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

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

It’s most often used when discussing velocity, however it has multiple uses throughout many industries. Because of its value, this formula is something that learners should understand.

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 denotes the change of one value in relation to another. In practical terms, it's employed to determine the average speed of a variation over a certain period of time.

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

R = Δy / Δx

This computes the change of y in comparison to the variation of x.

The change within the numerator and denominator is shown by the greek letter Δ, expressed as delta y and delta x. It is also 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 described as:

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

Average Rate of Change = Slope

Plotting out these numbers in a X Y axis, is helpful when working with differences 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 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 passing through two arbitrary 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 know the slope formula and what the figures mean, finding the average rate of change of the function is achievable.

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

Step 1: Understand Your Values

In these equations, math scenarios usually provide you with two sets of values, from which you will get x and y values.

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

In this case, next you have to search for the values along the x and y-axis. Coordinates are typically provided 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

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 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 numbers in place, all that remains is to simplify the equation by subtracting all the numbers. So, our equation will look something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As stated, by simply plugging in all our values and simplifying the equation, we get the average rate of change for the two coordinates that we were provided.

Average Rate of Change of a Function

As we’ve mentioned before, the rate of change is pertinent to numerous diverse scenarios. The aforementioned 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 obeys the same principle but with a different formula due to the distinct values that functions have. This formula is:

R = (f(b) - f(a)) / b - a

In this scenario, the values provided will have one f(x) equation and one X Y axis value.

Negative Slope

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

Sometimes, the equation results 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

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 last example, if an object has positive velocity and its position is increasing.

Examples of Average Rate of Change

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

Example 1

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

In this example, all we need to do is a simple 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 find 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 given, the average rate of change is equivalent to the slope of the line connecting two points.

Example 3

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

The final 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 case, 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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