Question

• a polynomial of degree 3
• a function with a positive leading coefficient
• $g(x) = -x^4 - 8x^3 - 24x^2 - 32x - 32$

this appears to be a polynomial of degree 3.
○ yes ○ no
this function appears to have a positive leading coefficient.
○ yes ○ no
this function appears to be $g(x)$
○ yes ○ no

this appears to be a polynomial of degree 3.
○ yes ○ no
this function appears to have a positive leading coefficient.
○ yes ○ no
this function appears to be $g(x)$.
○ yes ○ no

this appears to be a polynomial of degree 3.
○ yes ○ no
this function appears to have a positive leading coefficient.
○ yes ○ no
this function appears to be $g(x)$.
○ yes ○ no

Explanation:

Step1: Analyze the first graph

A degree 3 polynomial (cubic) has the end - behavior: as \(x
ightarrow+\infty\), if the leading coefficient \(a>0\), \(y
ightarrow+\infty\); as \(x
ightarrow-\infty\), \(y
ightarrow-\infty\) (or vice - versa for \(a < 0\)). The first graph has a "left - end down, right - end up" behavior, which is consistent with a cubic (\(n = 3\), odd) with positive leading coefficient. So for "This appears to be a polynomial of degree 3" the answer is Yes. For "This function appears to have a positive leading coefficient" the answer is Yes. Now, \(g(x)=-x^{4}-8x^{3}-24x^{2}-32x - 32\) is a degree 4 (quartic) function with leading coefficient \(- 1\) (negative). The first graph is a cubic, so it can't be \(g(x)\), so "This function appears to be \(g(x)\)" is No.

Step2: Analyze the second graph

The second graph has a vertical asymptote (since it has a break and the graph approaches a vertical line), so it's not a polynomial (polynomials are smooth and continuous everywhere). So "This appears to be a polynomial of degree 3" is No. Since it's not a polynomial, the question about positive leading coefficient is also No, and it can't be \(g(x)\) (which is a polynomial), so "This function appears to be \(g(x)\)" is No.

Step3: Analyze the third graph

The third graph has a vertical asymptote (non - polynomial behavior), so it's not a polynomial. So "This appears to be a polynomial of degree 3" is No. Since it's not a polynomial, the question about positive leading coefficient is No, and it can't be \(g(x)\) (a polynomial), so "This function appears to be \(g(x)\)" is No.

Answer:

First Graph:

• This appears to be a polynomial of degree 3: Yes
• This function appears to have a positive leading coefficient: Yes
• This function appears to be \(g(x)\): No

Second Graph:

• This appears to be a polynomial of degree 3: No
• This function appears to have a positive leading coefficient: No
• This function appears to be \(g(x)\): No

Third Graph:

• This appears to be a polynomial of degree 3: No
• This function appears to have a positive leading coefficient: No
• This function appears to be \(g(x)\): No