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

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

It’s most frequently used when discussing velocity, although it has numerous applications throughout many industries. Due to its usefulness, this formula is a specific concept that learners should grasp.

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 shows the change of one figure when compared to another. In practical terms, it's utilized to define the average speed of a change over a certain period of time.

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

R = Δy / Δx

This measures the variation of y in comparison to the change of x.

The variation through the numerator and denominator is shown by the greek letter Δ, read as delta y and delta x. It is additionally denoted as the variation 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 expressed as:

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

Average Rate of Change = Slope

Plotting out these figures in a X Y graph, is helpful when talking about dissimilarities in value A in comparison with value B.

The straight line that joins 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

In short, in a linear function, the average rate of change among two figures is equivalent 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. 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 know the slope formula and what the values mean, finding the average rate of change of the function is possible.

To make learning this concept easier, here are the steps you need to obey to find the average rate of change.

Step 1: Determine Your Values

In these equations, math scenarios typically offer you two sets of values, from which you extract x and y values.

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

In this instance, then you have to find the values via the x and y-axis. Coordinates are usually given in an (x, y) format, like this:

x1 = 1

x2 = 3

y1 = 2

y2 = 4

Step 2: Subtract The Values

Calculate 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 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 in place, 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 stated, 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 earlier, the rate of change is relevant to numerous different situations. The previous examples were more relevant to the rate of change of a linear equation, but this formula can also be relevant for functions.

The rate of change of function obeys an identical rule but with a different formula due to the distinct 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 Cartesian plane value.

Negative Slope

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

Sometimes, the equation results in a slope that is negative. This means that the line is descending from left to right in the Cartesian plane.

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

Positive Slope

At the same time, a positive slope shows 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 terms of our last example, if an object has positive velocity and its position is increasing.

Examples of Average Rate of Change

Now, we will run through 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 provided.

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 utilizing 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 equivalent 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 third 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 instance, we simply replace 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 need 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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