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

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

It’s most often applied when talking about velocity, though it has many uses across different industries. Due to its value, this formula is something that students 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 practice, it's employed to identify the average speed of a variation over a specified period of time.

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

R = Δy / Δx

This computes the variation of y in comparison to the change of x.

The change through the numerator and denominator is shown by the greek letter Δ, expressed as delta y and delta x. It is further expressed as the difference between 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 numbers in a X Y graph, is useful when working with dissimilarities in value A in comparison with value B.

The straight line that joins 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 the reason why the average rate of change of a function is the slope of the secant line passing through two arbitrary endpoints on the graph of the function. Simultaneously, 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 feasible.

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

Step 1: Find Your Values

In these types of equations, math scenarios usually provide you with two sets of values, from which you will get x and y values.

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

In this situation, then you have to find the values along the x and y-axis. Coordinates are generally provided 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 plug-in the values as follows.

R = 4 - 2 / 3 - 1

Step 3: Simplify

With all of our numbers plugged in, all that we have to do is to simplify the equation by deducting all the numbers. So, our equation will look something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As we can see, by replacing all our values and simplifying the equation, we get the average rate of change for the two coordinates that we were given.

Average Rate of Change of a Function

As we’ve stated before, the rate of change is pertinent to many diverse scenarios. 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 principle 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

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

Occasionally, the equation concludes in a slope that is negative. This denotes that the line is trending downward from left to right in the X Y axis.

This translates to the rate of change is decreasing 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 tells us that the object is gaining value, and the secant line is trending upward from left to right. In relation to our last example, if an object has positive velocity and its position is increasing.

Examples of Average Rate of Change

In this section, we will run through the average rate of change formula with some examples.

Example 1

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

In this example, all we have to do is a simple substitution due to the fact that the delta values are already provided.

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 Cartesian plane.

For this example, we still have to find 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 you can see, the average rate of change is equal to the slope of the line connecting two points.

Example 3

Extract 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 calculating the rate of change of a function, calculate the values of the functions in the equation. In this situation, we simply substitute the values on the equation using 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

With all our values, all we need 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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