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

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

It’s most frequently applied when discussing thrust, though it has numerous uses across many industries. Because of its usefulness, this formula is something that learners should grasp.

This article will discuss 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 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 calculates the change of y in comparison to the change of x.

The variation through the numerator and denominator is shown by the greek letter Δ, expressed as delta y and delta x. It is also portrayed 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 figures in a Cartesian plane, is helpful when discussing dissimilarities in value A versus value B.

The straight line that connects 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

To summarize, in a linear function, the average rate of change among two figures is equivalent to the slope of the function.

This is mainly why 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. In the meantime, 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 figures mean, finding the average rate of change of the function is feasible.

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

Step 1: Find Your Values

In these types of equations, math questions typically offer you two sets of values, from which you will get x and y values.

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

In this case, next you have to find the values via the x and y-axis. Coordinates are usually 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 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 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 numbers. Therefore, our equation becomes something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As stated, by simply plugging in 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 shared before, the rate of change is applicable to multiple diverse scenarios. The previous examples were applicable to the rate of change of a linear equation, but this formula can also be applied to functions.

The rate of change of function follows the same principle but with a distinct formula due to the different values that functions have. This formula is:

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

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

Negative Slope

As you might recall, the average rate of change of any two values can be plotted. 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 trending downward from left to right in the Cartesian plane.

This means that the rate of change is diminishing in value. For example, velocity can be negative, which results in a declining 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. With regards to our last example, if an object has positive velocity and its position is increasing.

Examples of Average Rate of Change

Now, we will review 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 need to do is a plain substitution since the delta values are already given.

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 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 equivalent 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 last 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 replace 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

Now that we have all our values, all we have 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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