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, most notably in physics, chemistry and accounting.

It’s most often utilized when talking about thrust, although it has multiple applications throughout various industries. Due to its value, this formula is something that students should grasp.

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

Average Rate of Change Formula

In math, the average rate of change formula denotes the change of one value in relation to another. In every day terms, it's utilized to determine the average speed of a change over a specific period of time.

To put it simply, the rate of change formula is expressed as:

R = Δy / Δx

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

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

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

Average Rate of Change = Slope

Plotting out these numbers in a Cartesian plane, is beneficial when discussing differences 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

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 passing through two random 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 understanding this topic easier, here are the steps you need to follow to find the average rate of change.

Step 1: Find 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 situation, next you have to find the values along the x and y-axis. Coordinates are generally provided in an (x, y) format, like this:

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 input the values as follows.

R = 4 - 2 / 3 - 1

Step 3: Simplify

With all of our numbers inputted, all that we have to do is to simplify the equation by subtracting all the numbers. Thus, our equation will look 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 mentioned before, the rate of change is applicable to many diverse 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 obeys the same principle but with a distinct formula due to the unique values that functions have. This formula is:

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

In this situation, the values provided will have one f(x) equation and one Cartesian plane value.

Negative Slope

If you can remember, the average rate of change of any two values can be graphed. The R-value, is, equivalent to its slope.

Every so often, the equation concludes in a slope that is negative. This indicates that the line is descending from left to right in the X Y graph.

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

Positive Slope

At the same time, a positive slope means that the object’s rate of change is positive. This means that the object is increasing in value, and the secant line is trending upward from left to right. With regards to our previous 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

Extract 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 since the delta values are already specified.

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 look for 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 you can see, the average rate of change is the same as the slope of the line joining two points.

Example 3

Find 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 calculating the rate of change of a function, calculate the values of the functions in the equation. In this case, we simply substitute the values on the equation using 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

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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