# How do you describe the end behavior for f(x)=-x^5+4x^3-5x-4?

Oct 29, 2016

As $x \rightarrow - \infty$, $f \left(x\right) \rightarrow \infty$
As $x \rightarrow \infty$, $f \left(x\right) \rightarrow - \infty$

#### Explanation:

$f \left(x\right) = \textcolor{b l u e}{- 1} {x}^{\textcolor{red}{5}} + 4 {x}^{3} - 5 x - 4$

End behavior is determined by the degree of the polynomial and the leading coefficient (LC).

The degree of this polynomial is the greatest exponent, or $\textcolor{red}{5}$.

The leading coefficient is the coefficient of the term with the greatest exponent, or $\textcolor{b l u e}{- 1}$.

For polynomials of even degree, the "ends" of the polynomial graph point in the same direction as follows.

Even degree and positive LC:
As $x \rightarrow - \infty$, $f \left(x\right) \rightarrow \infty$
As $x \rightarrow \infty$, $f \left(x\right) \rightarrow \infty$

Even degree and negative LC:
As $x \rightarrow - \infty$, $f \left(x\right) \rightarrow - \infty$
As $x \rightarrow \infty$, $f \left(x\right) \rightarrow - \infty$

For polynomials of odd degree, the "ends" of the polynomial graph point in opposite directions as follows (note, there is a saying that Odd means Opposite when graphing).

Odd degree and positive LC:
As $x \rightarrow - \infty$, $f \left(x\right) \rightarrow - \infty$
As $x \rightarrow \infty$, $f \left(x\right) \rightarrow \infty$

Odd degree and negative LC:
As $x \rightarrow - \infty$, $f \left(x\right) \rightarrow \infty$
As $x \rightarrow \infty$, $f \left(x\right) \rightarrow - \infty$

In this example, the degree is odd and the leading coefficient is negative. Therefore,
As $x \rightarrow - \infty$, $f \left(x\right) \rightarrow \infty$
As $x \rightarrow \infty$, $f \left(x\right) \rightarrow - \infty$