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

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

It’s most frequently applied when discussing velocity, although it has multiple uses across many industries. Due to its usefulness, this formula is something that learners should understand.

This article will share 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 figure when compared to another. In every day terms, it's used to determine the average speed of a variation over a certain period of time.

To put it simply, 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 within the numerator and denominator is portrayed by the greek letter Δ, read as delta y and delta x. It is additionally portrayed as the variation 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 values in a X Y graph, is helpful when reviewing dissimilarities in value A in comparison with value B.

The straight line that connects 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 intersecting 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 understand the slope formula and what the figures mean, finding the average rate of change of the function is achievable.

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

Step 1: Determine Your Values

In these types of equations, math scenarios usually give you two sets of values, from which you solve to find x and y values.

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

In this instance, then you have to locate the values along the x and y-axis. Coordinates are typically given in an (x, y) format, as you see in the example below:

x1 = 1

x2 = 3

y1 = 2

y2 = 4

Step 2: Subtract The Values

Find 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 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 values in place, 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 shown, by plugging in 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 mentioned before, 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 used in functions.

The rate of change of function obeys a similar 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 scenario, the values provided will have one f(x) equation and one Cartesian plane value.

Negative Slope

If you can remember, the average rate of change of any two values can be plotted on a graph. The R-value, therefore is, equal to its slope.

Occasionally, 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 graph.

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

Positive Slope

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

Examples of Average Rate of Change

In this section, 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 must do is a straightforward substitution since the delta values are already given.

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 X Y axis.

For this example, we still have to look for the Δy and Δx values by employing 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 equivalent to the slope of the line connecting 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 extracting 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, determine the values of the functions in the equation. In this situation, we simply substitute the values on the equation with the values given 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 plug in 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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