A rectangular piece of cardboard is being folded into a box. A portion of the graph represents the volume of the box based on the length of the piece of cardboard.

A graph of a function which forms an S shape, with two bumps, one at approximately (1.5, 65), and the other one at approximately (5, negative 20). The left side of the function points down, while the right side points up.

This function could be modeled by a polynomial with what degree?

There may be more than one correct answer. Select all correct answers.

degree 3
degree 4
degree 2
degree 5
degree 6

I think it is 3 degrees

correct.

while any odd-degree polynomial may have that shape, since a box is a 3-D figure, a degree-3 polynomial is the best choice.

To determine the degree of the polynomial that could model the given function, we need to examine the number of bumps or turning points on the graph. Each bump or turning point corresponds to a zero of the derivative of the function.

In this case, the graph has two bumps or turning points. Therefore, the derivative of the polynomial will have two zeros, indicating two turning points. The degree of the polynomial is directly related to the number of zeros (or turning points) of its derivative.

Hence, the degree of the polynomial could be either 3 or 5.

To confirm this, we can examine the options given:

- Degree 3: A polynomial of degree 3 could have two turning points (zeros of its derivative), so it matches the behavior observed on the graph.
- Degree 4: A polynomial of degree 4 could have at most three turning points. Since the graph shows two turning points, a degree 4 polynomial may not be necessary to model it accurately.
- Degree 2: A quadratic polynomial of degree 2 typically has one turning point unless it is a perfect parabola, which doesn't match the given graph.
- Degree 5: A polynomial of degree 5 could have at most four turning points, which is more than what the graph shows.
- Degree 6: A polynomial of degree 6 could have at most five turning points, which is again more than what the graph shows.

Based on this analysis, the correct answers are degree 3 or degree 5.