Derive the equation of the parabola with a focus at (0, −4) and a directrix of y = 4.

f(x) = −16x2
f(x) = 16x2
f(x) = −one sixteenth x2
f(x) = one sixteenthx2

Question

Derive the equation of the parabola with a focus at (0, −4) and a directrix of y = 4.

 f(x) = −16x2
 f(x) = 16x2
 f(x) = −one sixteenth x2
 f(x) = one sixteenthx2

campbell_st · Answer

the focus is below the directrix... so its concave down 

the distance between the focus and directrix is 2 x focal length (a)

the vertex is halfway between the focus and directrix... on the line of symmetry

so all that said

the distance between focus and directrix is 8

half = 4 ... so the focal distance is 4

the vertex is 4 units above the focus so (0, 0)

then using the formula 

$$(x - h)^2 = -4a(y - k)$$

(h, k) is the vertex(0, 0) and a = 4

just substitute to find the equation...