



We have
seen problems where we needed to find the point



where two straight
lines crossed.






Now we're
going to find the points 


where a straight line and a parabola cross. 





Say, like
this one: 








We can
see that in this problem there are 2 places where they cross. 


One where
both X and Y are negative, 


and one
where they are both positive. 





So how do
we find these points??? 





We start
off the same way we did when we had 2 straight lines. 





Where the
lines cross, the value of Y is the same for both, 


so our
two equations can be joined together with the same Y. 





X^{2}
 1 = Y Y = 2X + 1



X^{2}
 1 = Y = 2X + 1






Lose the
Y ... 


X^{2}
 1 = 2X + 1






Now we
get all of the terms on one side ... 








Now we
use our old buddy the quadratic formula ... 





Have you
memorized it yet??? 





A
= 1 B = 2 C = 2



So: 








X =
2.732 or X = .732






These are
the X values of the two points where our equation graph lines cross. 





We could
put these X values back into either equation to find the two Y
values. 


The
straight line equation is easier to use ... 





Y = 2X + 1 
Y = 2X + 1 
Y = 2(2.732)
+ 1 
Y = 2(.732)
+ 1 
Y = 6.464 
Y = .464 






So the
two equation lines cross at the point X = 2.732, Y = 6.464 


and the
point X = 0.732, Y = 0.464. 








If we had
a different straight line that was a bit lower 


there
would only be one point where the two lines touch. 


We would
get only one answer ... 











and if we
had a straight line that was any lower than that, 


the two
graph lines would not touch at all. 


We get NO
answer ... 











The quick
way to tell if we will get 2, 1, or 0 answers 


is to use
the discriminant part of the quadratic formula. 











IF 
B^{2}
 4AC > 0 
we get 2
answers 
IF 
B^{2}
 4AC = 0 
we get 1
answer 
IF 
B^{2}
 4AC < 0 
we get 0
answers 






Since we
only need to know if the discriminant is positive, negative, or
zero, 


we don't
even need to bother with the radical just the terms under it... 





One of
the exciting things we do in calculus is figure out the equations 


of
straight lines that just
touch curved lines 





That is
just too exciting to do here ... 





(OK, well
somebody might think it's exciting.) 





copyright 2005 Bruce Kirkpatrick 
