Q1
We have the form
( y - k)^2 = 4a ( x - h)
The vertex is given by ( h, k) = (3, -1)
Since y is squared and we have no negatives on either side of the equation, it opens to the right
4a = 12
a = 3 which means that the focus is 3 units to the right of the vertex
The directrix is 2a = 2(3) = 6 units from the focus
The focus is at ( 3 + 3, -1) = (6 - 1)
The dirctrix has the equation x = (x coordinate of the vertex - a) = (3 - 3) = 0
So x = 0
Here's the graph : https://www.desmos.com/calculator/jrejmuykh9
Q2
The directrix is to the left of the focus...so....this parabola opens to the right....therefore...we will hav no negatives on either side of the equation
The form will be
(y - k)^2 = 4a (x - h)
The vertex is given by ( (sum of directrix + x coordinate of the focus ) / 2, y coordinate of focus ) =
( [ 0 + 2]/2 , 6) = (1, 6) = (h, k)
"a" is the distance from the vertex to the focus = 1
So we have
(y - 6)^2 = 4(1) ( x - 1)
( y - 6)^2 = 4 ( x - 1)
Here's the graph : https://www.desmos.com/calculator/1y3feymzzs