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

The rate of change formula is one of the most important mathematical formulas throughout academics, most notably in chemistry, physics and accounting.

It’s most frequently used when discussing momentum, although it has numerous applications across many industries. Because of its value, this formula is something that learners should learn.

This article will discuss the rate of change formula and how you can solve it.

Average Rate of Change Formula

In mathematics, the average rate of change formula shows the variation of one value when compared to another. In practice, it's employed to identify the average speed of a change over a specified period of time.

To put it simply, the rate of change formula is written 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 represented by the greek letter Δ, read as delta y and delta x. It is further denoted 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 figures in a X Y graph, is beneficial when discussing differences in value A versus value B.

The straight line that joins 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 figures is the same as the slope of the function.

This is mainly why average rate of change of a function is the slope of the secant line going through two random 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 achievable.

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

Step 1: Determine Your Values

In these types of equations, math scenarios generally 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 scenario, then you have to find the values on 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 recall, 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 inputted, all that remains is to simplify the equation by subtracting all the values. So, our equation becomes something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As stated, just by replacing 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 shared earlier, the rate of change is applicable to many diverse situations. The previous examples were more relevant 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 the same rule 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 axis value.

Negative Slope

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

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

Positive Slope

On the contrary, a positive slope denotes 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. With regards to our last example, if an object has positive velocity and its position is increasing.

Examples of Average Rate of Change

Next, we will discuss the average rate of change formula via some examples.

Example 1

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

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

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 Cartesian plane.

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 identical to the slope of the line connecting two points.

Example 3

Find 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 extracting the rate of change of a function, determine the values of the functions in the equation. In this case, we simply substitute the values on the equation with the values specified 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 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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