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

The rate of change formula is one of the most widely used mathematical principles throughout academics, specifically in chemistry, physics and accounting.

It’s most frequently applied when talking about velocity, however it has multiple uses across various industries. Due to its value, this formula is something that learners should grasp.

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

Average Rate of Change Formula

In math, the average rate of change formula describes the variation of one figure when compared to another. In practical terms, it's utilized to identify the average speed of a variation over a certain period of time.

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

R = Δy / Δx

This calculates the variation of y in comparison 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 additionally portrayed as the variation within 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 described as:

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

Average Rate of Change = Slope

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

The straight line that links 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 short, in a linear function, the average rate of change between two values is the same as 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. 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 have discussed the slope formula and what the figures mean, finding the average rate of change of the function is feasible.

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

Step 1: Determine Your Values

In these equations, math scenarios typically provide you with 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, next you have to search for 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 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 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 plugged in, all that remains is to simplify the equation by subtracting all the numbers. Therefore, our equation becomes something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

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

Average Rate of Change of a Function

As we’ve stated before, the rate of change is pertinent to many 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 follows an identical rule but with a unique formula due to the distinct 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 graph value.

Negative Slope

Previously if you remember, the average rate of change of any two values can be plotted on a graph. The R-value, then is, equivalent to its slope.

Every so often, the equation results in a slope that is negative. This denotes 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, rate of change can be negative, which means a decreasing 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 increasing.

Examples of Average Rate of Change

Next, 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 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

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

The final example will be finding the rate of change of a function with the formula:

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

When calculating the rate of change of a function, calculate the values of the functions in the equation. In this situation, we simply replace the values on the equation with the values given 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 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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