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

The rate of change formula is one of the most important mathematical principles throughout academics, especially in physics, chemistry and accounting.

It’s most often utilized when talking about momentum, although it has many applications across many industries. Due to its usefulness, this formula is a specific concept that students should learn.

This article will discuss the rate of change formula and how you should solve them.

Average Rate of Change Formula

In math, the average rate of change formula denotes the change of one value when compared to another. In every day terms, it's utilized to evaluate 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 in comparison 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 further denoted as the variation 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 described as:

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

Average Rate of Change = Slope

Plotting out these values in a Cartesian plane, is useful when reviewing differences in value A versus 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

In summation, in a linear function, the average rate of change between two values 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 going through two random 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 feasible.

To make studying this principle easier, here are the steps you need to follow to find the average rate of change.

Step 1: Understand Your Values

In these types of equations, math questions usually provide you with 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, then you have to locate the values on the x and y-axis. Coordinates are usually 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 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 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 values inputted, all that is left is to simplify the equation by deducting all the numbers. Thus, our equation becomes something like this.

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 achieve the average rate of change for the two coordinates that we were given.

Average Rate of Change of a Function

As we’ve stated previously, the rate of change is applicable to numerous different scenarios. The previous 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 obeys a similar rule but with a unique formula due to the unique values that functions have. This formula is:

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

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

Negative Slope

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

Occasionally, the equation concludes in a slope that is negative. This denotes that the line is trending downward from left to right in the X Y axis.

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

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 the given example, all we have to do is a straightforward substitution because 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 axis.

For this example, we still have to find the Δy and Δx values by employing 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 equal to 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 finding the rate of change of a function, calculate the values of the functions in the equation. In this instance, we simply substitute 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

With all our values, all we need 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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