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 throughout academics, specifically in chemistry, physics and accounting.

It’s most often used when talking about velocity, however it has multiple uses across different industries. Because of its value, this formula is something that students should understand.

This article will discuss 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 variation of one value in relation to another. In practical terms, it's utilized to evaluate the average speed of a change 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 variation through the numerator and denominator is portrayed by the greek letter Δ, expressed as delta y and delta x. It is further 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 numbers in a X Y graph, is helpful when discussing dissimilarities in value A in comparison with 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

To summarize, 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 going through 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 know the slope formula and what the values mean, finding the average rate of change of the function is possible.

To make understanding this concept easier, here are the steps you need to keep in mind to find the average rate of change.

Step 1: Understand Your Values

In these types of equations, math questions typically provide you with 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 case, then you have to locate the values via the x and y-axis. Coordinates are generally 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 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 figures plugged in, all that is left is to simplify the equation by deducting all the numbers. Thus, our equation then becomes the following.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As shown, just by replacing all our values and simplifying the equation, we obtain the average rate of change for the two coordinates that we were provided.

Average Rate of Change of a Function

As we’ve mentioned earlier, the rate of change is pertinent to numerous diverse scenarios. The previous examples were applicable to the rate of change of a linear equation, but this formula can also be relevant for functions.

The rate of change of function follows an identical 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 situation, the values provided will have one f(x) equation and one Cartesian plane value.

Negative Slope

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

Occasionally, the equation results in a slope that is negative. This means that the line is trending downward from left to right in the X Y graph.

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

In contrast, a positive slope means 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 last example, if an object has positive velocity and its position is increasing.

Examples of Average Rate of Change

In this section, we will talk about 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 need to do is a plain substitution due to the fact that the delta values are already given.

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 using 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

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

The third example will be calculating 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, calculate the values of the functions in the equation. In this instance, 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

With all our values, all we must do is replace 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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