I think this is what you want, but not sure
y=x^2 -4x +3 complete the square by adding and subtracting 1/2 the x
terms coefficient squared
y=[x^2-4x+4]-4+3 combine terms and factor
y=[x-2]^2-1 rewrite
[y+1] =[x-2]^2 a parabola, open up vertex at 2,-1
axis of symmetry x=2
the equation of left 1/2 would be all points x<2 y=[x-2]^2-1
I think this is what you need.