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

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

It’s most frequently used when talking about momentum, however it has many applications throughout various industries. Due to its usefulness, this formula is a specific concept that learners should learn.

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

Average Rate of Change Formula

In mathematics, the average rate of change formula shows the change of one figure in relation to another. In practice, it's utilized to evaluate the average speed of a variation over a certain period of time.

Simply put, the rate of change formula is expressed as:

R = Δy / Δx

This computes the variation of y compared 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 additionally portrayed 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 described as:

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

Average Rate of Change = Slope

Plotting out these figures in a Cartesian plane, is helpful when reviewing dissimilarities in value A when compared to value B.

The straight line that links these two points is 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 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. 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 achievable.

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

Step 1: Understand Your Values

In these equations, math questions typically offer you 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, then you have to search for the values via 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

Find 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 input the values as follows.

R = 4 - 2 / 3 - 1

Step 3: Simplify

With all of our values plugged in, all that is left is to simplify the equation by deducting all the values. 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 get the average rate of change for the two coordinates that we were provided.

Average Rate of Change of a Function

As we’ve stated earlier, the rate of change is applicable to numerous diverse scenarios. The aforementioned examples were more relevant to the rate of change of a linear equation, but this formula can also be used in functions.

The rate of change of function observes a similar 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 situation, the values given will have one f(x) equation and one X Y axis value.

Negative Slope

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

Every so often, the equation results in a slope that is negative. This denotes that the line is descending from left to right in the X Y axis.

This translates to the rate of change is decreasing in value. For example, rate of change can be negative, which results in a decreasing position.

Positive Slope

On the other hand, a positive slope denotes that the object’s rate of change is positive. This tells us that the object is gaining value, and the secant line is trending upward from left to right. With regards to our aforementioned example, if an object has positive velocity and its position is increasing.

Examples of Average Rate of Change

In this section, we will review the average rate of change formula via 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 plain 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 X Y axis.

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 given, the average rate of change is equal to 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 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, calculate the values of the functions in the equation. In this case, we simply replace the values on the equation with the values provided 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 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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