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 accounting.

It’s most often applied when discussing momentum, however it has many applications across different industries. Due to its utility, this formula is a specific concept that learners should grasp.

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 variation of one value in relation to another. In practical terms, it's used to define the average speed of a change 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 shown by the greek letter Δ, read as delta y and delta x. It is further denoted 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 portrayed as:

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

Average Rate of Change = Slope

Plotting out these values in a Cartesian plane, is useful when reviewing differences in value A in comparison with value B.

The straight line that links these two points is called the 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 equal to the slope of the function.

This is mainly why 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 values mean, finding the average rate of change of the function is possible.

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

Step 1: Find Your Values

In these sort of equations, math problems generally give you two sets of values, from which you will get x and y values.

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

In this situation, next you have to find the values on 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

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 found 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 figures in place, all that is left 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 stated, just 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 relevant to numerous diverse situations. The aforementioned examples were applicable to the rate of change of a linear equation, but this formula can also be applied to functions.

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

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

In this instance, the values provided will have one f(x) equation and one X Y axis value.

Negative Slope

If you can recall, the average rate of change of any two values can be plotted. The R-value, is, identical to its slope.

Every so often, the equation results in a slope that is negative. This indicates that the line is descending 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

On the other hand, a positive slope shows 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. In relation to our previous example, if an object has positive velocity and its position is increasing.

Examples of Average Rate of Change

Next, we will talk about 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 need to do is a plain 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 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 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 finding 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 replace 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

Once we have all our values, all we must 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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