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

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

It’s most often used when discussing velocity, though it has many uses across different 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 work with them.

Average Rate of Change Formula

In math, the average rate of change formula describes the change of one value in relation to another. In practice, it's employed to identify 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 measures the change of y in comparison to the variation of x.

The change through the numerator and denominator is represented 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

As a result, 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 graph, is helpful when talking about dissimilarities 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 among two values is equal 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 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 figures mean, finding the average rate of change of the function is feasible.

To make studying this concept less complex, here are the steps you need to follow to find the average rate of change.

Step 1: Determine Your Values

In these equations, mathematical scenarios typically 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 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

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 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 numbers in place, all that remains is to simplify the equation by deducting all the numbers. So, our equation then becomes the following.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As shown, 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 mentioned before, the rate of change is applicable to many different situations. 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 observes an identical rule 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 instance, the values given will have one f(x) equation and one X Y graph value.

Negative Slope

As you might remember, the average rate of change of any two values can be plotted. The R-value, then is, equivalent to its slope.

Every so often, the equation concludes in a slope that is negative. This indicates that the line is trending downward from left to right in the X Y graph.

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

Positive Slope

In contrast, a positive slope indicates that the object’s rate of change is positive. This tells us that the object is gaining value, and the secant line is trending upward from left to right. In terms of our last example, if an object has positive velocity and its position is increasing.

Examples of Average Rate of Change

Next, we will run through the average rate of change formula through some examples.

Example 1

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

In this example, all we need to do is a straightforward substitution due to the fact that the delta values are already specified.

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 Cartesian plane.

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

Calculate 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 extracting the rate of change of a function, determine the values of the functions in the equation. In this situation, we simply replace 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 need to 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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