Subject: graphing equations

If, for example, 4x^{2}-12x+9=0 and -4x^{2}+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?

Hi,
The two equations

1) 4x^{2} - 12x + 9 = 0
and
2) -4x^{2} + 12x - 9 = 0
are equivalent, as you suggest. You can easily get from 1) to 2) by multiplying both sides by -1. Since
4x^{2} - 12x + 9 = (2x - 3)^{2}
they are each equivalent to
(2x - 3)^{2} = 0.
Hence the only x that satisfies 4x^{2} - 12x + 9 = 0 is x = ^{3}/_{2}.
However, the functions

3) y = 4x^{2} - 12x + 9
and
4) y = -4x^{2} + 12x - 9
are different. If you multiply both sides of equation 3) by -1 you get
-y = -4x^{2} + 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,

Penny