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 accounting.

It’s most frequently applied when discussing momentum, although it has multiple uses throughout different industries. Due to its utility, this formula is something that learners should grasp.

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

Average Rate of Change Formula

In math, the average rate of change formula describes the change of one figure in relation to another. In every day terms, it's employed to determine the average speed of a change over a specific period of time.

Simply put, the rate of change formula is expressed as:

R = Δy / Δx

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

The change through the numerator and denominator is represented by the greek letter Δ, expressed as delta y and delta x. It is also 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 portrayed as:

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

Average Rate of Change = Slope

Plotting out these figures in a Cartesian plane, is beneficial when working with 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

To summarize, in a linear function, the average rate of change among two values is equivalent to the slope of the function.

This is the reason why the 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 understand the slope formula and what the figures mean, finding the average rate of change of the function is feasible.

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

Step 1: Determine Your Values

In these sort of equations, math scenarios typically 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 instance, next you have to find the values along the x and y-axis. Coordinates are usually provided 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 in place, all that we have to do is to simplify the equation by deducting all the values. Thus, our equation then becomes the following.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As we can see, by 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 applicable to multiple diverse situations. The previous 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 observes a similar rule 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 case, the values given 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 plotted. The R-value, is, equal to its slope.

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

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

Positive Slope

At the same time, 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. In terms of our last example, if an object has positive velocity and its position is ascending.

Examples of Average Rate of Change

In this section, we will run through the average rate of change formula with some examples.

Example 1

Extract 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 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 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 you can see, the average rate of change is identical 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 final 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, determine the values of the functions in the equation. In this situation, we simply substitute the values on the equation using 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

With all our values, all we must 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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