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 concepts across academics, most notably in physics, chemistry and accounting.

It’s most often utilized when discussing momentum, however it has many applications throughout many industries. Due to its value, this formula is something 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 math, the average rate of change formula denotes the change of one figure when compared to another. In practical terms, it's utilized to evaluate the average speed of a variation over a specific period of time.

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

R = Δy / Δx

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

The variation through the numerator and denominator is portrayed by the greek letter Δ, expressed as delta y and delta x. It is also denoted as the difference 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 shown as:

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

Average Rate of Change = Slope

Plotting out these figures in a X Y axis, is useful when reviewing dissimilarities in value A when compared to 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 short, 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 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 understand the slope formula and what the figures mean, finding the average rate of change of the function is achievable.

To make learning this topic easier, here are the steps you must obey to find the average rate of change.

Step 1: Find Your Values

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

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

In this instance, next you have to locate the values along the x and y-axis. Coordinates are usually 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 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 obtained 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 values inputted, all that we have to do is to simplify the equation by subtracting all the numbers. Thus, our equation then becomes the following.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As stated, 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 shared earlier, the rate of change is applicable to many diverse scenarios. The previous examples were applicable to the rate of change of a linear equation, but this formula can also be used in functions.

The rate of change of function observes an identical principle but with a unique formula due to the unique 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 axis value.

Negative Slope

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

Occasionally, the equation concludes in a slope that is negative. This means that the line is descending from left to right in the Cartesian plane.

This translates to the rate of change is decreasing in value. For example, velocity can be negative, which means a declining position.

Positive Slope

On the other hand, 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 previous example, if an object has positive velocity and its position is ascending.

Examples of Average Rate of Change

Now, 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 this example, all we have to do is a plain 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 X Y graph.

For this example, we still have to find the Δy and Δx values by using 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 equivalent to the slope of the line linking two points.

Example 3

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

The last example will be extracting 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, determine the values of the functions in the equation. In this instance, we simply substitute the values on the equation using the values specified 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 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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