Below are a few examples of the polynomial sketching ideas from class. I haven't attempted to hide the Mathematica code (which is written in this bold font) that produces the graphs, but you certainly needn't worry about it. In fact, you might find the examples useful to look at sometime, especially the easy ones.

Keep in mind what the basic power functions from section 9-1 looked like. For example, pictured below, in no particular order, are the graphs of , where is one of each of the functions

.

You should easily be able to tell which is which. I have shuffled them using the Mathematica code below.

Looks wicked, eh? But it just shuffles a list, as in the following example

The code that produces the graph is needlessly complicated by me having fun making an array of plots while not letting you see which is which. Here goes:

Okay, down to business. Check out the following graph of . The first graph wil have a "true aspect ratio" in that the lengths along the and -axes are equal. Obviously, it just won't do.

So we'll let Mathematica pick an aspect ratio it likes. It works well for many occasions. Just keep in mind that the graph is squished in the vertical direction.

How would we know what this looks like without a computer? The first thing I like to is ask, What happens when x is BIG? As we have said before, when is very large, then in some sense we can say that

  and  .

The implication this has for us is that if we "zoom out" from our graph, that is, look at it for BIG values of , then we should see

,

and so our graph should look rather like that of when we back up, or zoom out. Here is that graph, from farther away. (All these ideas: zooming out, backing out, looking from farther away, are all achieved by taking larger intervals for .)

So at least we expect our graph should go down on the right and up on the left. But actually, the similarity of the two graphs is much stronger. The graphs are asymptotic to one another for large . Have a look at them together:

Back up even further:

Eventually you can't tell the difference:

We pause just a moment with some details for the interested student: The mathematical meaning of the statement that " is asymptotic to for large " is that

for large .  Even more precisely, we mean that the ratio above gets arbitrarily close to 1 for sufficiently large . We observed this sort of asymptotic behavior when we looked at the asymptotes of hyperbolas. In this case it is even easier to understand. Let's multiply out the numerator. Better yet, we'll have Mathematica do it using the Expand function:

(The built-in functions TraditionalForm and HoldForm just let us keep Mathematica from displaying it the way she wants. Here is what she really wants to do:)

Now look atthe above expression and consider what happens for very large . It should be clear now that the three terms on the right become as small as you like for sufficiently large . The resulting quantity is therefore as close to 1 as you wish, for large enough . By the way, a simple way to write this uses some notation you'll meet in calculus:

We now return to our regularly scheduled program...

Let's move back in on the graph. Recall that we are interested in the zeros of our poly, which are at and . We'll highlight them with orange dots. Why? Because we can!

(Keep in mind that the aspect ratio is not a true one.)  Zoom in on the zero at :

By golly, that sure reminds me of a parabola. And in fact, it "almost" is. Furthermore, the parabola it looks most like, in one sense, is given by

.

Wanna see? Here they are together, still zoomed in near :

They're very close! How did we come up with ? Simple: if , then

.

So for , we have

.

Okay, let's zoom in on the other zero, at :

We can see it is not straight. But if we zoom in further it looks almost straight:

And if this indeed nearly a atraight line, which line might it most resemble? Do the same thing as above, namely, let . Then

,

so,

.

Let's see them together.

Yup. They're close.

Let's see all three graphs together (because we can). The parabola is red, the line is green.

You do agree that this cool, don't you?

To use these ideas when sketching factored polynomials, we don't actually have to go through all of the above. (If you are about to ask, "Then why did you go through all that stuff?" then either (1) you haven't been in class much, or (B) go jump in a lake.) We know that our graph

   1. looks like when is "far out",
   2. looks like a parabola very near , and
   3. looks like straight very near .
   
   We can start sketching, beginning with large positive . We
   
   a. start on the right like ,
   b. bounce off the -axis like a parabola when we get to ,
   c. slice through the -axis like a line when we get to , and
   d. finish like .

I hope this helps. Send me email if you have any suggestions.

R. Mabry