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 throughout academics, especially in physics, chemistry and accounting.

It’s most frequently utilized when talking about velocity, however it has numerous uses throughout many industries. Due to its value, this formula is something that learners should learn.

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

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

R = Δy / Δx

This measures the change of y in comparison to the change of x.

The change within the numerator and denominator is portrayed by the greek letter Δ, expressed as delta y and delta x. It is further 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 shown as:

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

Average Rate of Change = Slope

Plotting out these values in a X Y axis, is useful 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 summation, in a linear function, the average rate of change between two values is equivalent to the slope of the function.

This is mainly why average rate of change of a function is the slope of the secant line going through two random endpoints on the graph of the function. Meanwhile, 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 possible.

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

Step 1: Understand Your Values

In these types of equations, math scenarios typically 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 instance, next you have to search for the values on the x and y-axis. Coordinates are typically 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 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 inputted, all that we have to do is to simplify the equation by deducting all the values. Therefore, our equation becomes something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As we can see, by simply plugging in all our values and simplifying the equation, we obtain the average rate of change for the two coordinates that we were given.

Average Rate of Change of a Function

As we’ve shared previously, the rate of change is relevant to multiple diverse scenarios. The aforementioned examples focused on the rate of change of a linear equation, but this formula can also be relevant for functions.

The rate of change of function observes the same principle but with a unique formula due to the unique values that functions have. This formula is:

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

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

Negative Slope

As you might remember, the average rate of change of any two values can be plotted. The R-value, then is, equivalent 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 diminishing in value. For example, velocity can be negative, which results in a declining position.

Positive Slope

In contrast, a positive slope means 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. 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 discuss the average rate of change formula via 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 plain substitution because the delta values are already specified.

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 search 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 given, the average rate of change is identical to the slope of the line linking two points.

Example 3

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

The last example will be extracting 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, solve for the values of the functions in the equation. In this situation, 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 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

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