find the vertex, focus, and the equation of the directrix for the parabola y = 2x^2 - 8x + 3

Question

if someone could help that would be awesome, thanks!

Answer 1

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.