# Cubic Function: Definition, Formula & Examples Video

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• 0:00 Cubic Functions
• 0:45 Box Example
• 3:09 More Volume Examples
• 4:32 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we will explore a specific function called a cubic function in mathematics. We'll define this type of function and look at some various examples involving cubic functions to help us to understand what cubic functions look like and how they can be used.

## Cubic Functions

The Earth we live on is an incredibly large planet. Have you ever wondered how much space is inside the Earth? In other words, have you ever wondered what the volume of the Earth is?

The answer lies in what is called a cubic function in mathematics. A cubic function can be described in a few different ways. Technically, a cubic function is any function of the form y = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants and a is not equal to zero. If we wanted to describe this type of function in words rather than by formula, we would say that a cubic function is any polynomial function where the highest exponent is equal to 3.

## Box Example

Let's consider a classic example of a cubic function. Assume you are moving and you need to place some of your belongings in a box, but you've run out of boxes. However, you do have a large piece of cardboard with length 36 inches and width 30 inches, some scissors, and some boxing tape. You're in luck! You can make a box with these materials, by cutting squares in each of the corners of the piece of cardboard, and then folding the sides up and taping the sides.

Great! We know how to make a box now, but we realize that we need to know what the volume of our box is in order to know if our belongings will fit inside the box. We can figure this out using a cubic function that represents the volume of our box as a function of the length of the sides of the squares we cut out from each corner. Let's call the length of the sides of each of the squares cut out of the corners x.

When we cut the square corners out and fold up the sides, the base of our box has dimensions equal to the lengths of the sides of our cardboard piece lessened by 2x. This is because we took two lengths of x from each of the sides of the cardboard piece when we cut out the squares in each corner. Similarly, the height of our box is x, because when we fold up the sides of the box, the length of the sides of the squares we cut from each corner becomes our height. We see that we have a box with height x, length 36 - 2x, and width 30 - 2x. The volume of the box is given by the formula length times width times height, so we have:

V = (36 - 2x)(30 - 2x)(x)

Multiplying this out gives

V = 4x^3 - 132x^2 + 1080x

This function is a polynomial with its highest exponent equal to 3, so we have a cubic function. We can calculate our volume based on what size we decide to cut our square corners. For instance, if we decided to cut our squares to have side length 10 inches, then the volume of our box would be

V = 4(10)^3 - 132(10)^2 + 1080(10) = 4000 - 13200 + 10800 = 1600 in^3

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