



So we
know that if we have an equation



that
factors to look like this: 


Y
= (X  2)(X + 3)(X  1)






The graph
will cross the X axis (where Y = 0) 


when X =
2, X = 3, and X = 1. 





If we
multiply the terms together we get: 





Y = X^{3}
7X + 6






The
biggest exponent is a 3, 


and the
term with that exponent is positive. 





So that
means the graph goes up on the right 


and down
on the left, like Y = X ^{3}. 





But the
extra terms (7X + 6), might give it a hitch in the middle. 


Kind of
like this: 








Sometimes
the graph line doesn't cross the X axis, 


but just
touches it like this instead: 








What
would the factors of this one look like? 





We need
to get the graph to just touch the X axis where X = 1. 





There is
a trick to this (big surprise, eh?) 


The
factor that would make the graph 


cross the
X axis at X  1 is (X  1). 





To make
the graph just touch the X axis but not cross it, 


use that
factor TWICE! 





That
means the equation that goes with this graph: 











Is: 


Y
= (X + 3)(X  1)(X  1)






Sometimes
we write this equation like this: 





Y
= (X + 3)(X  1)^{2}






Example: 


Draw the
graph of ... 


Y
= (X + 1)(X + 1)(X + 3)






Y = 0
when X = 1 (the double factor) or when X = 3. 


If we
multiply these factors together we get: 





Y
= X^{3} + 5X^{2} + 7X + 3






The
biggest exponent is a 3. The term it is on is positive. 


Put all
this together and we get ... 








Now just
connect the lines and dots ... 








This
double factor stuff may seem all new, but it's not. 


The very
first equation we looked at in the second Algebra stuff 


was one
of those "double factor" things. 





Oh
yeah??? 





Yeah!! It
was: 


Y
= X^{2}






If we
want to be really technical, 


we can
write it like this: 





Y
= (X  0)^{2} or
Y = (X  0)(X  0)






It's a
double factor equation that touches the X axis when X = 0. 





If you
multiply it out, you get: 


Y
= X^{2} (big
surprise, eh?)






The
biggest exponent is a 2. That makes the graph a parabola. 


The term
with the biggest exponent is positive, 


that
means the parabola opens upward. 





Now we
can draw the graph ... 











Hmmm, I
wonder what would happen if we had a factor 


show up
THREE times? 





copyright 2005 Bruce Kirkpatrick 
