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

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

It’s most often used when talking about momentum, although it has numerous applications throughout various 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 describes the variation of one value when compared to another. In practical terms, it's used to define the average speed of a variation 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 change within the numerator and denominator is portrayed by the greek letter Δ, read as delta y and delta x. It is additionally expressed 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 shown as:

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

Average Rate of Change = Slope

Plotting out these figures in a X Y graph, is beneficial when working with differences in value A when compared to 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 between 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 random 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 possible.

To make understanding this principle less complex, here are the steps you must follow to find the average rate of change.

Step 1: Determine Your Values

In these types of equations, math scenarios typically 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 scenario, 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

Find 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 plug-in the values as follows.

R = 4 - 2 / 3 - 1

Step 3: Simplify

With all of our numbers plugged in, all that we have to do is to simplify the equation by subtracting all the values. Therefore, our equation then becomes the following.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As we can see, 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 provided.

Average Rate of Change of a Function

As we’ve shared earlier, the rate of change is pertinent to numerous different situations. 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 the same principle 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 situation, the values given will have one f(x) equation and one X Y axis value.

Negative Slope

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

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

This translates to the rate of change is diminishing in value. For example, rate of change can be negative, which results in a decreasing position.

Positive Slope

At the same time, a positive slope denotes that the object’s rate of change is positive. This means that the object is gaining 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

Now, we will talk about the average rate of change formula with some examples.

Example 1

Calculate 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 due to the fact that the delta values are already provided.

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 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 linking 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 finding 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, determine the values of the functions in the equation. In this situation, we simply replace the values on the equation with 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 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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