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

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

It’s most frequently applied when talking about momentum, although it has numerous uses throughout different industries. Because of its value, this formula is a specific concept that students should understand.

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

Average Rate of Change Formula

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

At its simplest, the rate of change formula is expressed as:

R = Δy / Δx

This measures the change 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 further expressed as the difference between 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 X Y axis, is beneficial when reviewing dissimilarities in value A versus value B.

The straight line that links 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

In short, in a linear function, the average rate of change among two figures is equivalent to the slope of the function.

This is mainly why average rate of change of a function is the slope of the secant line intersecting 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 achievable.

To make understanding this principle simpler, here are the steps you need to follow to find the average rate of change.

Step 1: Find Your Values

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

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

In this situation, next you have to search for the values on 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

Calculate 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 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 figures plugged in, all that is left is to simplify the equation by subtracting all the numbers. So, our equation becomes 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 obtain the average rate of change for the two coordinates that we were given.

Average Rate of Change of a Function

As we’ve shared previously, the rate of change is pertinent to numerous diverse scenarios. The aforementioned 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 the same 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 recollect, the average rate of change of any two values can be plotted on a graph. The R-value, is, identical to its slope.

Occasionally, the equation results in a slope that is negative. This means 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 other hand, 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. With regards to our last example, if an object has positive velocity and its position is increasing.

Examples of Average Rate of Change

Now, 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 the given example, all we must do is a straightforward substitution because 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 axis.

For this example, we still have to search 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 you can see, the average rate of change is equal to the slope of the line connecting two points.

Example 3

Extract 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 calculating 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 using 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

Once we have 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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