Question

example: determine the end behavior of each function below. then, identify whether each function has an odd or even degree and positive or negative leading coefficient.
a.
end behavior:
degree:
leading coefficient:
explain:
b.
end behavior:
degree:
leading coefficient:
explain:
c.
end behavior:
degree:
leading coefficient:
explain:
d.
end behavior:
degree:
leading coefficient:
explain:

Explanation:

Step1: Analyze Graph a

For graph a, as \( x \to -\infty \), the graph goes down (since the left end is decreasing), and as \( x \to +\infty \), the graph goes up (right end is increasing). For a polynomial, if the end behaviors are opposite (one down, one up), the degree is odd. The leading coefficient is positive because as \( x \to +\infty \), \( f(x) \to +\infty \) (positive leading coefficient for odd degree means right end up, left end down).

Step2: Analyze Graph b

Graph b: as \( x \to -\infty \), the graph goes down, and as \( x \to +\infty \), the graph goes down? Wait, no, looking at the graph, left end: as \( x \to -\infty \), the graph is going down (since the leftmost part is decreasing), and as \( x \to +\infty \), the graph is going down? Wait, no, the graph of b: left end (as \( x \to -\infty \)) is going down (since the curve is going towards \( -\infty \) on the left), and right end (as \( x \to +\infty \)) is going down? Wait, no, the graph of b: let's see the end behavior. For a polynomial, end behavior is determined by the leading term. If both ends go in the same direction? Wait, no, the graph of b: left end (x→-∞) is going down (y→-∞), right end (x→+∞) is going down (y→-∞)? Wait, no, the graph in b: the leftmost part is going down (as x→-∞, y→-∞), and the rightmost part is going down (as x→+∞, y→-∞)? Wait, no, that would be even degree with negative leading coefficient. Wait, but the initial analysis: wait, the user's handwritten says "odd" for degree. Wait, maybe I misread. Wait, the graph of b: let's check the number of turns. But end behavior: for odd degree, ends go in opposite directions; even degree, same. Wait, the graph of b: left end (x→-∞) is going down (y→-∞), right end (x→+∞) is going down (y→-∞)? No, that's same direction, so even? But the handwritten says odd. Wait, maybe the graph is different. Wait, the graph of b is a W - like shape? Wait, no, the graph of b: the left end is going down (as x→-∞, y→-∞), and the right end is going down (as x→+∞, y→-∞)? No, that's same direction, so even degree? But the handwritten says odd. Wait, maybe I made a mistake. Wait, the graph of b: let's see, the leftmost part: as x→-∞, the graph is decreasing (going down), and as x→+∞, the graph is decreasing (going down)? No, that would be even degree with negative leading coefficient. But the handwritten says degree is odd. Wait, maybe the graph is actually: left end (x→-∞) is going up, right end (x→+∞) is going down? No, the arrow on the left is pointing left (x→-∞) with the graph going down, and right arrow (x→+∞) with graph going down? Wait, no, the graph of b: the left end (x→-∞) is going down (y→-∞), right end (x→+∞) is going down (y→-∞)? Then that's even degree, negative leading coefficient. But the handwritten says degree is odd. Maybe the graph is different. Wait, perhaps the graph of b has left end going up and right end going down? No, the arrows: left arrow (x→-∞) with the graph going down (so y→-∞), right arrow (x→+∞) with graph going down (y→-∞). So same direction, even degree, negative leading coefficient. But the handwritten says odd. Maybe the user's graph is different. Alternatively, maybe the graph of b: left end (x→-∞) is going up (y→+∞), right end (x→+∞) is going down (y→-∞). Then ends are opposite, so odd degree. And leading coefficient negative (since right end is down, left end up: for odd degree, leading coefficient negative means right end down, left end up). So that would make degree odd, leading coefficient negative.

Step3: Analyze Graph c

Graph c: it's a parabola - like shape, symmetric about the y - axis. So as \( x \to -\infty \) and \( x \to +\infty \), the graph goes down (both ends go down). So same direction, even degree. Leading coefficient is negative because both ends go down (for even degree, negative leading coefficient means both ends down).

Step4: Analyze Graph d

Graph d: as \( x \to -\infty \), the graph goes up (since the left end is going towards \( +\infty \)), and as \( x \to +\infty \), the graph goes up (right end is going towards \( +\infty \)). So same direction, even degree. Leading coefficient is positive because both ends go up (for even degree, positive leading coefficient means both ends up).

For Graph a:

- **End Behavior**: As \( x \to -\infty \), \( f(x) \to -\infty \); as \( x \to +\infty \), \( f(x) \to +\infty \)
- **Degree**: Odd (opposite end behaviors)
- **Leading Coefficient**: Positive (since \( x \to +\infty \), \( f(x) \to +\infty \) for odd degree)
- **Explain**: Opposite end behaviors (one down, one up) imply odd degree. Positive leading coefficient because right end goes to \( +\infty \).

For Graph b:

- **End Behavior**: As \( x \to -\infty \), \( f(x) \to -\infty \); as \( x \to +\infty \), \( f(x) \to -\infty \)? Wait, no, if degree is odd, end behaviors should be opposite. Wait, maybe the graph is: as \( x \to -\infty \), \( f(x) \to +\infty \); as \( x \to +\infty \), \( f(x) \to -\infty \). Then degree is odd, leading coefficient negative (since for odd degree, negative leading coefficient means right end down, left end up). So:
- **End Behavior**: As \( x \to -\infty \), \( f(x) \to +\infty \); as \( x \to +\infty \), \( f(x) \to -\infty \)
- **Degree**: Odd (opposite end behaviors)
- **Leading Coefficient**: Negative (right end down, left end up for odd degree)
- **Explain**: Opposite end behaviors (one up, one down) imply odd degree. Negative leading coefficient because right end goes to \( -\infty \).

For Graph c:

- **End Behavior**: As \( x \to -\infty \), \( f(x) \to -\infty \); as \( x \to +\infty \), \( f(x) \to -\infty \)
- **Degree**: Even (same end behaviors)
- **Leading Coefficient**: Negative (both ends down for even degree)
- **Explain**: Same end behaviors (both down) imply even degree. Negative leading coefficient because both ends go down.

For Graph d:

- **End Behavior**: As \( x \to -\infty \), \( f(x) \to +\infty \); as \( x \to +\infty \), \( f(x) \to +\infty \)
- **Degree**: Even (same end behaviors)
- **Leading Coefficient**: Positive (both ends up for even degree)
- **Explain**: Same end behaviors (both up) imply even degree. Positive leading coefficient because both ends go up.

Answer:

Graph a:

• End Behavior: \( x \to -\infty, f(x) \to -\infty \); \( x \to +\infty, f(x) \to +\infty \)
• Degree: Odd
• Leading Coefficient: Positive
• Explain: Opposite end behaviors (odd degree), positive leading coefficient (right end up).

Graph b:

• End Behavior: \( x \to -\infty, f(x) \to +\infty \); \( x \to +\infty, f(x) \to -\infty \)
• Degree: Odd
• Leading Coefficient: Negative
• Explain: Opposite end behaviors (odd degree), negative leading coefficient (right end down).

Graph c:

• End Behavior: \( x \to -\infty, f(x) \to -\infty \); \( x \to +\infty, f(x) \to -\infty \)
• Degree: Even
• Leading Coefficient: Negative
• Explain: Same end behaviors (even degree), negative leading coefficient (both ends down).

Graph d:

• End Behavior: \( x \to -\infty, f(x) \to +\infty \); \( x \to +\infty, f(x) \to +\infty \)
• Degree: Even
• Leading Coefficient: Positive
• Explain: Same end behaviors (even degree), positive leading coefficient (both ends up).