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

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

It’s most often applied when discussing momentum, though it has many applications throughout various industries. Due to its usefulness, this formula is a specific concept that students should understand.

This article will share the rate of change formula and how you can solve it.

Average Rate of Change Formula

In mathematics, the average rate of change formula describes the change of one value when compared to another. In practical terms, it's used to define the average speed of a variation over a certain period of time.

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

R = Δy / Δx

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

The variation within the numerator and denominator is portrayed by the greek letter Δ, expressed as delta y and delta x. It is also portrayed as the variation between the first point and the second point of the value, or:

Δy = y2 - y1

Δx = x2 - x1

Because of this, the average rate of change equation can also be shown as:

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

Average Rate of Change = Slope

Plotting out these figures in a X Y graph, is helpful when reviewing dissimilarities in value A when compared to value B.

The straight line that joins 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 short, in a linear function, the average rate of change among two values is equal to the slope of the function.

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

To make learning this topic simpler, here are the steps you need to follow to find the average rate of change.

Step 1: Find Your Values

In these sort of equations, math problems usually offer you two sets of values, from which you extract x and y values.

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

In this case, next you have to find the values along the x and y-axis. Coordinates are generally 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

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 found 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 values plugged in, all that remains is to simplify the equation by deducting all the values. So, our equation then becomes the following.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As shown, just 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 stated previously, the rate of change is relevant to numerous 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 follows an identical rule but with a different formula because of the distinct values that functions have. This formula is:

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

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

Negative Slope

As you might remember, the average rate of change of any two values can be plotted on a graph. The R-value, is, identical to its slope.

Sometimes, the equation results in a slope that is negative. This means that the line is descending from left to right in the X Y graph.

This means that the rate of change is decreasing in value. For example, velocity can be negative, which means a decreasing position.

Positive Slope

In contrast, 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 previous example, if an object has positive velocity and its position is ascending.

Examples of Average Rate of Change

Next, we will run through the average rate of change formula via some examples.

Example 1

Extract the rate of change of the values where Δy = 10 and Δx = 2.

In this example, all we need to do is a plain substitution due to the fact that the delta values are already provided.

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 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 given, the average rate of change is the same as 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 third example will be extracting 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, calculate the values of the functions in the equation. In this situation, we simply substitute the values on the equation using 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

With all our values, all we must 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

Grade Potential Can Help You Get a Grip on Math

Mathematical can be a demanding topic to grasp, but it doesn’t have to be.

With Grade Potential, you can get set up with an expert teacher that will give you personalized support depending on your capabilities. With the quality of our teaching services, getting a grip on equations is as easy as one-two-three.

Call us now!