Q&A #820

Explaining the relationship between the degree of a function and its shape

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From: Marielouise (for Teacher2Teacher Service)
Date: Nov 21, 1998 at 13:38:25
Subject: Re: Explaining the relationship between the degree of a function and its shape

Hi, Paul,

There is a fantastic video called "Polynomials" that comes from Project
Mathematics! at California Institute of Technology. It is available from

Aside from the video: start with the idea of a power function y = x^n  and
look at the two cases when n = 2k , an even number or when n = 2k + 1, an odd
number. All even power functions reside in the I and II quadrants and all
odd power functions reside in the I and III. From this perspective look at
each y = (x - 0)^ n  and start changing the 0 to a different root. Therefore,
 y = (x - 1 )( x + 3 ) ( x - 4) in the macro or big picture still looks like
a cubic with three roots at zero. However, now between -3 and 4 it has some
hills and valleys.

I tend to teach this material as a game of croquet where the roots and the
y-intercept are the wickets and I am the ball. The only problem you have
with uniqueness is when the y-intercept is one of the roots. Otherwise you
have only one path to take. You cannot tell how high the mountains or
valleys are until you actually compute them in calculus.

I hope that I have given you some ideas.

 -Marielouise, for the Teacher2Teacher service

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