Question

select one
① both down ② both up
③ left down, right up ④ left up, right down
⑤ as x approaches positive infinity, y approaches positive infinity; as x approaches negative infinity, y approaches negative infinity. ⑥ as x approaches positive infinity, y approaches negative infinity; as x approaches negative infinity, y approaches positive infinity.
⑦ as x approaches positive or negative infinity, y approaches 0. ⑧ as x approaches positive or negative infinity, y approaches a constant value.

Explanation:

Step1: Recall End Behavior Rules

For a polynomial function, the end - behavior is determined by the leading term (the term with the highest degree). If the degree of the polynomial is odd:
- If the leading coefficient is positive, as \(x ightarrow+\infty\), \(y ightarrow+\infty\) and as \(x ightarrow-\infty\), \(y ightarrow-\infty\) (left down, right up? No, wait: when \(x\) is very large positive (\(x ightarrow+\infty\)) and leading coefficient positive and odd degree, \(y\) goes to \(+\infty\); when \(x\) is very large negative (\(x ightarrow-\infty\)), \(y\) goes to \(-\infty\). The graph will have left - down (as \(x ightarrow-\infty\), \(y ightarrow-\infty\)) and right - up (as \(x ightarrow+\infty\), \(y ightarrow+\infty\))? Wait, no, let's correct:
For a polynomial \(y = a_nx^n+\cdots+a_0\), if \(n\) is odd:
- If \(a_n>0\): \(\lim_{x ightarrow+\infty}y = +\infty\) and \(\lim_{x ightarrow-\infty}y=-\infty\) (so as \(x\) goes to positive infinity, \(y\) goes up; as \(x\) goes to negative infinity, \(y\) goes down. So the left end (as \(x ightarrow-\infty\)) is down, right end (as \(x ightarrow+\infty\)) is up? Wait, no, when \(x\) is negative and very large in magnitude (\(x ightarrow-\infty\)), if \(n\) is odd and \(a_n>0\), \(x^n\) is negative (since odd power of negative number is negative), so \(y=a_nx^n+\cdots\) will be negative (approaching \(-\infty\)) when \(x ightarrow-\infty\). When \(x ightarrow+\infty\), \(x^n\) is positive (odd power of positive number is positive) and \(a_n>0\), so \(y ightarrow+\infty\). So the left end (as \(x ightarrow-\infty\)) is down ( \(y ightarrow-\infty\)) and the right end (as \(x ightarrow+\infty\)) is up ( \(y ightarrow+\infty\))? Wait, no, the description "left down, right up" would mean as \(x\) moves to the left (towards \(-\infty\)), the graph goes down ( \(y\) decreases to \(-\infty\)) and as \(x\) moves to the right (towards \(+\infty\)), the graph goes up ( \(y\) increases to \(+\infty\)). But the option 5 says "As \(x\) approaches positive infinity, \(y\) approaches positive infinity; as \(x\) approaches negative infinity, \(y\) approaches negative infinity". Let's check the options:

Option 5: "As \(x\) approaches positive infinity, \(y\) approaches positive infinity; as \(x\) approaches negative infinity, \(y\) approaches negative infinity" - this is the end - behavior of a polynomial with odd degree and positive leading coefficient.

Let's analyze other options:

Option 1: "Both down" - this would be for a function where as \(x ightarrow\pm\infty\), \(y ightarrow-\infty\). For polynomials, this would require even degree and negative leading coefficient, but the end - behavior for even degree is both ends in the same direction.

Option 2: "Both up" - even degree and positive leading coefficient.

Option 3: "Left down, right up" - the description of the end - behavior in words, but option 5 is the more precise (using limit - like language) description of the same end - behavior as "left down, right up" (since as \(x ightarrow-\infty\), \(y ightarrow-\infty\) (left down) and as \(x ightarrow+\infty\), \(y ightarrow+\infty\) (right up)). But option 5 is the one that states the limits: as \(x\) approaches positive infinity, \(y\) approaches positive infinity; as \(x\) approaches negative infinity, \(y\) approaches negative infinity.

Option 4: "Left up, right down" - this would be for odd degree and negative leading coefficient (\(\lim_{x ightarrow+\infty}y = -\infty\) and \(\lim_{x ightarrow-\infty}y = +\infty\))

Option 6: "As \(x\) approaches positive infinity, \(y\) approaches negative infinity; as \(x\) approaches negative infinity, \(y\) approaches positive infinity" - this is odd degree and negative leading coefficient.

Option 7: "As \(x\) approaches positive or negative infinity, \(y\) approaches 0" - this is the end - behavior of a rational function with horizontal asymptote \(y = 0\), not a polynomial.

Option 8: "As \(x\) approaches positive or negative infinity, \(y\) approaches a constant value" - this is a function with a horizontal asymptote (like rational functions or exponential functions approaching a constant), not a polynomial.

Step2: Match the End - Behavior

The end - behavior described in option 5 is consistent with a polynomial of odd degree with a positive leading coefficient. For example, \(y=x\) (degree 1, leading coefficient 1): as \(x ightarrow+\infty\), \(y ightarrow+\infty\); as \(x ightarrow-\infty\), \(y ightarrow-\infty\). Or \(y = x^3\): as \(x ightarrow+\infty\), \(y ightarrow+\infty\); as \(x ightarrow-\infty\), \(y ightarrow-\infty\).

Answer:

1. As x approaches positive infinity, y approaches positive infinity; as x approaches negative infinity, y approaches negative infinity.