Question

the function $f(x)$ is defined below. what is the end behavior of $f(x)$?
$f(x) = -1362x^2 - 3276x - 24x^5 + 3920 + 740x^3 + 2x^6$
answer
\\(\circ\\) as $x\to\infty, y\to\infty$ and
as $x\to -\infty, y\to -\infty$
\\(\circ\\) as $x\to\infty, y\to\infty$ and
as $x\to -\infty, y\to\infty$
\\(\circ\\) as $x\to\infty, y\to -\infty$ and
as $x\to -\infty, y\to -\infty$
\\(\circ\\) as $x\to\infty, y\to -\infty$ and
as $x\to -\infty, y\to\infty$

Explanation:

Step1: Identify the leading term

First, we need to find the leading term of the polynomial \( f(x) \). The leading term is the term with the highest degree. Let's combine like terms for the highest degree. The terms with \( x^6 \) are \( -24x^6 \) and \( 2x^6 \). Combining them: \( -24x^6 + 2x^6 = -22x^6 \). So the leading term is \( -22x^6 \), with degree 6 (even) and leading coefficient -22 (negative).

Step2: Analyze end behavior for even degree

For a polynomial with an even degree, the ends of the graph (as \( x \to \infty \) and \( x \to -\infty \)) both go in the same direction. The direction is determined by the sign of the leading coefficient. If the leading coefficient is negative, as \( x \to \infty \), \( x^6 \) is positive, so \( -22x^6 \to -\infty \). As \( x \to -\infty \), \( (-x)^6 = x^6 \) (since 6 is even), so \( -22x^6 \) is also \( -22(\text{positive}) \to -\infty \)? Wait, no, wait: Wait, the leading term is \( -22x^6 \). Let's re - check:

Wait, when \( x \to \infty \), \( x^6 \to \infty \), so \( -22x^6 \to -\infty \) (because of the negative coefficient). When \( x \to -\infty \), \( x^6 = (-x)^6 \), and since 6 is even, \( (-x)^6=x^6\to\infty \), so \( -22x^6\to -\infty \)? Wait, no, that can't be. Wait, no, wait: Wait, the leading term is \( -22x^6 \). Let's take the limit as \( x \to \infty \): \( \lim_{x\to\infty} - 22x^6=-\infty \) (because \( x^6 \) grows to infinity and we multiply by - 22). As \( x\to -\infty \), \( x^6 = (-x)^6=x^6 \) (since 6 is even), so \( \lim_{x\to -\infty}-22x^6 = - 22\lim_{x\to -\infty}x^6=-22\times\infty = -\infty \)? Wait, no, that's not right. Wait, no, I made a mistake. Wait, the leading term is \( -22x^6 \). Let's think again:

Wait, the general rule for end - behavior of a polynomial \( a_nx^n+\cdots+a_0 \):

• If \( n \) is even:
• If \( a_n>0 \), as \( x\to\infty \), \( y\to\infty \); as \( x\to -\infty \), \( y\to\infty \)
• If \( a_n < 0 \), as \( x\to\infty \), \( y\to -\infty \); as \( x\to -\infty \), \( y\to -\infty \)

Wait, but in our case, the leading term is \( -22x^6 \), so \( n = 6 \) (even) and \( a_n=-22<0 \). So as \( x\to\infty \), \( f(x)\to -\infty \) and as \( x\to -\infty \), \( f(x)\to -\infty \)? But that's one of the options? Wait, no, wait, let's check the options again.

Wait, the options are:

1. as \( x\to\infty,y\to\infty \) and as \( x\to -\infty,y\to -\infty \)

1. as \( x\to\infty,y\to\infty \) and as \( x\to -\infty,y\to\infty \)

1. as \( x\to\infty,y\to -\infty \) and as \( x\to -\infty,y\to -\infty \)

1. as \( x\to\infty,y\to -\infty \) and as \( x\to -\infty,y\to\infty \)

Wait, according to the rule, for even degree \( n \) and negative leading coefficient \( a_n \), both ends go to \( -\infty \). So the end - behavior is as \( x\to\infty,y\to -\infty \) and as \( x\to -\infty,y\to -\infty \)

Answer:

as \( x \to \infty, y \to -\infty \) and as \( x \to -\infty, y \to -\infty \) (the third option in the list of choices)