1. y = x^2 - 8x + 12
y - 12 = x^2 - 8x complete the square on x
y - 12 + 16 = x^2 - 8x + 16 simplify
(y + 4) = (x - 4)^2
We have the form (4p)(y - k) = (x - h)^2
The vertex = (h, k) = ( 4, - 4)
Axis of symmetry x = 4
p = (1/4)
The focus = ( 4 , - 4 + p) = ( 4, - 4 + 1/4) = ( 4, -15/4)
The directrix = y = (k - p) = - 4 - 1/4 = -17/4
Here's a graph : https://www.desmos.com/calculator/am3svgafgw
x = 2y^2 -8y + 3
x - 3 = 2(y^2 - 4y) complete the square on y
x - 3 + 8 = 2(y^2 - 4y + 4) simplify
(x + 5) = 2(y - 2)^2
(1/2)(x + 5) = (y - 2)^2
Vertex = ( -5, 2)
We have the form (4p)(x + 5) = (y - 2)^2
4p = 1/2
p = 1/8
This parabola opens to the right
The axis of symmetry is y = 2
The focus = ( -5 + p , 2) = (-5 + 1/8, 2) = ( -39/8, 2)
The directrix is x = -5 - p = -5 - 1/8 = -41/8
Here's the graph :