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.

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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.