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

f(x) = −(x + 2)2 − three fourths

f(x) = (x + 2)2 + three fourths

f(x) = −(x − 2)2 + three fourths

f(x) = −(x − 2)2 − three fourths

Vertical parabola opens down so
(y-k)^2=4P(x-h)
(h,k) is center
P is distance from directix to center and from center to focus (it is 1/2 distance between focus and directix)
-1 to -1/2 is a distance of 1/2
1/2 of 1/2=1/4
P=1/4

so move 1/4  down from -1/2
-3/4
center is (2,-3/4)

so
y+3/4=(x-2)^2
minus 3/4
y=(x-2)^2-3/4
f(x)=(x-2)^2-3/4

you made an error, either the last or first option doesn't have -(x+2)^2