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

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

It’s most often utilized when discussing velocity, however it has numerous uses throughout various industries. Due to its value, this formula is a specific concept that students should understand.

This article will discuss the rate of change formula and how you can work with them.

Average Rate of Change Formula

In mathematics, the average rate of change formula denotes the change of one value in relation to another. In every day terms, it's utilized to define the average speed of a change over a certain period of time.

At its simplest, the rate of change formula is written as:

R = Δy / Δx

This computes the change of y compared to the variation of x.

The change through the numerator and denominator is represented by the greek letter Δ, read as delta y and delta x. It is also portrayed 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 values in a Cartesian plane, is helpful when talking about dissimilarities in value A versus 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 equal 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 random 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 possible.

To make studying this concept easier, here are the steps you must keep in mind to find the average rate of change.

Step 1: Determine Your Values

In these types of equations, math problems 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 locate the values along the x and y-axis. Coordinates are usually provided 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 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 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 numbers plugged in, all that is left is to simplify the equation by deducting all the numbers. Therefore, our equation then becomes the following.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As stated, by simply replacing all our values and simplifying the equation, we get the average rate of change for the two coordinates that we were given.

Average Rate of Change of a Function

As we’ve mentioned before, the rate of change is pertinent to many diverse situations. The aforementioned examples were more relevant to the rate of change of a linear equation, but this formula can also be applied to functions.

The rate of change of function obeys an identical principle but with a different formula because of the distinct 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

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 indicates that the line is trending downward from left to right in the X Y graph.

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

Positive Slope

On the other hand, 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. With regards to our previous example, if an object has positive velocity and its position is ascending.

Examples of Average Rate of Change

In this section, we will talk about 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 have to do is a plain substitution since the delta values are already specified.

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 graph.

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 extracting the rate of change of a function, calculate the values of the functions in the equation. In this case, we simply substitute 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

Now that we have all our values, all we have 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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