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, specifically in physics, chemistry and finance.

It’s most frequently utilized when talking about velocity, however it has multiple applications across different industries. Due to its utility, this formula is something that learners should grasp.

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 in relation to another. In practice, it's utilized to evaluate the average speed of a variation over a specific period of time.

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

R = Δy / Δx

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

The variation within the numerator and denominator is portrayed by the greek letter Δ, read as delta y and delta x. It is further expressed as the difference within 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 expressed as:

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

Average Rate of Change = Slope

Plotting out these numbers in a X Y graph, is helpful when reviewing dissimilarities in value A when compared to value B.

The straight line that joins 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 between 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 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 understand the slope formula and what the values mean, finding the average rate of change of the function is achievable.

To make grasping 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 sort of equations, math questions usually offer you two sets of values, from which you will get x and y values.

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

In this instance, next you have to find the values via the x and y-axis. Coordinates are typically provided in an (x, y) format, as in this example:

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 plug-in the values as follows.

R = 4 - 2 / 3 - 1

Step 3: Simplify

With all of our figures inputted, all that is left is to simplify the equation by subtracting all the values. So, our equation then becomes the following.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As we can see, by simply replacing all our values and simplifying the equation, we obtain the average rate of change for the two coordinates that we were provided.

Average Rate of Change of a Function

As we’ve shared earlier, the rate of change is pertinent to multiple different 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 an identical rule but with a different formula because of the different values that functions have. This formula is:

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

In this scenario, the values provided will have one f(x) equation and one Cartesian plane value.

Negative Slope

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

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

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

Positive Slope

At the same time, a positive slope shows that the object’s rate of change is positive. This tells us that the object is increasing in 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 increasing.

Examples of Average Rate of Change

Now, we will run through the average rate of change formula with some examples.

Example 1

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

In this example, all we need to do is a simple 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 search 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 given, the average rate of change is equal 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 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 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

With all our values, all we have to 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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