Convert the equation to standard form by completing the square on x or y. Then find the vertex, focus?

Question

and directrix of the parabola. Please show work so I know you got your answer and I can learn, please.

1. x^2+6x+8y+1=0

2. y^2-2y-8x+1=0

Answer 1

1.

The standard equation of a parabola with a vertical axis is:

(x - h)^2 = 4a(y - k) ...(1)

The vertex is (h, k).

The focus is (h,  k + a).

The directrix is y = k - a.

x^2 + 6x + 8y + 1 = 0

x^2 + 6x = - 8y - 1

x^2 + 6x + 3^2 = - 8y + 8

(x + 3)^2 = - 8(y - 1)

Comparing with (1):

h = - 3, k = 1, a = - 2.

The vertex is (- 3,  1).

The focus is (- 3,  - 1).

The directrix is y = 3.

2.

The standard equation of a parabola with a horizontal axis is:

(y - k)^2 = 4a(x - h) ...(1)

The vertex is (h, k).

The focus is (h + a,  k).

The directrix is x = h - a.

y^2 - 2y - 8x + 1 = 0

y^2 - 2y + 1 = 8x

(y - 1)^2 = 8x

h = 0, k = 1, a = 2.

Vertex (0, 1).

Focus (2, 1).

Directrix x =  - 2.