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Graphing Polynomial Functions - Problem 2
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 another polynomial function this time g of x equals 1/3, x² minus 16 times x² minus 6x. Now in order to graph this, we're going to need to factor it completely that way we can figure out what the x intercepts are going to be.
This is a difference of squares so our factors is x minus 4, x plus 4 and this one has a common factor of x, so we can pull the x out and we're left with x minus 6 and that identifies 4,-4, 0, and 6 as the x intercepts. So let me plot those, but first I need to 1, 2, 3, 4 1, 2, 3, 4, 5, 6, 5, 6. All right we've got 6, we've got 4, we have 0 and we have -4. Those are our intercepts.
Then we have to determine the end behavior. The end behavior comes from the leading term which you would get if you multiply this out. Think about what product would give you the highest degree? X² times x² times a third, so our leading term is 1/3 x to the fourth and that means that the end behavior is going to look like an x to the fourth graph, both ends going up like this. Sometimes I just draw the ends just to remind myself the two ends go up, so go up to the left and up to the right.
Now to know what happens in between, I should plot a few points, so let's plot something in between -4 and 0 like -2 and I'll use the top. We have 1/3, -2² is 4, 4 minus 16 is -12, -2², 4, minus 6 times -2, so +12 so this is 1/3 and -12 times 16. The 1/3 and the -12 cancel to give me -4 times 16 that's -64. I don't want to plot too many points because the numbers are going to get crazy, so -64 let's go by 16s. 16, 32, 48, 64. So -2, -64, this is -64.
Let's plot 2, we have 1/3, 2² minus 16 again -12, 2 squares again 4 minus 6 times 2 minus 12 this time. So we have 1/3 times -12 times -8 and again the 3 and the 12 cancel leaving a 4. We have -4 times -8, 32 so we're going up 16, 32 and that's going to be a point and I'll plot one point in here x equals 5.
At 5 we have 1/3 of 25 minus 16, 25 minus 6 times 5 minus 30, so this is 1/3 of 25 minus 16 is 9 and this is -5, so we have the 3's canceling -15. Now -15 is pretty close to -16 so just a little bit more, so I'm going to be just a little above this line right there. That's probably enough to get a decent graph of my fourth degree function, remember the ends go up so I'm going to come in through the top here and I got to go up.
Now remember the maxs don't necessarily coincide with the points I have plotted so I don't want to make it look like that happens necessarily and that's a pretty good graph of my fourth degree polynomial function.
Just remember first you need to factor to find the x intercepts, second determine the end behavior by looking at the leading term, third plot some points and then fill in your graph.
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