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

The rate of change formula is one of the most used math principles across academics, particularly in physics, chemistry and finance.

It’s most frequently applied when talking about momentum, however it has numerous applications across various industries. Because of its usefulness, this formula is a specific concept 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 figure in relation to another. In practice, it's utilized to identify the average speed of a change over a specified period of time.

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

R = Δy / Δx

This calculates the variation of y in comparison to the variation 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 additionally denoted as the variation within 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 beneficial when working with differences in value A versus 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

To summarize, in a linear function, the average rate of change between 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 arbitrary endpoints on the graph of the function. Meanwhile, 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 feasible.

To make studying this topic easier, here are the steps you should obey to find the average rate of change.

Step 1: Determine Your Values

In these sort of equations, math problems generally offer you two sets of values, from which you extract x and y values.

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

In this situation, then you have to locate the values via the x and y-axis. Coordinates are typically given in an (x, y) format, like this:

x1 = 1

x2 = 3

y1 = 2

y2 = 4

Step 2: Subtract The Values

Find the Δx and Δy values. As you may remember, 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 numbers inputted, all that we have to do is to simplify the equation by subtracting all the values. So, our equation will look something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As we can see, just by replacing all our values and simplifying the equation, we get the average rate of change for the two coordinates that we were given.

Average Rate of Change of a Function

As we’ve stated before, the rate of change is relevant to many different scenarios. The aforementioned 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 obeys the same principle 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 Cartesian plane value.

Negative Slope

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

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

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

Positive Slope

On the contrary, a positive slope indicates that the object’s rate of change is positive. This tells us that the object is increasing in value, and the secant line is trending upward from left to right. In relation to our previous example, if an object has positive velocity and its position is ascending.

Examples of Average Rate of Change

Now, we will review the average rate of change formula through some examples.

Example 1

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

In the given example, all we have to do is a plain substitution since 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 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 given, the average rate of change is equal to the slope of the line joining two points.

Example 3

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

The last example will be calculating 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, solve for the values of the functions in the equation. In this case, we simply substitute the values on the equation using 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 must 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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