Question

1. identify whether each function graphed has an odd or even degree and a positive or negative leading coefficient.

first graph:
degree: __________
leading coefficient: ______

second graph:
degree: __________
leading coefficient: ______

third graph:
degree: __________
leading coefficient: ______

Explanation:

First Graph (Left)

Step1: Analyze End Behavior (Degree)

For a polynomial function, the end - behavior is determined by the degree (odd/even) and the leading coefficient. If the ends of the graph go in the same direction (both up or both down), the degree is even. If the ends go in opposite directions, the degree is odd. In the left - most graph, as \(x
ightarrow-\infty\), the graph goes down, and as \(x
ightarrow+\infty\), the graph goes up. So the degree is odd.

Step2: Analyze End Behavior (Leading Coefficient)

For an odd - degree polynomial, if as \(x
ightarrow+\infty\), \(y
ightarrow+\infty\), the leading coefficient is positive. In the left - most graph, as \(x
ightarrow+\infty\), \(y
ightarrow+\infty\), so the leading coefficient is positive.

Second Graph (Middle)

Step1: Analyze End Behavior (Degree)

In the middle graph, as \(x
ightarrow-\infty\), the graph goes up, and as \(x
ightarrow+\infty\), the graph goes up. Since the ends of the graph go in the same direction, the degree is even.

Step2: Analyze End Behavior (Leading Coefficient)

For an even - degree polynomial, if as \(x
ightarrow+\infty\), \(y
ightarrow+\infty\), the leading coefficient is positive. In the middle graph, as \(x
ightarrow+\infty\), \(y
ightarrow+\infty\), so the leading coefficient is positive.

Third Graph (Right)

Step1: Analyze End Behavior (Degree)

In the right - most graph, as \(x
ightarrow-\infty\), the graph goes up, and as \(x
ightarrow+\infty\), the graph goes down. Since the ends of the graph go in opposite directions, the degree is odd.

Step2: Analyze End Behavior (Leading Coefficient)

For an odd - degree polynomial, if as \(x
ightarrow+\infty\), \(y
ightarrow-\infty\), the leading coefficient is negative. In the right - most graph, as \(x
ightarrow+\infty\), \(y
ightarrow-\infty\), so the leading coefficient is negative.

Answer:

• First Graph: Degree: Odd, Leading Coefficient: Positive
• Second Graph: Degree: Even, Leading Coefficient: Positive
• Third Graph: Degree: Odd, Leading Coefficient: Negative