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

The rate of change formula is one of the most used math principles throughout academics, most notably in physics, chemistry and accounting.

It’s most often used when discussing velocity, however it has many uses across different industries. Because of its usefulness, this formula is something that students should grasp.

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 shows the variation of one figure in relation to another. In every day terms, it's used to identify the average speed of a variation over a certain period of time.

At its simplest, 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 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

Because of this, 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 X Y axis, is helpful when discussing differences in value A versus value B.

The straight line that joins 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

In short, in a linear function, the average rate of change between two figures is equal to the slope of the function.

This is why the 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. Simultaneously, 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 figures mean, finding the average rate of change of the function is achievable.

To make grasping this concept simpler, here are the steps you need to follow to find the average rate of change.

Step 1: Understand Your Values

In these equations, mathematical scenarios usually offer 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 case, next you have to locate the values on the x and y-axis. Coordinates are usually given in an (x, y) format, as you see in the example below:

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 found all the values of x and y, we can plug-in the values as follows.

R = 4 - 2 / 3 - 1

Step 3: Simplify

With all of our figures inputted, all that is left 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 shown, by simply 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 mentioned previously, the rate of change is pertinent to multiple diverse scenarios. The aforementioned examples were more relevant to the rate of change of a linear equation, but this formula can also be used in functions.

The rate of change of function follows 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

As you might remember, the average rate of change of any two values can be plotted. The R-value, then is, equivalent 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 diminishing in value. For example, velocity can be negative, which means a declining position.

Positive Slope

At the same time, a positive slope shows that the object’s rate of change is positive. This means that the object is gaining value, and the secant line is trending upward from left to right. With regards to our previous example, if an object has positive velocity and its position is ascending.

Examples of Average Rate of Change

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

Example 1

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

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

R = Δy / Δx

R = 10 / 2

R = 5

Example 2

Extract 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 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 equivalent 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 situation, 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 have 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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