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 throughout academics, specifically in chemistry, physics and finance.

It’s most frequently utilized when talking about momentum, although it has numerous applications across various industries. Because of its utility, this formula is something that learners should grasp.

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 change of one value when compared to another. In practice, it's employed to define the average speed of a variation over a specific period of time.

To put it simply, the rate of change formula is written as:

R = Δy / Δx

This calculates the change of y in comparison 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 also portrayed as the difference 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 values in a X Y graph, is beneficial when working with dissimilarities in value A when compared to 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 values 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 intersecting two random 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 understand the slope formula and what the values mean, finding the average rate of change of the function is feasible.

To make learning this topic simpler, here are the steps you need to keep in mind to find the average rate of change.

Step 1: Understand Your Values

In these equations, math scenarios usually 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, next you have to search for the values via the x and y-axis. Coordinates are typically given in an (x, y) format, as in this example:

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 found 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 values plugged in, all that remains is to simplify the equation by deducting all the values. Therefore, our equation will look something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As we can see, by plugging in 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 shared before, the rate of change is pertinent to many diverse situations. The aforementioned examples focused on the rate of change of a linear equation, but this formula can also be used in functions.

The rate of change of function observes an identical rule but with a distinct formula due to the distinct values that functions have. This formula is:

R = (f(b) - f(a)) / b - a

In this scenario, the values provided will have one f(x) equation and one Cartesian plane value.

Negative Slope

Previously if you recall, the average rate of change of any two values can be plotted on a graph. The R-value, therefore is, equivalent to its slope.

Sometimes, 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 axis.

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

Positive Slope

On the contrary, a positive slope indicates that the object’s rate of change is positive. This shows us 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 ascending.

Examples of Average Rate of Change

Now, we will run through 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 the given example, all we need to do is a simple substitution since the delta values are already specified.

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 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 the same as the slope of the line linking two points.

Example 3

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

The final 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 instance, 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

Now that 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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