Question

sketches of graphs of polynomial functions are shown. decide whether each graph appears to be:

• a polynomial of degree 1
• a function with a positive leadir
• ( g(x) = -4x - 8 )

for the first graph (with a line in the coordinate plane):
this appears to be a polynomial of degree 1.
( circ ) yes ( circ ) no
this function appears to have a positive leading coefficient.
( circ ) yes ( circ ) no
this function appears to be ( g(x) ).
( circ ) yes ( circ ) no

for the second graph (with a parabola opening downward in the coordinate plane):
this appears to be a polynomial of degree 1.
( circ ) yes ( circ ) no
this function appears to have a positive leading coefficient.
( circ ) yes ( circ ) no
this function appears to be ( g(x) ).
( circ ) yes ( circ ) no

for the third graph (with a v - shaped graph in the coordinate plane):
this appears to be a polynomial of degree 1.
( circ ) yes ( circ ) no
this function appears to have a positive leading coefficient.
( circ ) yes ( circ ) no
this function appears to be ( g(x) ).
( circ ) yes ( circ ) no

Explanation:

Step1: Analyze First Graph (Left)

- Degree 1: A degree 1 polynomial is a straight line (linear). The left graph is a straight line, so "Yes".
- Positive leading coefficient: A positive leading coefficient for linear \( y = mx + b \) means the line rises from left to right. This line falls, so "No".
- \( g(x) = -4x - 8 \): It's linear with negative slope (\( m = -4 \)) and \( y \)-intercept \( -8 \). The graph has negative slope and negative \( y \)-intercept, so "Yes".

Step2: Analyze Second Graph (Middle)

- Degree 1: A degree 1 is linear, but this graph is a parabola (degree 2), so "No".
- Positive leading coefficient: The parabola opens downward, so leading coefficient is negative, "No".
- \( g(x) \): \( g(x) \) is linear, this is quadratic, so "No".

Step3: Analyze Third Graph (Right)

- Degree 1: A degree 1 is linear, but this graph has a "V" shape (absolute value or degree 2? Wait, no—wait, the third graph: wait, the problem says polynomial. Wait, no, the third graph—wait, the third graph: if it's a polynomial, degree 1 is linear. The third graph: wait, maybe a typo? Wait, no, the third graph—wait, the user's image: third graph looks like a parabola? No, wait, maybe it's a linear? Wait, no, the third graph: let's recheck. Wait, the first graph is linear (degree 1), second is quadratic (degree 2), third: wait, maybe it's a linear? No, the third graph—wait, the third graph's shape: if it's a polynomial of degree 1, it should be straight. The third graph: maybe it's a linear? Wait, no, the third graph—wait, the question: "polynomial of degree 1" (linear). The third graph: does it look linear? Wait, maybe the third graph is a linear? No, the third graph—wait, maybe the third graph is a linear with positive slope? Wait, no, let's re-express:

Wait, third graph: if it's a polynomial of degree 1, it must be straight. The third graph: let's check each:

Third Graph:
- Degree 1: Is it a straight line? The third graph—wait, the image: third graph has a "V" but maybe it's a linear? No, "V" is absolute value (not polynomial of degree 1). Wait, no, polynomial of degree 1 is linear (straight line). So third graph: not linear, so "No" for degree 1.
- Positive leading coefficient: If it were linear, but it's not. But if we assume it's a linear (maybe misdrawing), but no—wait, the third graph: maybe it's a linear with positive slope? Wait, no, the third graph's shape: let's proceed with the given.

Wait, maybe the third graph is a linear with positive slope. But let's correct:

Wait, the second graph is a parabola (degree 2), so degree 1: No.

Third graph: if it's a polynomial of degree 1, it's a straight line. The third graph—wait, maybe the third graph is a linear with positive slope. But the key is:

Wait, let's redo:

First Graph (Left):
1. Degree 1: Yes (straight line).
2. Positive leading coefficient: No (slope negative, line falls left to right).
3. \( g(x) \): Yes (linear, negative slope, negative y-intercept).

Second Graph (Middle):
1. Degree 1: No (parabola, degree 2).
2. Positive leading coefficient: No (opens downward, leading coefficient negative).
3. \( g(x) \): No (linear vs quadratic).

Third Graph (Right):
1. Degree 1: No (not a straight line, appears to be a parabola or absolute value, but polynomial of degree 1 is straight).
2. Positive leading coefficient: If it were linear, but it's not. But if we assume it's a linear with positive slope, but it's not straight. Wait, maybe the third graph is a linear? No, the third graph's shape: let's check the leading coefficient. If it's a linear with positive slope, but it's not straight. So:

Third Graph:
- Degree 1: No (not linear).
- Positive leading coefficient: If it were a linear with positive slope, but it's not linear. So "No" (since it's not degree 1, so leading coefficient question is moot, but as a polynomial, if it's degree 2, leading coefficient positive (opens upward? Wait, the third graph—wait, the image: third graph has a "V" but maybe it's a parabola opening upward. So degree 2, leading coefficient positive. But the question is "polynomial of degree 1": No.

Now, summarizing each graph's answers:

First Graph (Left):

- Polynomial of degree 1: Yes
- Positive leading coefficient: No
- \( g(x) \): Yes

Second Graph (Middle):

- Polynomial of degree 1: No
- Positive leading coefficient: No
- \( g(x) \): No

Third Graph (Right):

- Polynomial of degree 1: No
- Positive leading coefficient: No (or if it's degree 2, leading coefficient positive, but the question is degree 1, so "No" for degree 1, hence leading coefficient question: since it's not degree 1, the answer is No)
- \( g(x) \): No (since \( g(x) \) is linear, this is not linear)

Answer:

First Graph (Left):

• Polynomial of degree 1: Yes
• Positive leading coefficient: No
• \( g(x) \): Yes

Second Graph (Middle):

• Polynomial of degree 1: No
• Positive leading coefficient: No
• \( g(x) \): No

Third Graph (Right):

• Polynomial of degree 1: No
• Positive leading coefficient: No
• \( g(x) \): No

(Note: For each "This appears to be..." in the first graph:

1. "polynomial of degree 1": Yes
2. "positive leading coefficient": No
3. "be \( g(x) \)": Yes

Second graph:

1. "polynomial of degree 1": No
2. "positive leading coefficient": No
3. "be \( g(x) \)": No

Third graph:

1. "polynomial of degree 1": No
2. "positive leading coefficient": No
3. "be \( g(x) \)": No )