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 principles across academics, especially in chemistry, physics and finance.

It’s most frequently applied when discussing thrust, though it has numerous uses across many industries. Because of 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 math, the average rate of change formula shows the change of one value when compared to another. In practical terms, it's employed to define the average speed of a variation over a specific period of time.

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

R = Δy / Δx

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

The change through the numerator and denominator is represented by the greek letter Δ, read as delta y and delta x. It is additionally denoted 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 described as:

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

Average Rate of Change = Slope

Plotting out these values in a Cartesian plane, is beneficial when talking about 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

In summation, in a linear function, the average rate of change among two values 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 going through two random 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 know the slope formula and what the values mean, finding the average rate of change of the function is achievable.

To make learning this concept simpler, here are the steps you must follow to find the average rate of change.

Step 1: Find Your Values

In these types of equations, mathematical scenarios usually offer you 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, then you have to locate 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 numbers plugged in, all that we have to do is to simplify the equation by deducting all the values. So, our equation then becomes the following.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

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

Average Rate of Change of a Function

As we’ve stated previously, the rate of change is applicable to many different situations. The previous examples were more relevant to the rate of change of a linear equation, but this formula can also be used in functions.

The rate of change of function obeys an identical rule 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 case, the values given will have one f(x) equation and one X Y graph value.

Negative Slope

As you might remember, the average rate of change of any two values can be graphed. The R-value, 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 translates to the rate of change is decreasing in value. For example, rate of change can be negative, which means a declining position.

Positive Slope

On the contrary, a positive slope shows that the object’s rate of change is positive. This means that the object is gaining value, and the secant line is trending upward from left to right. In relation 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 through some examples.

Example 1

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

In the given example, all we have to do is a straightforward substitution due to the fact that 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 look 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

Find the rate of change of function f(x) = x2 + 5x - 3 on the interval [3, 5].

The final example will be extracting 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 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 need 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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