# How do you describe the end behavior of a cubic function?

Mar 15, 2018

The end behavior of cubic functions, or any function with an overall odd degree, go in opposite directions.

#### Explanation:

Cubic functions are functions with a degree of 3 (hence cubic ), which is odd. Linear functions and functions with odd degrees have opposite end behaviors. The format of writing this is:
$x \to \infty$, $f \left(x\right) \to \infty$
$x \to - \infty$, $f \left(x\right) \to - \infty$

For example, for the picture below, as x goes to $\infty$ , the y value is also increasing to infinity. However, as x approaches -$\infty$, the y value continues to decrease; to test the end behavior of the left, you must view the graph from right to left!!

graph{x^3 [-10, 10, -5, 5]}

Here is an example of a flipped cubic function, graph{-x^3 [-10, 10, -5, 5]}
Just as the parent function ($y = {x}^{3}$) has opposite end behaviors, so does this function, with a reflection over the y-axis.
The end behavior of this graph is:
$x \to \infty$, $f \left(x\right) \to - \infty$
$x \to - \infty$, $f \left(x\right) \to \infty$

Even linear functions go in opposite directions, which makes sense considering their degree is an odd number: 1.