# How do you determine the end behavior of f(x)=-2.1x^5+4x^3-2?

Nov 7, 2017

$x \rightarrow \infty , y \rightarrow - \infty$
$x \rightarrow - \infty , y \rightarrow \infty$

#### Explanation:

This is a negative graph, as your first term has a negative coefficient of $- 2.1$. Secondly, it will be an odd graph, since the degree of your equation, or the highest exponent, is an odd number ($5$).

We can do this without writing a graph, and this will work for any polynomial for determining end behaviors. However, I will be using graphs to help visualize for future reference.

Since your function is odd, the ends of the line formed by your function will point in opposite directions. One up, one down. Since it is a negative function, the right-most arrow will be pointing down. Here is your graph, to use as an example: graph{-2.1x^5+4x^3-2 [-10, 10, -5, 5]}

Note how this followed the same rules we went over earlier. You can do this with any polynomial.

As your $x$ increases into $\infty$, your $y$ decreases into $- \infty$.
As your $x$ decreases into $- \infty$, your $y$ increases into $\infty$.

Thus, your end behavior can be written as follows:

$x \rightarrow \infty , y \rightarrow - \infty$
$x \rightarrow - \infty , y \rightarrow \infty$