Question

write a polynomial function that imitates the end behavior of the graph shown to the right. the dashed portion of the graph indicates that you should focus only on predicting the left and right behavior of the graph and can be flexible about what occurs between the left and right ends.
which of the following polynomial functions imitates the end behavior of the graph shown above?
\\( f(x) = -7x^6 + 3x + 4 \\)
\\( f(x) = -4x^7 - 9x - 5 \\)
\\( f(x) = 10x^2 - 4x + 2 \\)
\\( f(x) = 2x^5 + 4x^2 - 4 \\)

Explanation:

To determine which polynomial function imitates the end - behavior of the given graph, we use the following rules for the end - behavior of a polynomial function \(f(x)=a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_1x + a_0\) (where \(a_n
eq0\)):

Step 1: Recall the end - behavior rules

• If the degree \(n\) of the polynomial is even:
• If the leading coefficient \(a_n>0\), as \(x

ightarrow+\infty\), \(f(x)
ightarrow+\infty\) and as \(x
ightarrow-\infty\), \(f(x)
ightarrow+\infty\).

• If the leading coefficient \(a_n < 0\), as \(x

ightarrow+\infty\), \(f(x)
ightarrow-\infty\) and as \(x
ightarrow-\infty\), \(f(x)
ightarrow-\infty\).

• If the degree \(n\) of the polynomial is odd:
• If the leading coefficient \(a_n>0\), as \(x

ightarrow+\infty\), \(f(x)
ightarrow+\infty\) and as \(x
ightarrow-\infty\), \(f(x)
ightarrow-\infty\).

• If the leading coefficient \(a_n < 0\), as \(x

ightarrow+\infty\), \(f(x)
ightarrow-\infty\) and as \(x
ightarrow-\infty\), \(f(x)
ightarrow+\infty\).

Step 2: Analyze each option

• Option 1: \(f(x)=-7x^{6}+3x + 4\)
• The degree \(n = 6\) (even) and the leading coefficient \(a_n=-7<0\).
• For even degree with negative leading coefficient, as \(x

ightarrow+\infty\), \(f(x)
ightarrow-\infty\) and as \(x
ightarrow-\infty\), \(f(x)
ightarrow-\infty\).

• Option 2: \(f(x)=-4x^{7}-9x - 5\)
• The degree \(n = 7\) (odd) and the leading coefficient \(a_n=-4<0\).
• For odd degree with negative leading coefficient, as \(x

ightarrow+\infty\), \(f(x)
ightarrow-\infty\) and as \(x
ightarrow-\infty\), \(f(x)
ightarrow+\infty\).

• Option 3: \(f(x)=10x^{2}-4x + 2\)
• The degree \(n = 2\) (even) and the leading coefficient \(a_n = 10>0\).
• For even degree with positive leading coefficient, as \(x

ightarrow+\infty\), \(f(x)
ightarrow+\infty\) and as \(x
ightarrow-\infty\), \(f(x)
ightarrow+\infty\).

• Option 4: \(f(x)=2x^{5}+4x^{2}-4\)
• The degree \(n = 5\) (odd) and the leading coefficient \(a_n=2>0\).
• For odd degree with positive leading coefficient, as \(x

ightarrow+\infty\), \(f(x)
ightarrow+\infty\) and as \(x
ightarrow-\infty\), \(f(x)
ightarrow-\infty\).

Assuming the graph (not fully shown but from the end - behavior analysis of the options) has the end - behavior of an odd - degree polynomial with a negative leading coefficient (as \(x
ightarrow+\infty\), \(y
ightarrow-\infty\) and as \(x
ightarrow-\infty\), \(y
ightarrow+\infty\)), the function \(f(x)=-4x^{7}-9x - 5\) (Option 2) matches this end - behavior.

Answer:

\(\boldsymbol{f(x)=-4x^{7}-9x - 5}\) (the second option in the given list of polynomial functions)