Question

polynomial e
find your answer from the choices and color the
1
white
as ( x \to -infty, f(x) \to )
as ( x \to infty, f(x) \to infty )
yellow
as ( x \to -infty, f(x) \to -infty )
as ( x \to infty, f(x) \to infty )
red
as ( x \to -infty, f(x) \to infty )
as ( x \to infty, f(x) \to -infty )
orange
as ( x \to -infty, f(x) \to -infty )
as ( x \to infty, f(x) \to -infty )

Explanation:

Step1: Analyze end - behavior of the polynomial graph

To determine the end - behavior of a polynomial function, we look at the leading term (the term with the highest degree). The end - behavior is described by what happens to \(f(x)\) as \(x
ightarrow+\infty\) and as \(x
ightarrow-\infty\).

For the given graph, as \(x
ightarrow-\infty\), we observe the left - hand end of the graph. The left - most part of the graph is going downwards (towards \(-\infty\)). As \(x
ightarrow+\infty\), the right - hand end of the graph is going upwards (towards \(+\infty\)).

Step2: Match with the given options

• White option: As \(x

ightarrow-\infty,f(x)
ightarrow\infty\) and as \(x
ightarrow\infty,f(x)
ightarrow\infty\). This does not match our graph's end - behavior.

• Yellow option: As \(x

ightarrow-\infty,f(x)
ightarrow-\infty\) and as \(x
ightarrow\infty,f(x)
ightarrow\infty\). This matches the end - behavior we observed from the graph.

• Red option: As \(x

ightarrow-\infty,f(x)
ightarrow\infty\) and as \(x
ightarrow\infty,f(x)
ightarrow-\infty\). This does not match.

• Orange option: As \(x

ightarrow-\infty,f(x)
ightarrow-\infty\) and as \(x
ightarrow\infty,f(x)
ightarrow-\infty\). This does not match.

Answer:

Yellow (with the description "as \(x
ightarrow-\infty,f(x)
ightarrow-\infty\); as \(x
ightarrow\infty,f(x)
ightarrow\infty\)")