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Graphing Polynomial Functions with Repeated Factors - Problem 2

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

Let’s graph a slightly harder example. Y equals 2 times the quantity x plus 1 times the quantity x minus 2 to the third power. First of all since this is factored it’s really easy to find the x intercepts. They are (-1,0), for this factor and (2,0) for this one. And then the end behavior comes from the leading term. The leading term is the term with the highest power of x and that’s going to be 2 times x times x³ so 2x to the 4th. That’s a 4th degree and the coefficient’s positive so the two tails are going to go up.

Now all we have to do is let’s plot the -1, 2, 0 those are the intercepts, let’s plot some points between the intercepts to see what’s happening in between. For example let’s see what happens at zero. We’ve got x, 2 x plus 1, (x minus 2)³. At zero we have 2 times 1, times -2³. That’s -8 times 2, -16. Let’s plot a few more points before I decide what the scale is.

How about 1? 2 times 1 plus 1 is 2, 1 minus 2, -1³, looks like we’re going to get -1 times 4, -4. Okay let’s make this -16, so this will be -4 and at zero we get -16, so down here and at 1 we get -4, so here.

Let’s recall that we’re going to get some interesting behavior near the intercept that comes from this factor because this factor is cubed. So when we graph this, when we graph the curve it’s going to look a little bit like a cubic as it passes through x equals 2. So kind of like this. It will flatten out right at x equals 2 then come up again. It might be a good idea to actually plot a point at 3 just to see what happens there. I’m going to do that. 3, 2 times 3, 3 plus 1 is 4, 3 minus 2, 1³, I’m going to get 8. Let me plot that, (3, 8). I think I’ve got enough. Now just remember left tail will go up, right tail will go up. I think I’m ready to draw this.

I'll come in from up here, through that first intercept, down to here, I think I’m going to draw it from here, because remember it’s still a cubic. Something like that. There, how about that? You want the graph to reflect the behavior through the intercepts here. We have a simple intercept. The graph’s going to pass right through because we have an x plus 1, the power is 1 here. But here the graph behaves like a cubic. Like y equals x³. It flattens out before increasing again. Make sure you pay close attention to the powers of these repeated factors when you’re graphing.

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