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

# 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, specifically in chemistry, physics and finance.

It’s most frequently used when talking about velocity, however it has many applications throughout many industries. Because of its utility, this formula is a specific concept that students should grasp.

This article will go over the rate of change formula and how you should solve them.

## Average Rate of Change Formula

In mathematics, the average rate of change formula describes the variation of one value in relation to another. In practical terms, it's employed to define the average speed of a change over a certain period of time.

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

R = Δy / Δx

This computes the change of y compared to the change of x.

The change through the numerator and denominator is portrayed by the greek letter Δ, expressed as delta y and delta x. It is additionally expressed as the difference within 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 working with differences in value A versus value B.

The straight line that links 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 summation, in a linear function, the average rate of change among 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. 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 have discussed the slope formula and what the values mean, finding the average rate of change of the function is possible.

To make grasping this principle less complex, here are the steps you must obey to find the average rate of change.

### Step 1: Find Your Values

In these equations, mathematical scenarios usually give you 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 instance, then you have to find the values along the x and y-axis. Coordinates are usually 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 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 subtracting all the numbers. So, our equation will look something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As stated, 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 previously, the rate of change is relevant to multiple different situations. The previous examples focused on 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 distinct formula because of the distinct 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 plotted. The R-value, therefore is, equivalent to its slope.

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

This translates to 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 indicates that the object’s rate of change is positive. This means that the object is increasing in value, and the secant line is trending upward from left to right. In terms of our previous example, if an object has positive velocity and its position is ascending.

## Examples of Average Rate of Change

In this section, we will review the average rate of change formula via some examples.

### Example 1

Extract the rate of change of the values where Δy = 10 and Δx = 2.

In this example, all we have to do is a straightforward substitution since the delta values are already provided.

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 axis.

For this example, we still have to find 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 you can see, the average rate of change is equivalent to 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 third example will be extracting 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, 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 have to 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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