Translating a parabola

Translate the parabola $y = x^2$ by $2$ in the $x$-direction and $1$ in the $y$-direction. Replacing $x$ with $x - 2$ and adding $1$ gives $y = (x - 2)^2 + 1$.

In general, shifting $y = x^2$ by $p$ horizontally and $q$ vertically yields $y = (x - p)^2 + q$, moving the vertex from the origin $(0, 0)$ to $(p, q)$. Here the vertex moves from $(0, 0)$ to $(2, 1)$; the large dots mark the two vertices.