



OK,
the graph of ...



X^{2}
+ Y^{2} = 25






is a
circle, centered at (0,0) with a radius of 5 (5
^{2} = 25) 











But what
do we get when we have a minus sign between the terms? 





I could
make you build T tables and figure this out bit by bit, 





But hey,
life is too short as it is ... 





So here's
the deal, 


When you
have a minus between the terms you get this: 











This is
called a hyperbola. 


It kind
of looks like two parabolas back to back. 





It is
actually more like an outward reflection of the circle 


that has
the same equation 


but with
a plus instead of a minus between the terms ... 











The lines
of the hyperbola also keep getting closer and closer 


to a pair
of diagonal lines. 


But they
never touch or cross them. 


That
makes these lines asymptotes. 











The slope
of these asymptote lines is: 











Just like
all the rest of this junk, 


if you
put a number inside the parentheses with the X or the Y, 


it moves
the center of the graph in the opposite direction. 





For
example, 


(X + 2)^{2}
+ (Y  3)^{2} = 25






Everything
keeps the same shape, 


but moves
3 units up and 2 units to the left: 











copyright 2005 Bruce Kirkpatrick 
