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

The rate of change formula is one of the most important mathematical concepts across academics, specifically in chemistry, physics and finance.

It’s most often applied when discussing thrust, although it has numerous uses throughout many industries. Due to its utility, this formula is a specific concept that students should learn.

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

Average Rate of Change Formula

In math, the average rate of change formula describes the change of one value when compared to another. In practice, it's utilized to determine the average speed of a change over a specific period of time.

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

R = Δy / Δx

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

The variation 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 difference between 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 expressed as:

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

Average Rate of Change = Slope

Plotting out these figures in a Cartesian plane, is useful when reviewing differences in value A versus value B.

The straight line that joins 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

In summation, in a linear function, the average rate of change among 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 intersecting 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 understand the slope formula and what the figures mean, finding the average rate of change of the function is achievable.

To make studying this topic simpler, here are the steps you should follow to find the average rate of change.

Step 1: Understand Your Values

In these types of equations, math problems generally offer you two sets of values, from which you solve to find x and y values.

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

In this case, next you have to search for the values on the x and y-axis. Coordinates are typically provided 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 recall, 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 numbers in place, all that is left is to simplify the equation by subtracting all the values. Therefore, our equation becomes something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As we can see, by simply plugging in 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 multiple diverse situations. The previous examples focused on the rate of change of a linear equation, but this formula can also be relevant for functions.

The rate of change of function observes an identical principle but with a unique formula due to the distinct 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 X Y graph value.

Negative Slope

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

Sometimes, the equation concludes in a slope that is negative. This means that the line is descending from left to right in the X Y graph.

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

On the other hand, a positive slope indicates that the object’s rate of change is positive. This means 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

Next, we will run through 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 plain substitution due to the fact that 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 X Y axis.

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 provided, the average rate of change is equivalent to 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 extracting 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, calculate the values of the functions in the equation. In this instance, we simply replace 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 must 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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