Question

answer as $x \
ightarrow \infty, y \
ightarrow \infty$ and as $x \
ightarrow -\infty, y \
ightarrow -\infty$ as $x \
ightarrow \infty, y \
ightarrow \infty$ and as $x \
ightarrow -\infty, y \
ightarrow \infty$ as $x \
ightarrow \infty, y \
ightarrow -\infty$ and as $x \
ightarrow -\infty, y \
ightarrow \infty$ as $x \
ightarrow \infty, y \
ightarrow -\infty$ and as $x \
ightarrow -\infty, y \
ightarrow -\infty$

Explanation:

Step1: Analyze the graph's end - behavior

The graph shown has a leading term with an odd degree and a negative leading coefficient? Wait, no, looking at the options, we need to check the end - behavior. Wait, actually, from the graph, when we look at the right - hand side (as \(x
ightarrow\infty\)), the graph is going down (since the arrow is pointing down), and when we look at the left - hand side (as \(x
ightarrow-\infty\)), the graph is going up? Wait, no, wait the options: Let's recall the end - behavior of polynomials. For a polynomial, the end - behavior is determined by the leading term \(a_nx^n\). If \(n\) is odd:

• If \(a_n>0\), as \(x

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

• If \(a_n < 0\), as \(x

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

Looking at the graph, when \(x
ightarrow\infty\), the graph is going down (so \(y
ightarrow-\infty\)) and when \(x
ightarrow-\infty\), the graph is going up (so \(y
ightarrow\infty\)). So we need to find the option that has as \(x
ightarrow\infty\), \(y
ightarrow-\infty\) and as \(x
ightarrow-\infty\), \(y
ightarrow\infty\).

Looking at the options:

• First option: as \(x

ightarrow\infty\), \(y
ightarrow\infty\) and as \(x
ightarrow-\infty\), \(y
ightarrow-\infty\) (wrong, this is for odd degree with positive leading coefficient)

• Second option: as \(x

ightarrow\infty\), \(y
ightarrow\infty\) and as \(x
ightarrow-\infty\), \(y
ightarrow\infty\) (this is for even degree with positive leading coefficient)

• Third option: as \(x

ightarrow\infty\), \(y
ightarrow-\infty\) and as \(x
ightarrow-\infty\), \(y
ightarrow\infty\) (matches our analysis)

• Fourth option: as \(x

ightarrow\infty\), \(y
ightarrow-\infty\) and as \(x
ightarrow-\infty\), \(y
ightarrow-\infty\) (this is for even degree with negative leading coefficient)

Answer:

C. as \(x
ightarrow \infty, y
ightarrow -\infty\) and as \(x
ightarrow -\infty, y
ightarrow \infty\) (assuming the third option is labeled C, but in the given options, the third option is "as \(x
ightarrow \infty, y
ightarrow -\infty\) and as \(x
ightarrow -\infty, y
ightarrow \infty\)")