Question

a function with a negative leading coefficient

• $f(x)=x^3 + 12x^2 + 21x + 10$

first graph:
this appears to be a linear function.
yes no
this function appears to have a negative leading coefficient.
yes no
this function appears to be $f(x)$.
yes no

second graph:
this appears to be a linear function.
yes no
this function appears to have a negative leading coefficient.
yes no
this function appears to be $f(x)$.
yes no

third graph:
this appears to be a linear function.
yes no
this function appears to have a negative leading coefficient.
yes no
this function appears to be $f(x)$.
yes no

Explanation:

Step1: Analyze the function type

The given function is \( f(x) = x^3 + 12x^2 + 21x + 10 \), which is a cubic function (degree 3), while a linear function has degree 1. So for the "This appears to be a linear function" question, the answer is No.

Step2: Analyze the leading coefficient

The leading term of \( f(x) \) is \( x^3 \) with a leading coefficient of 1 (positive). So for "This function appears to have a negative leading coefficient", the answer is No.

Step3: Analyze the graph of \( f(x) \)

A cubic function \( f(x)=x^3 + 12x^2 + 21x + 10 \) will have a graph with end - behaviors: as \( x
ightarrow+\infty \), \( f(x)
ightarrow+\infty \) (since the leading coefficient is positive) and as \( x
ightarrow-\infty \), \( f(x)
ightarrow-\infty \). The first two graphs are not cubic (they look like even - degree functions, maybe quadratic with a repeated root) and the third graph is linear. So for "This function appears to be \( f(x) \)", the answer is No.

For the first set of questions (left graph):

• "This appears to be a linear function": No
• "This function appears to have a negative leading coefficient": No (the graph has a U - shape, suggesting a positive leading coefficient for a quadratic, but our function is cubic)
• "This function appears to be \( f(x) \)": No

For the second set of questions (middle graph):

• "This appears to be a linear function": No
• "This function appears to have a negative leading coefficient": No (the graph has a U - shape, positive leading coefficient for a quadratic - like graph)
• "This function appears to be \( f(x) \)": No

For the third set of questions (right graph):

• "This appears to be a linear function": Yes (it's a straight line)
• "This function appears to have a negative leading coefficient": No (the slope is positive, so leading coefficient of the linear function \( y = mx + b \) is positive)
• "This function appears to be \( f(x) \)": No ( \( f(x) \) is cubic, not linear)

Answer:

For the left - most graph:

• "This appears to be a linear function": No
• "This function appears to have a negative leading coefficient": No
• "This function appears to be \( f(x) \)": No

For the middle graph:

• "This appears to be a linear function": No
• "This function appears to have a negative leading coefficient": No
• "This function appears to be \( f(x) \)": No

For the right - most graph:

• "This appears to be a linear function": Yes
• "This function appears to have a negative leading coefficient": No
• "This function appears to be \( f(x) \)": No