



We have
graphed stuff like Y = X ^{2} ...












How about
Y = 2 ^{X}? 





Well, we
know that when X = 0, Y = 1 


since
anything to the zero power equals 1. 





We also
know that when X = 1, Y = 2. 


Lets see
what happens at even bigger X values ... 





X 
Y 
0 
1 
1 
2 
2 
4 
3 
8 
4 
16 
5 
32 
6 
64 
7 
128 
8 
256 
9 
512 
10 
1024 






So it
looks like this ... 








So for
positive values of X anyway, 


this sort
of looks like the graph of Y = X
^{2} 


I wonder
which one goes up faster? 





Y
= X^{2} 


X 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
Y 
0 
1 
4 
9 
16 
25 
36 
49 
64 
81 
100 






Y
= 2^{X} 


X 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
Y 
1 
2 
4 
8 
16 
32 
64 
128 
256 
512 
1024 






Wow! no
doubt about it. 


After a
close start and some lead changes, 


Y = 2
^{X}
goes up much faster on the right side of graph. 


I wonder
what happens on the left side of the graph of Y = 2
^{X}? 





Remember
that negative exponents don't mean negative numbers, 


they mean
"put this term in the denominator" 


So: 








So as X
values go off to the left, 


the
denominator keeps getting bigger and Y keeps getting smaller. 


But Y
always stays positive. 


That
means that this equation (and most other exponential equations), 


has a
horizontal asymptote at Y = 0. 








Don't
forget that an asymptote is just a really fancy name 


for some
value that a graph gets really close to. 


Here, our
graph line gets closer and closer to Y = 0 


but never
actually gets there. 


We KNOW
that it never gets there because ... 











Way back
when we learned a bunch of exponent rules. 


They
still apply when the exponent is an X (or any other letter) 


and not a
number. 








copyright 2005 Bruce Kirkpatrick 
