Derive the equation of the parabola with a focus at (6, 2) and a directrix of y = 1.

f (x) = - 1/2 times (x - 6) ^2 + 3/2

f (x) = 1/2 times (x - 6) ^2 + 3/2

f (x) = - 1/2 times (x + 3/2) ^2 + 6

f (x) = 1/2 times (x + 3/2) ^2 + 6

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

we know it is this one because the directix is y=something and not x=something

(h, k) is the vertex

p is the distance from the vertex to the focus

it is 1/2 the distance from the vertex to the directix

if p is positive, the focus is above the vertex

if p is negative, the focus is below the vertex

we have

focus is (6,2) and directix is y=1

distance from (6,2) to y=1 is the distance from 2 to 1 which is 1

1/2=p

since 1<2, we see that the focus is above the vertex (when the focus is greater than the directix, then the graph opens to the right or up)

p=1/2

1/2 below (6,2) is (6,1.5)

vertex is (6,1.5)

p=1/2

(x-6) ^2=4 (0.5) (y-1.5)

(x-6) ^2=2 (y-1.5)

(x-6) ^2=2y-3

2y = (x-6) ^2+3

divide both sides by 2

y=1/2 times (x-6) ^2+3/2

f (x) = 1/2 times (x-6) ^2+3/2

2nd option is answer