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

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

It’s most frequently utilized when talking about momentum, though it has many uses throughout different industries. Because of its utility, this formula is something that students should learn.

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

Average Rate of Change Formula

In math, the average rate of change formula denotes the variation of one figure when compared to another. In practice, it's employed to evaluate the average speed of a change over a specific period of time.

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

R = Δy / Δx

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

The change through the numerator and denominator is shown by the greek letter Δ, expressed as delta y and delta x. It is further denoted as the variation between 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 described as:

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

Average Rate of Change = Slope

Plotting out these figures in a X Y axis, is beneficial when working with differences in value A versus value B.

The straight line that connects 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 figures 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 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 figures mean, finding the average rate of change of the function is achievable.

To make studying this concept less complex, here are the steps you must keep in mind to find the average rate of change.

Step 1: Find Your Values

In these sort of equations, math problems typically provide you with 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, 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

Calculate 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 found all the values of x and y, we can add the values as follows.

R = 4 - 2 / 3 - 1

Step 3: Simplify

With all of our values inputted, all that is left is to simplify the equation by subtracting all the values. Thus, our equation becomes something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As we can see, just by replacing 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 previously, the rate of change is relevant to multiple different situations. The aforementioned examples were applicable to 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 distinct formula because of the unique values that functions have. This formula is:

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

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

Negative Slope

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

Sometimes, the equation concludes 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 results in a declining position.

Positive Slope

On the contrary, a positive slope shows that the object’s rate of change is positive. This means that the object is gaining value, and the secant line is trending upward from left to right. In terms of our aforementioned 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 through some examples.

Example 1

Calculate 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 since the delta values are already provided.

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 Cartesian plane.

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 you can see, the average rate of change is equal to the slope of the line connecting 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 finding 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 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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