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

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

It’s most often utilized when discussing velocity, although it has numerous uses across various industries. Because of its utility, this formula is a specific concept that learners should learn.

This article will go over 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 change of one value in relation to another. In practice, it's used to determine the average speed of a variation over a specific period of time.

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

R = Δy / Δx

This computes the change 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 also denoted as the difference within the first point and the second point of the value, or:

Δy = y2 - y1

Δx = x2 - x1

Because of this, the average rate of change equation can also be expressed as:

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

Average Rate of Change = Slope

Plotting out these figures in a Cartesian plane, is useful when reviewing differences in value A when compared to value B.

The straight line that links these two points is called the 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 know the slope formula and what the figures mean, finding the average rate of change of the function is possible.

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

Step 1: Understand Your Values

In these equations, math questions generally give 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 case, next you have to find the values on the x and y-axis. Coordinates are typically provided in an (x, y) format, as you see in the example below:

x1 = 1

x2 = 3

y1 = 2

y2 = 4

Step 2: Subtract The Values

Find 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 all the values of x and y, we can input 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 deducting all the values. Therefore, our equation becomes something like this.

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 previously, the rate of change is relevant to numerous different situations. The previous examples were more relevant to the rate of change of a linear equation, but this formula can also be relevant for functions.

The rate of change of function observes 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 instance, the values provided will have one f(x) equation and one X Y axis value.

Negative Slope

As you might remember, the average rate of change of any two values can be plotted. The R-value, then is, equivalent to its slope.

Occasionally, 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 axis.

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

Positive Slope

In contrast, a positive slope means 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 ascending.

Examples of Average Rate of Change

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

Example 1

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

In this example, all we must do is a straightforward substitution because the delta values are already provided.

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 graph.

For this example, we still have to look 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 equal to the slope of the line joining two points.

Example 3

Find the rate of change of function f(x) = x2 + 5x - 3 on the interval [3, 5].

The third example will be extracting 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, solve for 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

Now that we have all our values, all we need to do is substitute 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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