Find the focus and directrix of 8x=y^2+6y+9

We see it can be iinto form
4p(x-h)=(y-k)^2
4(2)(x-0)=(y+3)^2
this is a parabola that is facing left to right
when p is positive, then it faces to right
when p is negative, it faces to left
p is the distance from the vertex to the directix, also the distance from the focus to the directix
the directix is oposite the focus
the focus is on the axis of symmetry
so since this faces left to right, the focus is 2 units to the right
directix is 2 units to left

vertex is (h,k)
4(2)(x-0)=(y+3)^2
(0,-3)
2 to the right is (2,-3) that is focus
2 to the left is x=-2

focus=(2,-3)
directix is at x=-2

Period is the time it takes for one “cycle” to complete. in the parent sine function, period is 2*pi since sin(0) = 0, sin(pi/2) = 1, sin(pi) = 0, sin(3pi/2) = -1, and sin(2pi) = 0. at that point, the sequence repeats itself. i’m not sure if your graph is similar to the sine function since you didn’t provide a picture, but you can find the period by looking at a graph.