y=2x2-8x+3=
2(x2-4x+4-4)+3=
2[(x-2)2-4]+3=
2(x-2)2-5.
The standard vertex form of the parabola is y=2(x-2)2-5, vertex at (2,-5).
The axis of symmetry is x=2 (the x-coordinate of the vertex), and the focus lies on this axis.
If f is the focal distance then the coefficient of x2 (or, in this case, (x-2)2) is 1/(4f).
Therefore, 1/(4f)=2, 4f=½, f=⅛. The vertex is at (2,-5) and the focus is ⅛ from this point along the axis of symmetry. -5+⅛=-39/8, so the focus is at (2,-39/8).
The directrix is the same distance from the vertex as the focus but on the other side of the vertex, so the equation of the directrix is y=-5-⅛=-41/8.