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

The rate of change formula is one of the most widely used mathematical concepts throughout academics, specifically in chemistry, physics and accounting.

It’s most frequently applied when discussing momentum, however it has numerous applications across different industries. Due to its utility, this formula is something that students should understand.

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

Average Rate of Change Formula

In mathematics, the average rate of change formula describes the change of one figure in relation to another. In practice, it's used to determine the average speed of a variation over a specified period of time.

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

R = Δy / Δx

This measures the variation of y compared to the variation of x.

The variation through the numerator and denominator is portrayed by the greek letter Δ, read as delta y and delta x. It is additionally denoted as the difference between 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 described as:

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

Average Rate of Change = Slope

Plotting out these values in a Cartesian plane, is beneficial when reviewing dissimilarities in value A in comparison with value B.

The straight line that links 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

In short, in a linear function, the average rate of change among two values is the same as 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 random 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 values mean, finding the average rate of change of the function is achievable.

To make studying this principle simpler, 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 problems typically give 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, next you have to locate the values on the x and y-axis. Coordinates are typically provided 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 found all the values of x and y, we can add 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. Thus, our equation becomes something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As stated, 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 stated before, the rate of change is pertinent to multiple diverse situations. The aforementioned examples focused on the rate of change of a linear equation, but this formula can also be applied to functions.

The rate of change of function follows the same 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 case, the values provided will have one f(x) equation and one X Y graph value.

Negative Slope

If you can recollect, the average rate of change of any two values can be graphed. The R-value, is, identical to its slope.

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

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

Positive Slope

On the other hand, 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. In relation to our aforementioned example, if an object has positive velocity and its position is increasing.

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 this example, all we must do is a plain substitution because the delta values are already given.

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 using 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 equal to the slope of the line connecting two points.

Example 3

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

The final 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, calculate the values of the functions in the equation. In this case, we simply substitute the values on the equation with 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

Now that we have all our values, all we have 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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