Unit
Polynomial and Rational Functions
Cornell University
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
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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 harder example of a polynomial function. We have h of x equals minus 1 over 24 x plus 4, x plus 2, x minus 1, x minus 3 and x minus 5. It's got five factors, this is going to be a fifth degree polynomial function, a quintic function.
Now the first thing I want to do is plot the intercepts, he intercepts correspond to zeros of this function, so -4 and -2 1, 3, 5. -4, -2, 1, 3, 5 and then let's figure out what the end behavior is going to be and for this we need to look at the leading term which is going to be -1 over 24 times x, x, x, x, x minus 1 over 24 x to the fifth.
If you recall a power function with an odd degree has ends pointing in the opposite direction, but with a negative sign in front it's going to go like this. So the left hand is going to point up, the right hand is going to point down for our polynomial.
Now to find out what happens in between the intercepts, we have to plot some points, so let's plot between -4 and -2, let's plot -3 and you get -1 over 24 buckle up there's going to be a lot of arithmetic, -1 over 24 -3 plus 4 is -1, -3 plus 2 -1, -3 minus 1 is -4, -3 minus -6, -3 minus 5 -8 and we get a little cancellation because of the 24. Now these two guys make 24, so I can cancel that and I'm left with -1 times -1 times -8, -8, so -3 and -8 down here.
And then let's plot a point between -2 and 1, you know what let's plot the y intercept, so 0 we get -1 over 24 times 4, 2, -1, -3, -5 and we'll get some cancellation again let's see, 4 times 2 times 3 that's another 24 and the negatives will actually cancel, so this will be completely gone right? And we'll be left with -1 times -5, 5 so the y intercept is 5 right about here.
Okay and then how about right in here x equals 2? We have -1 over 24, 2 plus 4 6, 2 plus 2 is 4, 2 minus 1, 1, 2 minus 3 -1 and 2 minus 5 -3. Again nice cancellation 24 6 and 4. We have -1 times -1 times -3, this is -3, so 2 -3, it's a little short of -4 so right about here and finally x equals 4.
We have as usual -1 over 24 and then 4 plus 4, 8, 4 plus 2 6, 4 minus 1, 3, 4 minus 3, 1 and 4 minus 5 -1, so 8 and 3 make 24 so we have cancellation again -6 times -1, 6. So 4, 6 it's about here halfway between 4 and 8. Finally we have enough points we can actually draw a nice graph here. Remember the left tail goes up, we're going to want to go through the xs, turn around it's probably going to go way up and then come back down like this.
Now I wouldn't worry too much if you don't know exactly where the maxs and mins occurs, it turns out that you really need Calculus to find out the exact location of maxs and mins, but the teacher is going to be very happy if you can give good qualitative behavior like this, that includes the intercepts of both x intercepts and y intercepts and just the general behavior, the correct end behavior and the correct location approximate location of turning points.
That's it even though really difficult a polynomial function, a fifth degree is, not too bad if you actually have it factored, but to completely be able to graph polynomial functions, we'll need to be able to deal with both repeating factors and later on in Calculus you'll learn how to deal with situations where the graph doesn't cross the x axis at all.