+129852 a) y = (1/8) ( x - 6)^2 + 4
We are working backwards, here
(y - 4) = (1/8) ( x - 6)^2
8(y - 4) = ( x - 6)^2 the vertex is ( 6, 4)
4p = 8 ⇒ p = 2
The parabola turns upward so the focus is ( 6 , 4 + p) = ( 6 , 4 + 2) = (6, 6)
And the directrix is y = 4 - p = 4 - 2 = 2 ⇒ y = 2
+129852 (b) y = (-1/4) ( x + 2)^2 + 1
y - 1 = -(1/4)(x + 2)^2
4 ( y - 1) = - ( x + 2)^2 the parabola turns downward
The vertex is ( -2, 1)
4p = 4 so p = 1
The focus is ( -2, 1 - p) = (-2, 1 - 1 ) = (-2, 0)
The directrix is y = 1 + p = 1 + 1 = 2 ⇒ y = 2
+129852 (c) y = (1/16) ( x - 3)^2 easier!!!
16 y = (x - 3)^2
16 ( y - 0) = (x - 3)^2
The vertex is (3, 0) and this turns upward
4p = 16
p = 4
The focus is ( 3, 0 + p) = ( 3, 0 + 4) = ( 3, 4)
The directrix is y = 0 - p = 0 - 4 = -4 ⇒ y = -4
+129852 (d) y = 2 ( x + 1)^2 - 5 add 5 to both sides
y + 5 = 2 ( x + 1)^2 multiply both sides by 1/2
(1/2) ( y + 5) = ( x + 1)^2
The vertex is ( -1, -5) and this opens upward
4p = 1/2
p = 1/8
The focus is ( -1, -5 + 1/8) = (-1, -39/8 )
The directrix is y = -5 - p = - 5 - 1/8 = -41/8 ⇒ y = -41/8
