Question

the function $f(x)$ is defined below. what is the end behavior of $f(x)$?
$f(x) = -63x^5 - 77x^4 + 168x^2 - 7x^6 + 336 + 343x^3 - 700x$
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

The leading term of a polynomial is the term with the highest degree. First, we find the degree of each term:
- For \(-63x^5\), degree is \(5\)
- For \(-77x^4\), degree is \(4\)
- For \(168x^2\), degree is \(2\)
- For \(-7x^6\), degree is \(6\)
- For \(336\), degree is \(0\)
- For \(343x^3\), degree is \(3\)
- For \(-700x\), degree is \(1\)

The highest degree is \(6\) (from \(-7x^6\)), so the leading term is \(-7x^6\). The coefficient of the leading term is \(-7\) (negative) and the degree \(6\) is even.

Step2: Analyze end behavior based on leading term

For a polynomial \(f(x)=a_nx^n + \dots+a_0\):
- If \(n\) is even:
- If \(a_n>0\), as \(x ightarrow\infty\), \(f(x) ightarrow\infty\) and as \(x ightarrow-\infty\), \(f(x) ightarrow\infty\)
- If \(a_n<0\), as \(x ightarrow\infty\), \(f(x) ightarrow-\infty\) and as \(x ightarrow-\infty\), \(f(x) ightarrow-\infty\) (Wait, no, wait. Wait, for even degree: when \(x ightarrow\pm\infty\), the sign of \(x^n\) is positive (since even power). So if leading coefficient \(a_n\) is negative:
As \(x ightarrow\infty\), \(x^6 ightarrow\infty\), so \(-7x^6 ightarrow-\infty\)
As \(x ightarrow-\infty\), \(x^6 = (-x)^6=x^6 ightarrow\infty\), so \(-7x^6 ightarrow-\infty\)? Wait, no, wait the options have a case where as \(x ightarrow\infty,y ightarrow-\infty\) and \(x ightarrow-\infty,y ightarrow\infty\)? Wait, no, I made a mistake. Wait, the leading term is \(-7x^6\). Let's re - check:

Wait, the leading term is \(-7x^6\). The degree \(n = 6\) (even), coefficient \(a_n=-7\) (negative).

For \(x ightarrow\infty\): \(x^6 ightarrow\infty\), so \(-7x^6 ightarrow-\infty\) (because negative times positive large number is negative large number)

For \(x ightarrow-\infty\): \(x^6=(-x)^6 = x^6 ightarrow\infty\) (since even power), so \(-7x^6 ightarrow-\infty\)? Wait, but that's not matching the options. Wait, no, wait I think I messed up the leading term. Wait, let's re - order the polynomial:

\(f(x)=-7x^6-63x^5 - 77x^4+343x^3 + 168x^2-700x + 336\)

Leading term is \(-7x^6\) (degree 6, even), coefficient \(-7\) (negative).

So as \(x ightarrow\infty\): \(x^6 ightarrow\infty\), so \(-7x^6 ightarrow-\infty\)

As \(x ightarrow-\infty\): \(x^6 = (-\infty)^6=\infty\), so \(-7x^6 ightarrow-\infty\)? But the options:

Wait the options are:

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

2. as \(x ightarrow\infty,y ightarrow\infty\) and as \(x ightarrow-\infty,y ightarrow-\infty\)

3. as \(x ightarrow\infty,y ightarrow-\infty\) and as \(x ightarrow-\infty,y ightarrow\infty\)

4. as \(x ightarrow\infty,y ightarrow\infty\) and as \(x ightarrow-\infty,y ightarrow\infty\)

Wait, maybe I made a mistake in the leading term. Wait, let's check the degree again. Wait, the term \(-7x^6\) has degree 6, \(-63x^5\) degree 5, etc. So leading term is \(-7x^6\), degree 6 (even), coefficient - 7 (negative).

So as \(x ightarrow\infty\), \(f(x) ightarrow-\infty\) (because \(-7x^6\) dominates, \(x^6 ightarrow\infty\), times - 7 is \(-\infty\))

As \(x ightarrow-\infty\), \(x^6 = (-\infty)^6=\infty\), so \(-7x^6 ightarrow-\infty\)

Wait, but the first option is "as \(x ightarrow\infty,y ightarrow-\infty\) and as \(x ightarrow-\infty,y ightarrow-\infty\)"

Wait, but let's check the options again. Wait, maybe I made a mistake. Wait, no, let's re - evaluate.

Wait, the leading term is \(-7x^6\). So:

When \(x\) is very large positive (\(x ightarrow\infty\)): \(x^6\) is positive and large, multiply by - 7: \(f(x) ightarrow-\infty\)

When \(x\) is very large negative (\(x ightarrow-\infty\)): \(x^6=(-x)^6\) (since 6 is even), so \(x^6\) is positive and large, multiply by - 7: \(f(x) ightarrow-\infty\)

So the first option: "as \(x ightarrow\infty,y ightarrow-\infty\) and as \(x ightarrow-\infty,y ightarrow-\infty\)" is correct? Wait, but let's check the options again. Wait, the options are:

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

2. as \(x ightarrow\infty,y ightarrow\infty\) and as \(x ightarrow-\infty,y ightarrow-\infty\)

3. as \(x ightarrow\infty,y ightarrow-\infty\) and as \(x ightarrow-\infty,y ightarrow\infty\)

4. as \(x ightarrow\infty,y ightarrow\infty\) and as \(x ightarrow-\infty,y ightarrow\infty\)

Wait, but according to the leading term \(-7x^6\) (even degree, negative coefficient), the end behavior should be as \(x ightarrow\pm\infty\), \(y ightarrow-\infty\). So the first option is correct? Wait, no, wait I think I messed up. Wait, no, the degree is 6 (even), coefficient is negative. So for even degree:

- If coefficient positive: both ends go to \(+\infty\)

- If coefficient negative: both ends go to \(-\infty\)

Yes, that's correct. So as \(x ightarrow\infty\), \(y ightarrow-\infty\) and as \(x ightarrow-\infty\), \(y ightarrow-\infty\)

Answer:

as \(x
ightarrow \infty, y
ightarrow -\infty\) and as \(x
ightarrow -\infty, y
ightarrow -\infty\) (the first option in the list of options)