



In
general, when the number that we put in for X



makes the
denominator of a fraction be zero, 


we get a
vertical (up and down) asymptote. 





But not
always ... 





Say we
have the equation: 








When X =
1 the denominator is zero. 





BUT
WAIT!!! 





Before we
do anything, lets try to simplify this. 


We can
factor the top part, X ^{2}  1 factors to (X + 1) times (X 
1). 


That
means: 








and this
we can simplify ... 








We can
graph Y = X + 1 ... 








This is
the graph of a straight line. 


But in
the original equation, 


the
denominator was zero when X = 1! 





So what
do we do? What does the graph of 











look
like? 





Does it
have an asymptote where X = 1 or what? 





Here's
the deal. 





If we
have X values where the denominator equals zero, 


but we
can factor away those places. 


The graph
DOES NOT have an asymptote at these places. 


Graph the
simplified equation BUT at the point where the denominator is zero, 


the graph
has a hole. 


It sounds
harder than it really is, watch ... 





We have
the equation: 





We factor
it to: 





That
simplifies to: 





Graph Y =
X + 1 





Put a
hole in the graph at the place where the denominator was zero. 








Example: 





This
factors to: 





This
simplifies to: 





We can
graph: 








In the
original equation, 


the
denominator equals zero when X = 0. 


That
means the graph has a hole where X = 0. 











copyright 2005 Bruce Kirkpatrick 
