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 concepts across academics, specifically in chemistry, physics and finance.

It’s most frequently applied when discussing momentum, however it has numerous uses throughout many industries. Because of its value, this formula is a specific concept that learners should learn.

This article will go over the rate of change formula and how you can solve it.

Average Rate of Change Formula

In mathematics, the average rate of change formula describes the change of one figure in relation to another. In practical terms, it's utilized to define 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 change of y in comparison to the change 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 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 portrayed as:

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

Average Rate of Change = Slope

Plotting out these numbers in a Cartesian plane, is beneficial when talking about differences in value A versus 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

To summarize, 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. 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 feasible.

To make grasping this topic less complex, here are the steps you need to obey to find the average rate of change.

Step 1: Understand Your Values

In these sort of equations, math scenarios usually give you two sets of values, from which you solve to find x and y values.

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

In this situation, then you have to locate the values via the x and y-axis. Coordinates are typically given in an (x, y) format, like this:

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 obtained 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 subtracting all the values. Thus, our equation then becomes the following.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As stated, by replacing 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 earlier, the rate of change is relevant 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 obeys an identical rule but with a different formula due to 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 Cartesian plane value.

Negative Slope

As you might recollect, the average rate of change of any two values can be plotted. The R-value, then is, identical to its slope.

Every so often, the equation concludes in a slope that is negative. This means that the line is descending from left to right in the X Y axis.

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

Positive Slope

In contrast, a positive slope shows that the object’s rate of change is positive. This tells us that the object is increasing in value, and the secant line is trending upward from left to right. In relation to our aforementioned example, if an object has positive velocity and its position is ascending.

Examples of Average Rate of Change

Next, we will run through the average rate of change formula via some examples.

Example 1

Extract 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 since the delta values are already given.

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 X Y axis.

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 you can see, the average rate of change is the same as the slope of the line linking two points.

Example 3

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

The third example will be calculating 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, solve for the values of the functions in the equation. In this instance, we simply replace the values on the equation with 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

Now that we have 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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