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 across academics, most notably in chemistry, physics and accounting.

It’s most frequently utilized when discussing velocity, although it has many uses across various industries. Due to its usefulness, this formula is a specific concept that students 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 shows the variation of one value 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.

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

R = Δy / Δx

This computes the variation 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 further portrayed as the difference between 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 numbers in a Cartesian plane, is useful when talking about differences in value A versus value B.

The straight line that connects 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 summation, in a linear function, the average rate of change between two figures 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 intersecting two arbitrary 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 have discussed the slope formula and what the values mean, finding the average rate of change of the function is feasible.

To make learning this principle less complex, here are the steps you should keep in mind to find the average rate of change.

Step 1: Find Your Values

In these types of equations, mathematical scenarios generally provide you with two sets of values, from which you solve to find 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 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

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 obtained 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 numbers inputted, all that is left is to simplify the equation by deducting 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 replacing all our values and simplifying the equation, we get the average rate of change for the two coordinates that we were provided.

Average Rate of Change of a Function

As we’ve shared before, the rate of change is pertinent to numerous diverse situations. The previous examples focused on the rate of change of a linear equation, but this formula can also be used in functions.

The rate of change of function obeys the same principle 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 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 plotted. The R-value, is, identical to its slope.

Every so often, the equation results in a slope that is negative. This indicates that the line is descending from left to right in the X Y axis.

This translates to 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 contrary, a positive slope shows that the object’s rate of change is positive. This shows us that the object is gaining value, and the secant line is trending upward from left to right. In terms of our aforementioned example, if an object has positive velocity and its position is ascending.

Examples of Average Rate of Change

Next, we will review 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 the given example, all we need to do is a simple substitution since the delta values are already specified.

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 Cartesian plane.

For this example, we still have to find 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 equivalent 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 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, determine the values of the functions in the equation. In this instance, we simply substitute the values on the equation using 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

Once we have all our values, all we must 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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