Question

which statement describes the end behavior of an odd - degree polynomial with a negative leading coefficient?
(options are shown with their end - behavior descriptions:

• green option: as ( x

ightarrow+infty ), ( f(x)
ightarrow-infty ); as ( x
ightarrow-infty ), ( f(x)
ightarrow+infty )

• blue option: as ( x

ightarrow+infty ), ( f(x)
ightarrow+infty ); as ( x
ightarrow-infty ), ( f(x)
ightarrow-infty )

• orange option: as ( x

ightarrow+infty ), ( f(x)
ightarrow-infty ); as ( x
ightarrow-infty ), ( f(x)
ightarrow-infty )

• light blue option: as ( x

ightarrow+infty ), ( f(x)
ightarrow+infty ); as ( x
ightarrow-infty ), ( f(x)
ightarrow+infty )
also, there is a table with functions ( y = -x^{2}), ( y = x^{3}), ( y=-x^{3}) and their end - behavior descriptions)

Explanation:

Step1: Recall End Behavior Rules

For a polynomial \( f(x) = a_nx^n + \dots + a_0 \), end behavior depends on degree (\( n \)) and leading coefficient (\( a_n \)):

• Odd degree (\( n \) odd): Ends go in opposite directions.
• Negative leading coefficient (\( a_n < 0 \)): If \( n \) odd, as \( x \to +\infty \), \( f(x) \to -\infty \); as \( x \to -\infty \), \( f(x) \to +\infty \) (since odd degree flips direction with negative coefficient).

Step2: Analyze Each Option

• Green option: \( x \to +\infty \), \( f(x) \to -\infty \); \( x \to -\infty \), \( f(x) \to +\infty \). Matches odd degree, negative leading coefficient.
• Blue option: \( x \to +\infty \), \( f(x) \to +\infty \); \( x \to -\infty \), \( f(x) \to -\infty \). This is odd degree with positive leading coefficient.
• Orange option: \( x \to +\infty \), \( f(x) \to -\infty \); \( x \to -\infty \), \( f(x) \to -\infty \). Even degree (both ends same) with negative coefficient (but degree is odd here, so invalid).
• Cyan option: \( x \to +\infty \), \( f(x) \to +\infty \); \( x \to -\infty \), \( f(x) \to +\infty \). Even degree with positive coefficient.

Answer:

The green - colored option (As \( x \to +\infty \), \( f(x) \to -\infty \); as \( x \to -\infty \), \( f(x) \to +\infty \))