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 Polynomial Functions - Concept

Norm Prokup
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.


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.

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