Polynomial and Rational Functions
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 simple polynomial function. I want to graph f of x equals -x times x minus 1 times x plus 3. Now when you have the polynomial factored like this, you can tell what the units are going to be right? Because you're going to see the zeros 0, 1 and -3 and we're going to actually plot those now.
So we've got 0, 1 and -3 and the next thing I want to do is determine the end behavior and the end behavior as you recall comes from the leading term of the polynomial. So you have to multiply it out just enough to know what the leading term is. You have -x times x times x, the end behavior terms will be -x³ and the graph of -x³ looks like this, the left hand will go up and the right end will go up so that's important to know. Just keep in mind when we graph, the left end is going to go up the right end will go down.
Now usually what I like to do is plot just a couple of points in between the intercepts just to get an idea of the shape. So that's x and y, now let's plot we have -3 here, why don't we don't we do -2. And our function will have the opposite of -2, 2 times -2 minus 1, -3 -2 plus 3, 1. This is -6, so we get -2, -6 and our scale of our y axis is different, where I'll make this -6 and then that gives us the point -2, -6.
Let's plot -1 just to get an idea. I have a feeling this is going to curve, I just want to know how far down it goes, -1 opposite of -1 is 1, -1 minus 1, -2, -1 plus 3, 2 and we get -4 so -1, -4 is another point that we have to plot and I'll go right here and I think that's probably good enough let's draw our curve.
Remember the left end goes up, so we're going to come in like this and we have to pass through our intercepted 0, 0 and then go pass it and down like that, something like that. All right and that's a good enough graph. Make sure that you've got your intercepts clearly marked, I'll put a 1 here -3 and that's a pretty good graph of my cubic.