Is this true? x^2=4py is the standard equation of a parabola with focuc (0,p) and directrix y=-p?

Answer 1

yes it is a parabola

Answer 2

Is this true? x^2=4py is the standard equation of a parabola with focuc (0,p) and directrix y=-p?

Yes it is a parabola, but its vertex is always on the origin.
You should use the form (x-h)^2= 4p(y-k) for a parabola that hyas a vertex at (h,k), axis of symmetry is x=h, focus at (h,k+p) and directrix at (h, k-p)

Answer 3

Yes. The vertex is (0,0). Usually the standard form refers to y^2=2px and the focus is (p/2,0) and the directrix is x=-p/2.Your equation is not standard. It is just the right equation given your focus and directrix.