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### Equation for a W-Shaped Graph

```
Date: 08/01/99 at 04:05:03
From: Laureen clarke
Subject: Functions

I need to know what sort of equation has a graph that is shaped like a
W.

Thanks.
```

```
Date: 08/02/99 at 21:32:53
From: Doctor Ian
Subject: Re: Functions

Hi Laureen,

Well, you know right away that the function has to change direction
exactly three times:

(2)
.            .            .
.         .   .         .
.       .     .       .
.     .       .     .
.   .         .   .
.             .
(1)           (3)

Polynomials change direction like this, and the number of changes for
any polynomial is one less than the degree of the polynomial, i.e., a
line (degree 1) changes direction zero times, a parabola (degree 2)
changes direction once, and so on.

Since we need three changes, we'll want a polynomial of degree four.

If we assume that the changes in direction take place on opposite
sides of the x-axis,

.            .
.         .   .         .
.       .     .       .
------------------------------
.   .         .   .
.             .

then we know that the function has to cross the x-axis in exactly four
places. That means that the function will look like this:

y = (x - a)(x - b)(x - c)(x - d)

Not surprisingly, this is a polynomial of degree four. The exact shape
of the function will depend on the values that you choose for a, b, c,
and d.

Normally, a W is symmetric. So we can simplify a little:

y = (x + b)(x + a)(x - a)(x - b)

Can you take it from here?

Of course, if you can limit the domain, you could also choose subsets
of sine(x) or cosine(x). Do you see why?

I hope this helps. If not, be sure to write back.

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Equations, Graphs, Translations
High School Functions

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