SOLUTION: Describe, sketch and label the focus, vertex, and directrix of the parabola 4𝑦² + 8𝑦 − 8𝑥 + 7 = 0.

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Describe, sketch and label the focus, vertex, and directrix of the parabola 4𝑦² + 8𝑦 − 8𝑥 + 7 = 0.      Log On


   

Question 1177687: Describe, sketch and label the focus, vertex, and directrix of the parabola 4𝑦² + 8𝑦 − 8𝑥 + 7 = 0.
Answer by MathLover1(20855) About Me  (Show Source):
You can put this solution on YOUR website!
4y%5E2+%2B+8y+%E2%88%92+8x+%2B+7+=+0
%284y%5E2+%2B+8y+%29=8x+-+7+
4%28y%5E2+%2B+2y+%29=8x+-+7+
%28y%5E2+%2B+2y+%29=2x+-+7%2F4 .........complete square
%28y%5E2+%2B+2y+%2Bb%5E2%29-b%5E2=2x+-+7%2F4.........b=2/2=1
%28y%5E2+%2B+2y+%2B1%5E2%29-1%5E2=2x+-+7%2F4
%28y%2B1%29%5E2-1=2x+-+7%2F4
%28y%2B1%29%5E2=2x+-+7%2F4%2B1
%28y%2B1%29%5E2=2x+-+3%2F4
%28y%2B1%29%5E2=2%28x+-+%283%2F4%29%2F2%29
%28y%2B1%29%5E2=2%28x+-+3%2F8%29
so you have
%28y-k%29%5E2=4p%28x-h%29+is the standard equation for a right-left facing parabola with vertex at (h,+k+),

%28y%2B1%29%5E2=2%28x-3%2F8%29-> h=3%2F8, k=-1, 4p=2+->p=1%2F2
vertex: (3%2F8,-1)
focus: (h%2Bp,k) = (7%2F8,-1)

directrix:
parabola is symmetric around the x-axis and so the directrix is a line parallel to the y-axis, a distance -p from the vertex ( 3%2F8,-1) x-coordinate
x=3%2F8-p.........since p=1%2F2
x=3%2F8-1%2F2
x=3%2F8-4%2F8
x=-1%2F8