Find the focus and directrix for each parabola below:

a)  

y = -(1/4)(x - 2)^2 + 5

y - 5 = (-1/4)(x - 2)^2       multiply through by 4

4(y - 5) = -(x - 2)^2

The vertex  is ( 2, 5)

4p = 4

p = 1

This parabola turns downward

The focus is  (2, 5 - p)  = (2, 5 - 1)   =  (2, 4)

The directrix is  y = 5 + p =  5 + 1  = 6   ⇒   y = 6

8y = (x + 1)^2 - 32

8y + 32 = (x + 1)^2

8(y + 4) = (x + 1)^2

The vertex is (-1, -4)

The parabola turns upward

8 = 4p

p = 2

The focus is  (-1, -4 + p)  =  (-1, -4 + 2)   = (-1, -2)

The directrix is  y = -4 - p = -4 - 2  =  -6  ⇒  y = -6