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

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

It’s most often used when talking about momentum, though it has many uses across various industries. Due to its usefulness, this formula is a specific concept that students should learn.

This article will go over 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 value in relation to another. In practical terms, it's employed to evaluate the average speed of a change over a certain period of time.

To put it simply, 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 change through the numerator and denominator is represented by the greek letter Δ, expressed as delta y and delta x. It is also denoted as the variation between 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 shown as:

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

Average Rate of Change = Slope

Plotting out these numbers in a X Y axis, is useful when discussing differences in value A versus 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 summation, in a linear function, the average rate of change between 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 going through two arbitrary endpoints on the graph of the function. At the same time, 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 possible.

To make studying this topic easier, here are the steps you need to keep in mind to find the average rate of change.

Step 1: Find Your Values

In these equations, math problems typically provide you with 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 scenario, next you have to find the values via the x and y-axis. Coordinates are typically 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 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 numbers plugged in, all that is left is to simplify the equation by subtracting all the numbers. Thus, our equation becomes something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As shown, just by plugging in 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 mentioned before, the rate of change is pertinent to many diverse situations. 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 observes a similar rule but with a unique formula due to the unique 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

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

Occasionally, the equation results 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, rate of change can be negative, which results in a declining position.

Positive Slope

On the other hand, a positive slope denotes 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. In relation to our previous example, if an object has positive velocity and its position is ascending.

Examples of Average Rate of Change

Now, we will talk about the average rate of change formula with some examples.

Example 1

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

In this example, all we must do is a straightforward 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 X Y axis.

For this example, we still have to search for 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 you can see, the average rate of change is equal to the slope of the line linking two points.

Example 3

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

The final example will be calculating 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, determine the values of the functions in the equation. In this instance, we simply replace the values on the equation with 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

Now that we have all our values, all we must do is substitute 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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