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

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

It’s most often used when discussing momentum, though it has multiple applications throughout many industries. Due to its usefulness, this formula is a specific concept that students should learn.

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 change of one figure when compared to another. In practical terms, it's employed to determine 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 computes 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 portrayed as the variation within 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 expressed as:

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

Average Rate of Change = Slope

Plotting out these figures in a Cartesian plane, is beneficial when discussing 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 summation, in a linear function, the average rate of change among two values is the same as the slope of the function.

This is why the average rate of change of a function is the slope of the secant line going 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 understand the slope formula and what the figures mean, finding the average rate of change of the function is achievable.

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

Step 1: Understand Your Values

In these sort of equations, math problems typically give you 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 case, next you have to find 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

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 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 figures plugged in, all that we have to do is to simplify the equation by subtracting all the numbers. So, our equation becomes something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

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

Average Rate of Change of a Function

As we’ve stated earlier, the rate of change is applicable to many diverse scenarios. The previous 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 the same principle but with a unique formula because of the unique 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

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

Occasionally, the equation concludes in a slope that is negative. This indicates that the line is descending from left to right in the X Y graph.

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

Positive Slope

On the contrary, a positive slope denotes 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 terms of our previous example, if an object has positive velocity and its position is ascending.

Examples of Average Rate of Change

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

Example 1

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

In the given example, all we must do is a straightforward substitution because the delta values are already specified.

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 the same as 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 extracting the rate of change of a function, determine the values of the functions in the equation. In this instance, we simply substitute the values on the equation using the values specified 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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