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

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

It’s most frequently utilized when discussing momentum, although it has multiple applications across many industries. Because of its value, this formula is something that students should grasp.

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 describes the variation of one value in relation to another. In every day terms, it's employed to evaluate the average speed of a variation over a certain period of time.

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

R = Δy / Δx

This computes the variation of y compared to the variation of x.

The change through the numerator and denominator is represented by the greek letter Δ, expressed as delta y and delta x. It is also expressed 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 values in a Cartesian plane, is helpful when reviewing dissimilarities in value A in comparison with value B.

The straight line that connects 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

To summarize, in a linear function, the average rate of change among two figures 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 passing 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 have discussed the slope formula and what the values mean, finding the average rate of change of the function is achievable.

To make grasping this principle less complex, here are the steps you must keep in mind to find the average rate of change.

Step 1: Determine Your Values

In these equations, math questions typically provide you with 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 situation, then you have to locate the values along the x and y-axis. Coordinates are usually given in an (x, y) format, as in this example:

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 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 numbers plugged in, all that remains is to simplify the equation by subtracting all the values. So, our equation becomes something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

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

Average Rate of Change of a Function

As we’ve stated before, the rate of change is pertinent to multiple diverse scenarios. The previous examples were applicable 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 the same rule but with a different formula due to the different 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

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

Occasionally, 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 axis.

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 contrary, a positive slope denotes 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. In relation to our last example, if an object has positive velocity and its position is ascending.

Examples of Average Rate of Change

Now, we will discuss the average rate of change formula with some examples.

Example 1

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

In this example, all we must do is a plain substitution due to the fact that the delta values are already specified.

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 employing 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 the same as the slope of the line joining 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 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

With all our values, all we must do is replace 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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