July 18, 2022

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

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

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

It’s most often used when talking about thrust, however it has multiple applications throughout many industries. Because of its utility, this formula is something that students should grasp.

This article will discuss 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 variation of one value in relation to another. In every day terms, it's utilized to determine the average speed of a change over a certain period of time.

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

R = Δy / Δx

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

The variation within the numerator and denominator is portrayed 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

Consequently, 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 X Y axis, is helpful when working with 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

To summarize, in a linear function, the average rate of change among two figures is the same as 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. 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 grasping this principle less complex, here are the steps you must follow to find the average rate of change.

Step 1: Understand Your Values

In these types of equations, math problems generally provide you with two sets of values, from which you solve to find x and y values.

For example, let’s assume 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 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 numbers inputted, all that is left is to simplify the equation by subtracting all the values. Therefore, our equation becomes something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As shown, by simply replacing all our values and simplifying the equation, we achieve 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 applicable to multiple diverse scenarios. The aforementioned 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 follows the same principle but with a distinct formula due to the unique values that functions have. This formula is:

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

In this scenario, the values given will have one f(x) equation and one X Y axis 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.

Every so often, 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 decreasing in value. For example, velocity can be negative, which results in a decreasing position.

Positive Slope

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

Examples of Average Rate of Change

Now, we will review the average rate of change formula with 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 because the delta values are already provided.

R = Δy / Δx

R = 10 / 2

R = 5

Example 2

Find 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 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 equivalent to the slope of the line connecting two points.

Example 3

Find 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 finding the rate of change of a function, solve for the values of the functions in the equation. In this case, we simply substitute 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

Once we have 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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