The left translation p unit length, that is, the analytical formula of the original parabola is
y=a(x+p)^2+b(x+p)+c
=ax^2+2apx+ap^2+bx+bp+c
=ax^2+(2ap+b)x+ap^2+bp+c
So the value of a is constant, and the value of a b is constant when translating up and down.