Graphing Polynomial Functions

Graphing higher degree polynomial functions can be more complicated than graphing linear and quadratic functions. When graphing polynomial functions, we can identify the end behavior, shape and turning points if we are given the degree of the highest term.

I want to talk about graphing higher degree polynomial functions. Now, you've probably already graphed lots and lots of linear functions and quadratics and those are polynomial functions, but I want to talk about cubic, quartic and higher degree polynomial functions, what's their shape, what should we expect when we're graphing them.

Let's take a look at a demonstration on Geometer Sketch pad to see. Okay. We're in Geometer Sketch Pad.

What we're looking at right now is a cubic function F of X equals X cubed plus 1. I want show you a bunch of cubic functions so we can get some intuition about what their shape is.

These are all four cubic functions. And you can see that one of the things about cubic functions is that they have this sort of pattern of end behavior. If the left end goes down, the right end goes up. You see the blue one and the purple one have that property. The left end goes down, the right end goes up and the red and orange have the opposite property. The left goes up the right end goes down. To generalize this, opposite ends go in opposite directions. That's really important.

The other thing about cubics is that they all have this thing called an inflexion point, it's a twist in the graph right in the center right here or here or here for the blue one. And somewhere in here for the red one.

It's where the graph goes from curved upward to curved downward. And that's going to be a really important feature of these graphs.

So again opposite end behavior and an inflection point. Oh, one more thing, notice that some of these graphs have turning points, right? We have a minimum here and a maximum here. And the blue one has a maximum here and a minimum here. Whereas there's no max or min on the purple graph. No max or min on the red graph. As it's going to turn out there can only be two turning points or 0 in a cubic function. So that will be important.

Let's take a look at fourth degree polynomial functions which are called quartic functions. Here we have F of X equals X plus 1 quantity of the fourth plus 3. Here's another. Another and another.

So we have four quartic, four fourth degree polynomial functions. And notice that their end behavior, both ends do the same thing. In this case the left and right sides go up on the purple graph and the blue one, both ends go up. But in the orange and red ones both ends go down. So whatever the right end does the left end also does. That's important for fourth degree polynomials.

Now what about turning points? You'll notice that this graph, the purple one, only has one turning point. It's got a minimum here at the bottom whereas the blue one has a minimum, a maximum and another minimum. Three turning points. Now, this graph, the orange one, looks like it has one and it almost has more but it doesn't. It just kind of levels off. Looks like an inflection point and then it goes down. I would count this as one turning point. The same with the red one, one turning point.

Looks like these guys can have one turning point or three. So that's sort of interesting. And, again, note the end behavior. Both ends go in the same direction. Let's reveal what we've learned here about higher degree polynomial functions. All right.

So the graphs of higher degree polynomial functions first and foremost the graphs are always smooth and continuous. Now that just means there aren't going to be any corners and there aren't going to be any breaks in the graph as you draw them. They'll always have nice curves and be nice and smooth.

Maximum number of turning points, remember that the cubic had at most two and the quartic had at most 3. So it's going to be the degree of the polynomial minus 1. As far as end behavior goes, let me just draw a sample.

There's a cubic. And a quartic might look something like this. Remember the cubics, the ends go in opposite directions and with quartics the ends go in the same direction. I want to explain end behavior a little more closely. So let's take a look at this PowerPoint.

I've got two functions graphed here. One of them is the polynomial F. And the other is a power function G. Let's see what these look like when we zoom in on them. Actually, we're zooming out. Notice that as we zoom out, the details of the polynomial function are completely gone. The little wobble it had. And the ends are starting to look more and more alike.

And as we zoom out more, now we're on the order of a million, the two graphs are really close together. They're so close that they are almost indistinguishable. And zooming out a little more, they become completely indistinguishable from one another.

So this tells us that the determining factor in the end behavior of a polynomial function is the leading term. Notice that the leading term, that's exactly what we have functioned here, graphed here in blue.

The leading term is a power function. It's the power function with the highest degree. That's going to determine your end behavior completely. Both ends will go up or both ends will go down or everything is determined by the leading term and its coefficient.

So that's one more thing I want to write down here. Leading term. Does what? It determines end behavior. Really important.

So knowing these things, the graphs are smooth and continuous, the max number of turning points is the degree of the polynomial minus 1. Remembering that the ends of a cubic go in opposite directions the ends of a quartic go in the same direction and that the leading term determines end behavior that is enough to get us started graphing polynomials.