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

The rate of change formula is one of the most important math principles throughout academics, especially in chemistry, physics and accounting.

It’s most frequently used when discussing thrust, although it has multiple applications throughout many industries. Due to its usefulness, this formula is something 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 shows the variation of one value in relation to another. In practice, it's used to define the average speed of a variation over a certain period of time.

At its simplest, the rate of change formula is written as:

R = Δy / Δx

This computes the change of y compared to the change of x.

The variation through the numerator and denominator is portrayed by the greek letter Δ, read as delta y and delta x. It is further expressed as the variation within the first point and the second point of the value, or:

Δy = y2 - y1

Δx = x2 - x1

Consequently, 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 useful when talking about differences in value A when compared to 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 between 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 passing through two arbitrary endpoints on the graph of the function. At the same time, 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 possible.

To make grasping this concept easier, here are the steps you must obey to find the average rate of change.

Step 1: Understand Your Values

In these types of equations, mathematical scenarios typically provide you with 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 instance, next you have to search for the values along the x and y-axis. Coordinates are typically given in an (x, y) format, as in this example:

x1 = 1

x2 = 3

y1 = 2

y2 = 4

Step 2: Subtract The Values

Find the Δx and Δy values. As you can recollect, 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 plug-in the values as follows.

R = 4 - 2 / 3 - 1

Step 3: Simplify

With all of our values in place, all that we have to do is to simplify the equation by subtracting all the values. Therefore, our equation then becomes the following.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As we can see, by simply plugging in all our values and simplifying the equation, we get 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 different situations. The previous 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 obeys the same principle but with a different formula due to the distinct values that functions have. This formula is:

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

In this situation, the values provided will have one f(x) equation and one Cartesian plane value.

Negative Slope

If you can recollect, the average rate of change of any two values can be plotted on a graph. The R-value, then is, equivalent 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 Cartesian plane.

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

Positive Slope

On the contrary, a positive slope denotes that the object’s rate of change is positive. This means 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 ascending.

Examples of Average Rate of Change

Next, we will talk about 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 the given example, all we need to do is a straightforward substitution because 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 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 the same as the slope of the line linking 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 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 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

With all our values, all we have 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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