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 formulas throughout academics, most notably in chemistry, physics and finance.

It’s most frequently used when discussing momentum, though it has many uses throughout different industries. Due to its usefulness, this formula is something 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 mathematics, the average rate of change formula shows the change of one value when compared to another. In every day terms, it's employed to evaluate the average speed of a change over a specified period of time.

Simply put, the rate of change formula is expressed as:

R = Δy / Δx

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

The change within the numerator and denominator is shown by the greek letter Δ, expressed as delta y and delta x. It is additionally 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 described as:

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

Average Rate of Change = Slope

Plotting out these figures in a X Y graph, is helpful when working with differences in value A in comparison with value B.

The straight line that joins these two points is 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 why the 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. 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 possible.

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

Step 1: Understand Your Values

In these sort of equations, mathematical problems typically give you two sets of values, from which you solve to find x and y values.

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

In this instance, then you have to find the values along the x and y-axis. Coordinates are usually 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 remember, 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 input 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 subtracting all the numbers. Thus, our equation then becomes the following.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As stated, just by plugging in 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 shared previously, the rate of change is relevant to multiple diverse situations. The aforementioned examples were more relevant to the rate of change of a linear equation, but this formula can also be relevant for functions.

The rate of change of function observes a similar 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 case, the values given will have one f(x) equation and one Cartesian plane value.

Negative Slope

Previously if you remember, the average rate of change of any two values can be plotted. The R-value, is, equal to its slope.

Sometimes, the equation results in a slope that is negative. This denotes that the line is trending downward from left to right in the Cartesian plane.

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

Positive Slope

In contrast, 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 last 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 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 need to do is a plain substitution since 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 look 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 you can see, the average rate of change is equal to 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 third 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 instance, we simply replace the values on the equation using 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

With all our values, all we have 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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