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 across academics, specifically in chemistry, physics and accounting.

It’s most often applied when discussing velocity, however it has multiple uses across many industries. Because of its usefulness, 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 mathematics, the average rate of change formula shows the change of one figure when compared to another. In practical terms, it's utilized to define the average speed of a variation over a certain period of time.

To put it simply, the rate of change formula is expressed as:

R = Δy / Δx

This measures the change of y in comparison to the change 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 denoted as the variation within the first point and the second point of the value, or:

Δy = y2 - y1

Δx = x2 - x1

As a result, the average rate of change equation can also be described as:

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

Average Rate of Change = Slope

Plotting out these numbers in a X Y graph, is useful when discussing differences in value A in comparison with value B.

The straight line that links 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

To summarize, in a linear function, the average rate of change between 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 going through two arbitrary endpoints on the graph of the function. Meanwhile, 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 possible.

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

Step 1: Understand Your Values

In these equations, mathematical scenarios typically offer you two sets of values, from which you extract x and y values.

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

In this scenario, then you have to search for the values via the x and y-axis. Coordinates are generally 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 input 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 values. Therefore, our equation then becomes the following.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As shown, 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 mentioned before, the rate of change is pertinent to numerous diverse scenarios. The previous examples were more relevant to the rate of change of a linear equation, but this formula can also be applied to functions.

The rate of change of function observes an identical principle but with a distinct formula because of the different values that functions have. This formula is:

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

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

Negative Slope

If you can recall, the average rate of change of any two values can be graphed. The R-value, is, equivalent to its slope.

Occasionally, the equation concludes in a slope that is negative. This indicates 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 means a decreasing position.

Positive Slope

On the other hand, a positive slope means 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. 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 talk about 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 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 Cartesian plane.

For this example, we still have to search for the Δy and Δx values by utilizing the average rate of change formula.

R = y2 - y1 / x2 - x1

R = (14 - 6) / (3 - 1)

R = 8 / 2

R = 4

As given, the average rate of change is identical to 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 last example will be finding 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, solve for the values of the functions in the equation. In this instance, we simply substitute 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

Once we have all our values, all we must 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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