write an equation of a parabola with vertex at the origin and the given directrix y=3

Question

Find the standard form of the equation of the parabola with vertex a the origin directrix y =3.

Answer 1

Since the directrix is the line y = 3, this is a horizontal line ... thus the parabola opens up or down.

 

Since the vertex is at the origin (0,0), we know that the parabola opens downward, because the vertex is between the directrix and the focus ... and the focus is on the inside of the open parabola

 

The distance from the focus to the vertex, is also equal to the distance from the directrix to the vertex ... which is 3.

 

Here is some base info and equations about parabolas that open up and down

 

     (x - h)² = 4p (y - k), where (h, k) is the vertex, the focus is (h, k + p), and the directrix is y = k - p.

 

The vertex os (h, k), or (0,0)

The directrix is y = 3 ... so k - p = 3 ... 0 - p = 3 ... p = -3

The focus is (h, k + p) ... (0, 0 - 3) ... (0, -3)

 

Thus substituting the values into the base equation,

   

    (x - h)² = 4p (y - k)

    (x - 0)² = 4(-3)(y - 0)

    x²  = -12y

 

The above is the standard form ... however

if we solve for y, we get

 

    y = (-1/12)x²