly in physics, chemistry and finance.

It’s most often applied when talking about velocity, however it has multiple uses throughout many industries. Because of its utility, this formula is a specific concept that students should grasp.

This article will discuss 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 value in relation to another. In practice, it's utilized to define the average speed of a change over a certain period of time.

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

R = Δy / Δx

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

The variation 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

Because of this, 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 beneficial when discussing dissimilarities 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

To summarize, in a linear function, the average rate of change among two values is the same as 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 passing 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 values mean, finding the average rate of change of the function is achievable.

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

Step 1: Understand Your Values

In these equations, math scenarios 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 along the x and y-axis. Coordinates are usually provided in an (x, y) format, like this:

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 found 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 subtracting all the numbers. Thus, our equation will look something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

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

Average Rate of Change of a Function

As we’ve shared earlier, the rate of change is relevant to numerous 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 observes an identical 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 instance, the values provided will have one f(x) equation and one X Y axis value.

Negative Slope

Previously if you recall, the average rate of change of any two values can be plotted on a graph. The R-value, therefore is, identical to its slope.

Occasionally, the equation results 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 diminishing in value. For example, velocity can be negative, which means a decreasing position.

Positive Slope

At the same time, a positive slope indicates that the object’s rate of change is positive. This tells us that the object is gaining value, and the secant line is trending upward from left to right. With regards to our previous example, if an object has positive velocity and its position is ascending.

Examples of Average Rate of Change

In this section, we will talk about 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 the given example, all we need to do is a straightforward substitution because 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 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 the same as the slope of the line connecting two points.

Example 3

Calculate the rate of change of function f(x) = x2 + 5x - 3 on the interval [3, 5].

The third example will be finding 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, calculate the values of the functions in the equation. In this instance, we simply replace 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

Once 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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