



Look at
this equation:









This
looks kind of simple, 


but look
what happens when X = 1 ... 








We have a
big problem. 


We can't
deal with a zero in the denominator. 


That
means that X can't equal 1. 





Let's
make a T table to see what happens to Y 


when X
gets close to 1. 





X 
Y 
0 
1 
.5 
2 
.75 
4 
.9 
10 
.95 
20 
.99 
100 
1.01 
100 
1.05 
20 
1.1 
10 
1.25 
4 
1.5 
2 
2 
1 






Now we
can draw the graph. 


Since X
can't equal 1, draw a dotted line where X = 1 ... 











Now we
put the X, Y points from the T table into the graph. 


Look at
what happens when the graph gets close to X = 1 ... 








When X
gets close to X = 1 from either side 


the graph
line shoots off the drawing. 





When X is
smaller than 1, 


the
values of Y get to be really big negative numbers 


the
closer we get to X = 1. 





When X is
bigger than 1, 


the
values of Y get to be really big positive numbers 


the
closer we get to X = 1. 





Math
people really get excited about strange stuff like this. 


They
usually give strange stuff strange names. 


That's
what they did here. 


They call
lines like the one at X = 1, an asymptote. 


Since the
line goes up and down, they call it a vertical asymptote. 


A
vertical asymptote only happens 


at an X
value that makes the denominator equal zero, 


that we
can't get rid of by simplifying the fraction. 





Examples: 











Can we
have more than one vertical asymptote 


in the
same equation??? 





SURE! 





We can
have lots! 





Here's an
equation with two: 











This
equation will have a vertical asymptote where X = 1, 


and
another one where X = 1. 











Let's
build a T table and see what this puppy looks like ... 





X 
Y 
3 
.125 
2 
.333 
1.5 
.8 
1.1 
4.76 
1.01 
49.8 
0.99 
50.2 
0.9 
5.02 
0 
1 
0.9 
5.02 
0.99 
50.2 
1.01 
49.8 
1.1 
4.76 
1.5 
.8 
2 
.333 
3 
.125 






Now draw
the graph ... 








copyright 2005 Bruce Kirkpatrick 
