Question

the function $f(x)$ is defined below. what is the end - behavior of $f(x)$?
$f(x)=514x^{2}+9x^{5}+x^{6}-405x^{3}-47x^{4}+2016x + 1152$
answer
attempt 1 out of 2
the degree of the polynomial is
, and the leading coefficient is
.
choose which pair of limits below represents the end - behavior:
$lim_{x
ightarrow-infty}f(x)=infty,lim_{x
ightarrowinfty}f(x)=infty$
$lim_{x
ightarrow-infty}f(x)=-infty,lim_{x
ightarrowinfty}f(x)=infty$
$lim_{x
ightarrow-infty}f(x)=infty,lim_{x
ightarrowinfty}f(x)=-infty$
$lim_{x
ightarrow-infty}f(x)=-infty,lim_{x
ightarrowinfty}f(x)=-infty$

Explanation:

Step1: Identify degree of polynomial

The highest - power of \(x\) in \(f(x)=514x^{2}+9x^{5}+x^{6}-405x^{3}-47x^{4}+2016x + 1152\) is \(6\), so the degree \(n = 6\).

Step2: Identify leading coefficient

The coefficient of the term with the highest - power of \(x\) (the \(x^{6}\) term) is \(1\).

Step3: Determine end - behavior

For a polynomial \(y = a_{n}x^{n}+a_{n - 1}x^{n - 1}+\cdots+a_{0}\) with even degree \(n\) and positive leading coefficient \(a_{n}>0\), as \(x\to-\infty\), \(y\to\infty\) and as \(x\to\infty\), \(y\to\infty\). That is, \(\lim_{x\to-\infty}f(x)=\infty\) and \(\lim_{x\to\infty}f(x)=\infty\).

Answer:

The degree of the polynomial is \(6\), and the leading coefficient is \(1\).
\(\lim_{x\to-\infty}f(x)=\infty,\lim_{x\to\infty}f(x)=\infty\)