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

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

It’s most frequently used when discussing momentum, however it has many uses across various industries. Due to its value, this formula is a specific concept that students should learn.

This article will go over the rate of change formula and how you should solve them.

Average Rate of Change Formula

In mathematics, the average rate of change formula denotes the variation of one figure in relation to another. In practical terms, it's used to identify 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 measures the change of y in comparison to the variation of x.

The variation through the numerator and denominator is shown by the greek letter Δ, read as delta y and delta x. It is also expressed 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 shown as:

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

Average Rate of Change = Slope

Plotting out these values in a Cartesian plane, is beneficial when working with differences 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 summation, in a linear function, the average rate of change between 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. In the meantime, 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 have discussed the slope formula and what the figures mean, finding the average rate of change of the function is possible.

To make studying this topic simpler, here are the steps you must keep in mind to find the average rate of change.

Step 1: Determine Your Values

In these types of equations, mathematical questions generally give you two sets of values, from which you extract x and y values.

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

In this instance, then 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 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 found 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 figures inputted, all that we have to do is to simplify the equation by deducting all the numbers. So, our equation then becomes the following.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As we can see, by plugging in 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 mentioned before, the rate of change is applicable to numerous different 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 obeys a similar rule but with a unique formula due to the unique values that functions have. This formula is:

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

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

Negative Slope

If you can remember, the average rate of change of any two values can be graphed. The R-value, then 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 X Y graph.

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

Positive Slope

At the same time, a positive slope shows 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. With regards to our previous example, if an object has positive velocity and its position is increasing.

Examples of Average Rate of Change

Now, we will talk about 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 need to do is a straightforward substitution since the delta values are already given.

R = Δy / Δx

R = 10 / 2

R = 5

Example 2

Calculate 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 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

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 finding 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 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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