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

The rate of change formula is one of the most used mathematical principles across academics, most notably in physics, chemistry and finance.

It’s most often utilized when talking about thrust, although it has multiple applications across various industries. Because of its value, this formula is something that students should understand.

This article will share the rate of change formula and how you can work with them.

Average Rate of Change Formula

In mathematics, the average rate of change formula denotes the variation of one figure 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 variation of x.

The variation within the numerator and denominator is represented by the greek letter Δ, expressed as delta y and delta x. It is additionally 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 expressed as:

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

Average Rate of Change = Slope

Plotting out these numbers in a X Y axis, is useful when discussing dissimilarities in value A in comparison with value B.

The straight line that links 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

To summarize, in a linear function, the average rate of change among 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 arbitrary 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 values mean, finding the average rate of change of the function is achievable.

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

Step 1: Determine Your Values

In these equations, math questions typically give you two sets of values, from which you solve to find x and y values.

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

In this scenario, then you have to search for the values along the x and y-axis. Coordinates are usually given in an (x, y) format, as in this example:

x1 = 1

x2 = 3

y1 = 2

y2 = 4

Step 2: Subtract The Values

Find 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 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 figures plugged in, all that we have to do is to simplify the equation by subtracting all the values. Therefore, our equation then becomes the following.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As stated, by replacing all our values and simplifying the equation, we get 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 relevant to many different situations. The aforementioned examples were more relevant to the rate of change of a linear equation, but this formula can also be used in functions.

The rate of change of function obeys the same rule but with a different formula due to the distinct values that functions have. This formula is:

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

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

Negative Slope

Previously if you recollect, the average rate of change of any two values can be plotted on a graph. 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 descending 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

At the same time, a positive slope denotes 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 last example, if an object has positive velocity and its position is ascending.

Examples of Average Rate of Change

In this section, we will review 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 must do is a straightforward substitution due to the fact that 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 X Y graph.

For this example, we still have to find 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 equal 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 last example will be calculating 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, solve for the values of the functions in the equation. In this case, we simply substitute the values on the equation using 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

With 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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