What is the equation of the quadratic graph with a focus of (5, - 1) and a directrix of y = 1?

The answer is directrix is horizontal, so the parabola is vertical

focus lies below the directrix, so the parabola opens downwards.

General equation for down-opening parabola:

y = a (x-h) ²+k

with

a<0

vertex (h, k)

focal length p = 1/|4a|

focus (h, k-p)

directrix (h, k+p)

Apply your data and solve for h, k, and a.

vertex is halfway between focus and directrix: (3,3)

h = 3

k = 3

distance between focus and vertex is 2, so p = 2.

a = - 1 / (4p) = - ⅛

y = - ⅛ (x-3) ²+3

(x-h) ^2=4p (y-k)

vertex is (h, k)

p is the distance from focus to vertex and distance from vertex to directix (vertex is in middle of directix and focus)

if p is positive, the parabola opens up and the focus is above the directix

if p is negative, the parabola opens down and the focus is below the directix

we see the directix is over the focus (1>-1) so the parabola opens down and p is negative

distance from (5,-1) to y=1 is 2 units

2/2=1

p=-1 since p is negative

1 up from (5,-1) is (5,0)

veretx is (5,0)

(x-5) ^2=4 (-1) (y-0)

(x-5) ^2=-4y is the equation