# What is the end behavior of the graph of f(x)=-2x^4+7x^2+4x-4?

May 31, 2018

${\lim}_{x \to \infty} f \left(x\right) = - \infty , {\lim}_{x \to - \infty} f \left(x\right) = - \infty$

#### Explanation:

To determine the end behavior, let's take the limit as $x \to \infty$ and $x \to - \infty$.

In our polynomial, $f \left(x\right)$, the first term is what will dominate the end behavior, because it has the highest degree. So we can find the limit of that:

${\lim}_{x \to \infty} \textcolor{red}{- 2} \textcolor{b l u e}{{x}^{4}} = - \infty$

As $x$ gets very large, the blue term will always be positive, but the $- 2$ (red) will turn it negative. This is why our limit evaluates to $- \infty$.

${\lim}_{x \to - \infty} \textcolor{red}{- 2} \textcolor{b l u e}{{x}^{4}} = - \infty$

As $x$ gets very negative, the even exponent will make the term positive, but the red $- 2$ on the outside will make it negative. Thus, this limit will also evaluate to $- \infty$.

In general, the function is downward opening because of the negative coefficient on the ${x}^{4}$ term.

Hope this helps!