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

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

It’s most often utilized when talking about thrust, though it has numerous applications throughout many industries. Because of 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 math, the average rate of change formula shows the variation of one value in relation to another. In practical terms, it's employed to define the average speed of a change over a specified period of time.

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

R = Δy / Δx

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

The variation within the numerator and denominator is portrayed by the greek letter Δ, read as delta y and delta x. It is further portrayed as the difference between 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 portrayed as:

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

Average Rate of Change = Slope

Plotting out these numbers in a X Y axis, is useful when reviewing dissimilarities in value A when compared to value B.

The straight line that joins these two points is 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 short, in a linear function, the average rate of change between two figures is the same as the slope of the function.

This is mainly why 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 understand the slope formula and what the values mean, finding the average rate of change of the function is possible.

To make grasping this principle simpler, here are the steps you must obey to find the average rate of change.

Step 1: Understand Your Values

In these equations, mathematical problems generally 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 generally provided in an (x, y) format, as in this example:

x1 = 1

x2 = 3

y1 = 2

y2 = 4

Step 2: Subtract The Values

Calculate the Δx and Δy values. As you may remember, 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 plug-in the values as follows.

R = 4 - 2 / 3 - 1

Step 3: Simplify

With all of our values in place, all that is left is to simplify the equation by deducting all the numbers. Thus, our equation becomes something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

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

Average Rate of Change of a Function

As we’ve mentioned earlier, the rate of change is applicable to multiple diverse scenarios. The previous 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 an identical principle but with a distinct formula because of 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 Cartesian plane value.

Negative Slope

Previously if you recall, the average rate of change of any two values can be graphed. The R-value, therefore is, identical 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 decreasing in value. For example, velocity can be negative, which results in a decreasing 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 relation to our last 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 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 due to the fact that the delta values are already provided.

R = Δy / Δx

R = 10 / 2

R = 5

Example 2

Extract 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 find 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 provided, the average rate of change is equivalent to 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 final example will be calculating 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 substitute the values on the equation with the values provided 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 need 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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