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Degrees, Turnings, and "Bumps" (page 4 of 5)

Sections: End behavior, Zeroes and their multiplicities, "Flexing", Turnings & "bumps", Graphing

Compare the numbers of bumps in the graphs below to the degrees of their polynomials:

You can see from these graphs that, for degree n, the graph will have, at most, n – 1 bumps. The bumps represent the spots where the graph turns back on itself and heads back the way it came. This change of direction often happens because of the polynomial's zeroes or factors. But extra pairs of factors don't show up in the graph as much more than just a little extra flexing or flattening in the graph.

Because pairs of factors have this habit of disappearing from the graph (or hiding as a little bit of extra flexture or flattening), the graph may have two fewer, or four fewer, or six fewer, etc, bumps than you might otherwise expect, or it may have flex points instead of some of the bumps. That is, the degree of the polynomial gives you the upper limit (the ceiling) on the number of bumps possible for the graph (this upper limit being one less than the degree of the polynomial), and the number of bumps gives you the lower limit (the floor) on degree of the polynomial.

• What is the minimum possible degree of the polynomial graphed below?

Since there are four bumps on the graph, and since the end-behavior says that this is an odd-degree polynomial, then the degree of the polynomial is 5, or 7, or 9, or... But:

The minimum possible degree is 5.

• Given that a polynomial is of degree six, which of the following could be its graph?
     

To answer this question, I have to remember that the polynomial's degree gives me the ceiling on the number of bumps. In this case, the degree is 6, so the highest number of bumps the graph could have would be 6 – 1 = 5. But the graph, depending on the multiplicities of the zeroes, might have only 3 or 1 bumps.

(I would add 1 or 3 or 5, etc, if I were going from the number of displayed bumps on the graph to the possible degree of the polynomial, but here I'm going from the known degree of the polynomial to the possible graph, so I subtract.)

Also, I'll want to check the zeroes (and their multiplicities) to see if they give me any additional information.

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).

Graph B: This has seven bumps, so this is a polynomial of degree at least 8, which is too high.

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, so this is probably a zero of multiplicity 4 or more. With the two other zeroes looking like multiplicity-1 zeroes, this is a likely graph for a sixth-degree polynomial.

Graph D: This has six bumps, which is too many. On top of that, this is an odd-degree graph, since the ends head off in opposite directions. This can't 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 probably more than just a quadratic. This might be 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, I've got an even-multiplicity zero, a zero that looks like multiplicity-1, a zero that looks like at least a multiplicity-3, and another even-multiplicity zero. That gives me a minimum of 2 + 1 + 3 + 2 = 8 zeroes, which is too many for a degree-six polynomial. The bumps were right, but the zeroes were wrong. This can't be a degree-six graph.

Graph G: This is another odd-degree graph.

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.

Graphs B, D, F, and G can't possibly be graphs of degree-six polynomials.
Graphs A and E might be degree-six, and Graphs C and H probably are.

To help you keep straight when to add and when to subtract, remember your graphs of quadratics and cubics. Quadratics are degree-two polynomials and have one bump (always); cubics are degree-three polynomials and have two bumps or none (having a flex point instead). So going from your polynomial to your graph, you subtract, and going from your graph to your polynomial, you add. If you know your quadratics and cubics very well, and if you remember that you're dealing with families of polynomials and their family characteristics, you shouldn't have any trouble with this sort of exercise.

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