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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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