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

The rate of change formula is one of the most important mathematical concepts throughout academics, especially in physics, chemistry and finance.

It’s most frequently utilized when discussing momentum, however it has multiple applications throughout various industries. Due to its usefulness, this formula is a specific concept that learners should grasp.

This article will share the rate of change formula and how you should solve them.

Average Rate of Change Formula

In math, the average rate of change formula shows the variation of one figure in relation to another. In practice, it's utilized to identify the average speed of a variation over a specific period of time.

At its simplest, 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 within the numerator and denominator is represented by the greek letter Δ, read as delta y and delta x. It is additionally portrayed 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 described as:

R = (y2 - y1) / (x2 - x1)

Average Rate of Change = Slope

Plotting out these figures in a X Y graph, is useful when reviewing 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 among two values 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 passing through two random 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 know the slope formula and what the figures mean, finding the average rate of change of the function is possible.

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

Step 1: Understand Your Values

In these equations, math scenarios generally offer 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 case, then you have to search for the values via the x and y-axis. Coordinates are generally provided 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 obtained all the values of x and y, we can input 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. So, our equation becomes 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 provided.

Average Rate of Change of a Function

As we’ve mentioned earlier, the rate of change is pertinent to many diverse 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 follows the same 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

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

Occasionally, the equation results 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 diminishing in value. For example, velocity can be negative, which results in a decreasing position.

Positive Slope

At the same time, a positive slope denotes 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. With regards to our previous 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 through some examples.

Example 1

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

In this example, all we must do is a simple substitution due to the fact that 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 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 provided, the average rate of change is equivalent to the slope of the line linking two points.

Example 3

Find 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 calculating the rate of change of a function, solve for the values of the functions in the equation. In this instance, we simply replace the values on the equation with 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

With 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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