Question 114527
The first thing we know about this parabola is that the axis of symmetry is the line x = 0.  We know this because the both the focus and the vertex have to lie on the same line and the only line that passes through both (0,2) and (0,0) is x = 0, or the y-axis.


The next thing is to determine the distance between the focus and the vertex.  We can really just tell by inspection that the distance is 2 because both points are on a vertical line with the y coordinates differing by 2.  But, just to show the general case, lets use the distance formula:


{{{d=sqrt((x[1]-x[2])^2+(y[1]-y[2])^2)=sqrt((0-0)^2+(2-0)^2)=sqrt(2^2)=2}}}


Now the equation for a parabola is {{{4p(y-k)=(x-h)^2}}} where p is the distance from the focus to the vertex, and the vertex is at point(h,k).  So,



{{{4(2)(y-0)=(x-0)^2}}}
{{{8y=x^2}}}
{{{y=(x^2)/8}}}


The directrix is a line perpendicular to the axis of symmetry -p units distant from the vertex.  Since our parabola has a vertical line as an axis of symmetry, the directrix must be a horizontal line.  The only horizontal line that is -2 units from the vertex (0,0) is y = -2.


The green line is the directrix


{{{graph(600,600,-10,10,-10,10,x^2/8,-2)}}}