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

The rate of change formula is one of the most important math principles across academics, most notably in physics, chemistry and accounting.

It’s most frequently applied when talking about velocity, though it has many applications throughout various industries. Due to its usefulness, this formula is something that students should understand.

This article will discuss 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 variation of one value when compared to another. In practical terms, it's utilized to define the average speed of a variation over a specific period of time.

To put it simply, the rate of change formula is written as:

R = Δy / Δx

This measures the variation of y in comparison to the variation of x.

The variation through the numerator and denominator is represented by the greek letter Δ, expressed 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

Consequently, the average rate of change equation can also be expressed 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 when compared to value B.

The straight line that joins 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

In short, in a linear function, the average rate of change between two figures 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 intersecting two arbitrary endpoints on the graph of the function. Simultaneously, 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 grasping this topic less complex, here are the steps you should follow to find the average rate of change.

Step 1: Determine Your Values

In these types of equations, math questions usually give you two sets of values, from which you solve to find x and y values.

For example, let’s assume the values (1, 2) and (3, 4).

In this case, then you have to search for the values via the x and y-axis. Coordinates are typically given in an (x, y) format, as in this example:

x1 = 1

x2 = 3

y1 = 2

y2 = 4

Step 2: Subtract The Values

Find 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 obtained 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 figures inputted, all that we have to do is to simplify the equation by subtracting all the numbers. Therefore, our equation then becomes the following.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As shown, by plugging in all our values and simplifying the equation, we obtain the average rate of change for the two coordinates that we were given.

Average Rate of Change of a Function

As we’ve shared before, the rate of change is relevant to multiple diverse scenarios. 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 follows a similar principle but with a unique formula because of the unique 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 remember, the average rate of change of any two values can be graphed. The R-value, is, equal to its slope.

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

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

Positive Slope

On the contrary, a positive slope denotes that the object’s rate of change is positive. This tells us that the object is gaining value, and the secant line is trending upward from left to right. In relation to 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

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 straightforward substitution since the delta values are already given.

R = Δy / Δx

R = 10 / 2

R = 5

Example 2

Find 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 provided, the average rate of change is the same as the slope of the line joining two points.

Example 3

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

The last example will be finding 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, calculate the values of the functions in the equation. In this case, we simply replace 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

Once 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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