



What's
the graph of Y = X ^{4} look like?



Well
pretty much like the graph of Y = X
^{2} but narrower. 





The graph
of Y = X ^{4} is narrower than the graph of Y = X
^{2} 


everywhere
except between X = 1 and X = 1. 


In that
area, the graph of Y = X ^{4} is wider. 





This is
because when you multiply a fraction times itself 


the
answer is smaller than the fraction you used. 











And the
more times you multiply the fraction times itself 


the
smaller the answer gets ... 











And a
funny thing happens when you put ANY exponent on 1 ... 


1^{5} 
=
1 
1^{1} 
=
1 
1^{0} 
=
1 
1^{3} 
=
1 
1^{2000} 
=
1 
1^{(any
exponent)} 
=
1 






So what
does that mean to us? 


It means
that if we have an equation like: 





Y = X^{(any
exponent)}



When
X = 1
, then Y = 1 too! 








OK, back
to the big exponents. 


What does
the graph of Y = X ^{5} look like? 


Pretty
much like the graph of Y = X
^{3} but narrower. 


Except
... wait for it ... 





in the
area between X = 1 and X = 1. 








So the
deal is ... 


Y = X^{(even
number)}






All of
these equations with even number exponents 


have the
same "U" shape as Y = X
^{2}. 


The
bigger the exponent, the narrower the "U" 


except
between X = 1 and X = 1. 





AND ... 


Y = X^{(odd
number)}






All of
these equations with odd number exponents 


have the
same shape as Y = X ^{3}. 


The
bigger the exponent, the narrower the "U" 


except
between X = 1 and X = 1. 





copyright 2005 Bruce Kirkpatrick 
