Parabola:

set of points that are the same distance from a point, the focus, and a line, the directrix.
==locus of points equi-distant from the focus and the directrix.

Each focus-directrix pair uniquely defines a parabola.
Vertical parabola is the graph of a quadratic function.
Given a horizontal directrix y=d and a focus (xf,yf)
the coefficients a, b, and c of the quadratic polynomial function are:
a= 1 / 2(yf-d)
b= xf / yf-d
c= xf2+yf2-d2 / 2(yf-d)

Vertical parabola with vertex (0,0), focus at (0,p) is x2=4py, or:

Vertical parabola with vertex (h,k), focus p=1/4a away is (x-h)2=4p(y-k)
Horizontal parabola with vertex (h,k), focus p away is (y-h)2=4p(x-k)

Latus rectum is line segment through the focus parallel to the directrix. Is |4p| in length.

Horizontal parabola. x as a function of y.:

Oblique parabolas. Ax2+Bxy+Cy2+Dx+Ey+F=0 with B2=4AC

How to find the angle of rotation?

The eccentricity e of every parabola is 1.