Degrees, Turnings, and "Bumps"

Graph A: This shows one bump (so not too many), but only two zeroes, each looking like a multiplicity-1 zero. This is probably just a quadratic, but it might possibly be a sixth-degree polynomial (with four of the zeroes being complex). I would have expected at least one of the zeroes to be repeated, thus showing flattening as the graph flexes through the axis. But this could maybe be a sixth-degree polynomial's graph.

Graph B: This has seven bumps, so this is a polynomial of degree at least 8, which is too high. This graph cannot possibly be of a degree-six polynomial.

Graph C: This has three bumps (so not too many), it's an even-degree polynomial (being "up" on both ends), and the zero in the middle is an even-multiplicity zero. Also, the bump in the middle looks flattened at the axis, so this is probably a repeated zero of multiplicity 4 or more. With the two other zeroes looking like multiplicity-1 zeroes, this is very likely a graph of a sixth-degree polynomial.

Graph D: This has six bumps, which is too many; this is from a polynomial of at least degree seven. On top of that, this is an odd-degree graph, since the ends head off in opposite directions. So this can't possibly be a sixth-degree polynomial.

Graph E: From the end-behavior, I can tell that this graph is from an even-degree polynomial. The one bump is fairly flat, so this is more than just a quadratic. This might be the graph of a sixth-degree polynomial.

Graph F: This is an even-degree polynomial, and it has five bumps (and a flex point at that third zero). But looking at the zeroes, the left-most zero is of even multiplicity; the next zero passes right through the horizontal axis, so it's probably of multiplicity 1; the next zero (to the right of the vertical axis) flexes as it passes through the horizontal axis, so it's of multiplicity 3 or more; and the zero at the far right is another even-multiplicity zero (of multiplicity two or four or...). Adding these up, the number of zeroes is at least 2 + 1 + 3 + 2 = 8 zeroes, which is way too many for a degree-six polynomial. The bumps were right, but the zeroes were wrong. This can't possibly be a degree-six graph.

Graph G: The graph's left-hand end enters the graph from above, and the right-hand end leaves the graph going down. Since the ends head off in opposite directions, then this is another odd-degree graph. As such, it cannot possibly be the graph of an even-degree polynomial, of degree six or any other even number.

Graph H: From the ends, I can see that this is an even-degree graph, and there aren't too many bumps, seeing as there's only the one. Looking at the two zeroes, they both look like at least multiplicity-3 zeroes. So this could very well be a degree-six polynomial.