Find the vertex, focus and directrix of the parabola y2-4y-2x-4=0

Question

Find the vertex, focus and directrix of the parabola y2-4y-2x-4=0

Answer 1

Greetings! Lets solve this shall we ?

So, we must find the vertex, focus and directrix of the given parabola y²-4y-2x-4=0. Then, we manipulate the parabola to put it into vertex form such that we first add -2x-4 to the right side to obtain

y²-4y = 2x+4............Then the not so obvious next step is to Complete the Square on the left side, dividing -4 by two then squaring it to get 4 such that

y²-4y+4 = 2x + 4 + 4...........We add what we squared on the right side too. Then we factor the left and right side to obtain

(y-2)² = 2(x+4).......We are now in Vertex Form where the vertex is (-4, 2) and the parabola will be turned to the right. We find the focus after we find p knowing that the parabola is in the form of (y-k)² = 4p(x-h). Then,

4p = 2; and p = 1/2......This means p > 0, which makes it open to the right instead of the left. Then the focus yields, (h+p, k) = (-4+1/2, 2) = (-7/2, 2) which is located inside the parabola. Finally, we can find the directrix using x = h - p located as a vertical line behind the parabola on the graph such that x = -4 - 1/2 = -9/2.

I hope this helped!