Write the equation, in standard form, of a parabola with a focus at (2,2) and a directrix of x=-2

Question

Write the equation, in standard form, of a parabola with a focus at (2,2) and a directrix of x=-2

Answer 1

(2,2) focus and directrix y=-2 is an upward opening parabola with vertex midway between focus and (2,-2) or (2,0)

y-k = a(x-h)^2 is the general equation for vertex (h,k)

plug in (2,0) for (h,k)

y=a(x-2)^2 there's a formula for a, the x^2 coefficient or just find another point on the parabola and plug it in, to solve for a. Another point is directly to the right of the focus, a distance = 4 = the vertical distance to the directrix, so (4,2) is another point on the parabola, so is (0,2) the y intercept, it's symmetric about x=2.

2=a(4-2)^2 = 4a

a =2/4 = 1/2 = (1/4)(distance from vertex to focus) = (1/4)(2) = 2/4=1/2

parabola equation is

y=(1/2)(x-2)^2