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

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

It’s most often used when talking about thrust, though it has multiple applications throughout various industries. Due to its value, this formula is a specific concept that learners should understand.

This article will share 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 shows the variation of one figure when compared to another. In practical terms, it's used to define 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 variation of y compared 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 further denoted as the difference 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 portrayed as:

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

Average Rate of Change = Slope

Plotting out these numbers in a X Y axis, is helpful when working with 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

To summarize, in a linear function, the average rate of change between two figures is equal 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 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 feasible.

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

Step 1: Determine Your Values

In these types of equations, math scenarios usually offer you 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 scenario, then you have to search for the values via the x and y-axis. Coordinates are usually 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

Calculate 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 obtained 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 plugged in, all that we have to do is to simplify the equation by deducting all the numbers. Thus, our equation will look something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As shown, by simply plugging in all our values and simplifying the equation, we get the average rate of change for the two coordinates that we were provided.

Average Rate of Change of a Function

As we’ve mentioned earlier, the rate of change is pertinent to numerous diverse 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 the same principle but with a distinct formula because of the different 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 graph value.

Negative Slope

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

Occasionally, the equation concludes in a slope that is negative. This means that the line is descending from left to right in the X Y axis.

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

Positive Slope

In contrast, a positive slope indicates that the object’s rate of change is positive. This shows us that the object is gaining value, and the secant line is trending upward from left to right. In relation to our aforementioned example, if an object has positive velocity and its position is ascending.

Examples of Average Rate of Change

In this section, we will review 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 the given example, all we must do is a plain substitution due to the fact that the delta values are already specified.

R = Δy / Δx

R = 10 / 2

R = 5

Example 2

Find 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 employing the average rate of change formula.

R = y2 - y1 / x2 - x1

R = (14 - 6) / (3 - 1)

R = 8 / 2

R = 4

As provided, the average rate of change is identical to the slope of the line joining two points.

Example 3

Extract 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 calculating the rate of change of a function, solve for the values of the functions in the equation. In this case, 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 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

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