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

The rate of change formula is one of the most widely used mathematical concepts across academics, especially in chemistry, physics and accounting.

It’s most often applied when discussing velocity, though it has multiple applications throughout different industries. Due to its usefulness, this formula is something that learners should understand.

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

Average Rate of Change Formula

In math, the average rate of change formula denotes the variation of one value in relation to another. In practical terms, it's employed to determine the average speed of a change over a specific period of time.

To put it simply, the rate of change formula is written as:

R = Δy / Δx

This calculates the change of y compared to the change of x.

The change within 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

As a result, the average rate of change equation can also be portrayed as:

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

Average Rate of Change = Slope

Plotting out these figures in a X Y graph, is useful when working with 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 summation, in a linear function, the average rate of change among two figures is equivalent 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. Meanwhile, 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 studying this principle easier, here are the steps you should keep in mind to find the average rate of change.

Step 1: Find Your Values

In these types of equations, mathematical scenarios generally give you two sets of values, from which you extract x and y values.

For example, let’s take 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 generally given in an (x, y) format, as in this example:

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 found 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 figures inputted, all that remains is to simplify the equation by deducting all the numbers. Therefore, our equation will look something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As we can see, by 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 previously, the rate of change is applicable to multiple diverse situations. The aforementioned examples focused on the rate of change of a linear equation, but this formula can also be relevant for functions.

The rate of change of function follows a similar principle but with a unique formula because of the different values that functions have. This formula is:

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

In this instance, the values provided will have one f(x) equation and one X Y graph value.

Negative Slope

As you might 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 denotes that the line is trending downward 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 means a declining position.

Positive Slope

On the contrary, a positive slope indicates 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 last example, if an object has positive velocity and its position is increasing.

Examples of Average Rate of Change

Next, 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 this example, all we must do is a straightforward substitution since 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 graph.

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 given, the average rate of change is the same as the slope of the line linking 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 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, determine the values of the functions in the equation. In this situation, we simply substitute 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

Once we have all our values, all we must 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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