# Using Nonlinear Functions in Real Life Situations

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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we will define non-linear functions and look at examples of non-linear functions being used in real-world situations. We will also touch on determining which non-linear function is best to use in different real-world scenarios.

## Non-Linear Functions

Suppose you are making a pendant necklace, and the pendant is going to be circular in shape. You're trying to decide how large you want the pendant to be, and while doing this, you think back to your math class and remember that the area of a circle is a function of the radius of the circle. This function can be represented as follows: A(r) = Ï€r2, where r is the radius of the circle.

This function is an example of a non-linear function. A non-linear function is a function that is not linear. Well then, it would probably be helpful to know what a linear function is! A linear function is a polynomial function in which the highest exponent of the variable is one. The graph of a linear function is a line. Since a non-linear function is a function that is not linear, the graph of a non-linear function is not a line. For instance, take a look at this graph of our area function of a circle.

We see that the graph is definitely not a line, so it's a non-linear function. We could also determine that it's non-linear by observing that the highest exponent on the variable is two, not one.

Back to your pendant necklace! You can use the function to determine what the radius of your pendant would be for different areas, or you can use it to determine the area of the pendant based on different radii. For example, if you decided to have a pendant with radius 3 centimeters, then you can calculate the area by finding A(3).

We see that when the radius is 3 centimeters, the area of the pendant is approximately 28.27 square centimeters.

This is a great example of using non-linear functions in the real world. There are many different real-life scenarios in which non-linear functions can be used! Let's look at some more!

## Non-Linear Functions in Real Life

Let's consider another example of a non-linear function lending itself to a real-world scenario. Suppose your city's parks and recreation department is constructing a park that will take up a square plot of land and have a trail that borders the entire park.

The foreman, Larry, on the construction project is trying to figure out how long the trail will be (one loop around) given different area options of the park. Larry knows that the area of a square is found by squaring the length of one of its sides. Based on this, he also realizes that he can find the length of one of the sides (or 1/4 of the trail) by taking the square root of the area. In other words, we can represent the length of one of the sides as a function of its area as follows: l(A) = âˆšA

Now, you may look at this and think that the highest exponent of the variable is one. However, that's not actually the case. It just so happens that âˆš(A) can actually be rewritten as A1/2, so we see that the highest exponent is 1/2, not one, so this is a non-linear function. Of course, we can verify this by looking at its graph as well.

We see that the graph is not a line, so the function is non-linear.

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