Final answer:
The graph of a polynomial with the given roots and multiplicities will touch the x-axis at x = -3 and x = 3 and cross the x-axis at x = 0 and x = 1. Considering the positive leading coefficient and even degree, the ends of the graph will face upwards, indicating that option A is the correct answer.
Step-by-step explanation:
Let's consider the characteristics of the polynomial function based on the given roots and their multiplicities. A root with even multiplicity means the graph of the function touches the x-axis at that root and turns around, while a root with odd multiplicity means the graph crosses the x-axis at that root.
Since the polynomial has a root of -3 with multiplicity 2, it will touch and turn at x = -3. The root of 0 with multiplicity 1 will have the graph crossing the x-axis at x = 0. The root of 1 with multiplicity 1 will also result in a crossing at x = 1. Lastly, the root of 3 with multiplicity 2 indicates another touch-and-turn at x = 3.
Given that the leading coefficient is positive and the polynomial degree is even, the ends of the polynomial will both point upwards. Hence, the correct graph of the function would begin with a positive slope, turning downwards at x = -3, crossing at x = 0, crossing again at x = 1, and finally turning upwards at x = 3.
The appropriate answer would be option A: A curve touching the x-axis at -3, 0, 1, and 3, as this reflects both the multiplicity of the roots and the positive leading coefficient of an even-degree polynomial.