2
claim: Let a and c be fixed undetermined real numbers in y  ax  bx  c . Let b vary as
a parameter and consider the locus of vertices of the resulting parabolas. I claim that the
2
locus will have the equation y  ax  c .
 b 2  b 
b
pf: The x-coordinate of the vertex is  . Thus, y  a   b  c . This gives
 2a 
 2a 
2a
2 
 b
b
(after simplifying y) the coordinates of a vertex as 
 2a ,c  4a 
. Just like in the special


b
(2ax)2
 c  ax 2 
case computation, I have x    b  2ax  y  c 
2a
4a