y-k=a(x-h)² where vertex is V(h,k) and focus is F(h,k+1/(4a)) for a vertical parabola.
x-h=a(y-k)² where vertex is V(h,k) and focus is F(h+1/(4a),k) for a horizontal parabola.
|1/(4a)| is the distance between the vertex and the focus and the perpendicular distance between the vertex and the directrix line.
(1) V(-6,-1), F(-6,2) Vertical parabola (red).
y+1=a(x+6)²
-1+1/(4a)=2, 1/(4a)=3, a=1/12.
y+1=(x+6)²/12,
y=(x²+12x+36)/12-1=x²/12+x+2.
y=x²/12+x+2.
(2) V(9,-2), F(5,-2) Horizontal parabola (blue).
x-9=a(y+2)²
9+1/(4a)=5, 1/(4a)=-4, a=-1/16.
x-9=-(y+2)²/16,
x=-(y²+4y+4)/16+9=-y²/16-y/4-1/4+9,
x=-y²/16-y/4+35/4.
Parabolas shown with focus and directrix in the same colour.
