Find the equation of the parabola with vertex at (0,0) and focus at (0,3)

Find the equation of the parabola with vertex at (0,0) and focus at (0,3).

In the equation  y = a·x², the value 'a' has the value '1/(4f)', where 'f' is the focal length.

Since the distance form (0,0) to (0,3) is 3, the focal length = 3   --->   f = 3   --->   a = 1/(4·3)   --->   a = 1/12

     --->     y  =  (1/12)·x²

 

Find the equation of the parabola with vertex at (5,4) and focus at (-3, 4).

In the equation x - h  =  a(y - k)², (h,k) is the vertex   --->   h = 5  and  k = 4  

Since the focal length is 8:  f = 8   --->   a = 1/(4·8)   --->   a = 1/32

Since the parabola opens to the left, a must be negative   --->   a = -1/32

     --->     x - 5  =  (-1/32)(y - 4)²