The graph in the image represents a polynomial function. Let's break down the information needed to determine the degree of the polynomial and the sign of the leading term.

1. Degree of the Polynomial:

• The degree of a polynomial is determined by the number of turning points and the overall behavior of the graph.
• A polynomial of degree $n$ can have up to $n-1$ turning points (local maxima and minima).
• In this graph, there are three turning points (two maxima and one minima).
• This suggests that the polynomial could be of degree 3, 4, or 5, since a cubic (degree 3) can have up to 2 turning points, a quartic (degree 4) can have up to 3, and a quintic (degree 5) can have up to 4.

2. Sign of the Leading Coefficient:

• The end behavior of the graph helps determine the sign of the leading coefficient.
• The graph starts low (as $x \to -\infty$, $f(x) \to -\infty$) and ends high (as $x \to +\infty$, $f(x) \to +\infty$).
• This indicates that the leading term has a positive coefficient for an odd-degree polynomial.

Conclusion:

• The most plausible degrees for this polynomial are 3, 4, or 5.
• The leading coefficient is positive, not negative as indicated in the answer choice. The graph increases as $x \to \infty$, which corresponds to a positive leading term in an odd-degree polynomial.

If you have any more questions or need further clarification, feel free to ask!

5 Related Questions:

1. How do you determine the degree of a polynomial from its graph?
2. What is the relationship between the degree of a polynomial and its turning points?
3. How does the leading coefficient of a polynomial affect the graph’s end behavior?
4. Why does an odd-degree polynomial exhibit different end behaviors on each side?
5. What are the characteristics of a polynomial function with an even degree?

Tip:

Remember, the number of turning points gives a clue about the degree of the polynomial, but the end behavior is crucial for determining the sign of the leading coefficient.