suppose we have following Parabola $y=x^2+4x+6$
our goal is to find vector which describes parallel translation of given parabola into new one $y=x^2$ to find vertex of the parabola, i set two equation
$x=-b/2a=-4/2=-2$
$y=(-2)^2+4*(-2)+6=2$
for the parabola $y=x^2$ we have following vertexes $(0,0$) , so does it means that our vectors coordinates are $(2,-2)$ ?
We look for $(a,b)$ such that
$$\forall (x,y)\in \mathbb R^2$$
$$(x,y)\in P_1\implies (x+a,y+b)\in P_2$$ or $$y=x^2+4x+6 \implies x^2+4x+6+b=(x+a)^2=x^2+2ax+a^2$$
$$\implies a=2, b+6=4.$$
the translation vector is $\vec{u}=(2,-2)$.