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

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

It’s most frequently utilized when discussing momentum, though it has multiple uses across various industries. Because of its usefulness, this formula is something that learners should understand.

This article will share the rate of change formula and how you should solve them.

Average Rate of Change Formula

In mathematics, the average rate of change formula describes the change of one figure in relation to another. In every day terms, it's employed to determine the average speed of a change over a specific period of time.

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

R = Δy / Δx

This computes the change of y compared to the change of x.

The variation within the numerator and denominator is shown by the greek letter Δ, read 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 expressed as:

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

Average Rate of Change = Slope

Plotting out these numbers in a X Y graph, is beneficial when working with differences in value A in comparison with 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 figures is equivalent to the slope of the function.

This is why the average rate of change of a function is the slope of the secant line going through two arbitrary 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 understand the slope formula and what the values mean, finding the average rate of change of the function is achievable.

To make grasping this concept less complex, here are the steps you must obey to find the average rate of change.

Step 1: Find Your Values

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

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

In this instance, then you have to locate the values on the x and y-axis. Coordinates are generally provided 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

Find the Δx and Δy values. As you can recollect, 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 figures in place, all that remains 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 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 stated previously, the rate of change is applicable to multiple different scenarios. The aforementioned examples were applicable to the rate of change of a linear equation, but this formula can also be used in functions.

The rate of change of function follows a similar rule but with a distinct formula because of 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

If you can recall, the average rate of change of any two values can be plotted on a graph. The R-value, therefore is, equal 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 decreasing in value. For example, rate of change can be negative, which results in a declining 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 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 ascending.

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 the given example, all we have to do is a simple substitution because the delta values are already specified.

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 axis.

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

Example 3

Extract 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, solve for the values of the functions in the equation. In this situation, we simply replace 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

Once we have all our values, all we need to 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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