Question

what is the degree of the polynomial graphed? negative odd even positive

Explanation:

Step1: Analyze end - behavior of polynomial

To determine if the degree of a polynomial is odd or even, we look at the end - behavior of the polynomial graph. For a polynomial function \(y = a_nx^n+a_{n - 1}x^{n - 1}+\cdots+a_1x + a_0\), as \(x\to+\infty\) and \(x\to-\infty\):

• If the degree \(n\) is even:
• If the leading coefficient \(a_n>0\), then as \(x\to+\infty\) and \(x\to-\infty\), \(y\to+\infty\).
• If the leading coefficient \(a_n < 0\), then as \(x\to+\infty\) and \(x\to-\infty\), \(y\to-\infty\).
• If the degree \(n\) is odd:
• If the leading coefficient \(a_n>0\), then as \(x\to+\infty\), \(y\to+\infty\) and as \(x\to-\infty\), \(y\to-\infty\).
• If the leading coefficient \(a_n < 0\), then as \(x\to+\infty\), \(y\to-\infty\) and as \(x\to-\infty\), \(y\to+\infty\).

From the given graph, we observe the end - behavior: as \(x\to+\infty\), the graph goes in one direction and as \(x\to-\infty\), it goes in the opposite direction. This is characteristic of a polynomial with an odd degree. The options "Negative" and "Positive" refer to the sign of the leading coefficient, not the degree. The option "Even" would imply that the ends of the graph go in the same direction (both up or both down), which is not the case here.

Step2: Conclusion

Based on the end - behavior analysis, the degree of the polynomial is odd.

Answer:

B. Odd