Subject: graphing equations
If, for example, 4x2-12x+9=0 and -4x2+12x-9=0, which I'm assuming it does
since you can derive that equation from the first, why do those two
equations have different graphs?
The two equations
1) 4x2 - 12x + 9 = 0
2) -4x2 + 12x - 9 = 0
are equivalent, as you suggest. You can easily get from 1) to 2) by multiplying both sides by -1. Since
4x2 - 12x + 9 = (2x - 3)2
they are each equivalent to
(2x - 3)2 = 0.
Hence the only x that satisfies 4x2 - 12x + 9 = 0 is x = 3/2.
However, the functions
3) y = 4x2 - 12x + 9
4) y = -4x2 + 12x - 9
are different. If you multiply both sides of equation 3) by -1 you get
-y = -4x2 + 12x - 9
Which is not the sames as 4).
As I think you have observed the graph of equation 3) is a parabola that opens upward while the graph of equation 4) is a parabola that opens downward.
I hope this helps,