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

The rate of change formula is one of the most widely used math formulas throughout academics, especially in chemistry, physics and accounting.

It’s most often applied when discussing momentum, although it has numerous uses throughout many industries. Because of its value, this formula is a specific concept that students should understand.

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 figure when compared to another. In every day terms, it's employed to define the average speed of a variation over a specified period of time.

At its simplest, the rate of change formula is expressed as:

R = Δy / Δx

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

The change within the numerator and denominator is shown by the greek letter Δ, read as delta y and delta x. It is also portrayed as the difference between 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 values in a X Y axis, is helpful when working with differences in value A versus value B.

The straight line that connects these two points is also known as 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 equal 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. 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 understand the slope formula and what the figures mean, finding the average rate of change of the function is feasible.

To make understanding this topic simpler, here are the steps you need to keep in mind to find the average rate of change.

Step 1: Understand Your Values

In these sort of equations, math questions typically offer 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, then you have to locate the values via 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 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 we have to do 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 stated, by 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 shared previously, the rate of change is applicable to many diverse scenarios. The aforementioned examples focused on the rate of change of a linear equation, but this formula can also be used in functions.

The rate of change of function follows the same rule but with a different formula due to the different 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

As you might recollect, the average rate of change of any two values can be graphed. The R-value, is, equivalent to its slope.

Every so often, the equation concludes in a slope that is negative. This means that the line is trending downward from left to right in the Cartesian plane.

This translates to the rate of change is diminishing in value. For example, velocity can be negative, which results in a declining position.

Positive Slope

At the same time, 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 aforementioned example, if an object has positive velocity and its position is increasing.

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 this 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 employing 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 equivalent to the slope of the line joining 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 extracting 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

Once we have all our values, all we have to 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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