Question

the function $f(x)$ is defined below. what is the end - behavior of $f(x)$?
$f(x)=-24 - 18x-3x^{2}$
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 \(f(x)=-24 - 18x-3x^{2}\) is 2. So the degree is 2, which is an even number.

Step2: Identify leading coefficient

The leading term is \(-3x^{2}\), so the leading coefficient is - 3.

Step3: Determine end - behavior

For a polynomial \(y = ax^{n}\) with \(n\) even and \(a<0\), 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 even, and the leading coefficient is - 3. The pair of limits representing the end - behavior is \(\lim_{x\to-\infty}f(x)=-\infty\), \(\lim_{x\to\infty}f(x)=-\infty\)