The affine symmetries of P_{0}.
Theorem AS3
The affine symmetry group of P_{0} is
E_{P}(2) = {t : t(x,y) = (e^{2}x+2efy+f^{2},ey+f), e ≠ 0}.
proof
The twopoint theorem for affine parabolas
If P and Q are points on an affine parabola P, then there is a unique affine transformation
which maps P to P_{0} and P to O(0,0), Q to U(1,1).
proof

For the web, it is easier to
give an affine transformation
in the linear form
t(x,y)=(ax+by+c,dx+ey+f)
This is equivalent to the
usual vector form
t(x) = Ax+b, with
A having rows (a,b), (c,d), and
b^{T} = (e,f). 