Translations of Graphs and Equations

Suppose a quadratic graph  
\[y=f(x)\]
  is translated by the vector  
\[\begin{pmatrix}1\\3\end{pmatrix}\]
. What will the equation of the resulting graph?
The vertex of the graph is initially at  
\[(0,0)\]
  and is moved to  
\[(1,2)\]
  by the translation. A translation of  
\[{} +2\]
  in the  
\[x\]
  direction results in the relabelling  
\[f(x) \rightarrow f(x-2)\]
  so  
\[f(x)=x^2\]
  becomes  
\[f(x-2)=(x-2)^2\]
  and the translation of  
\[{} +3\]
  in the  
\[y\]
  direction results in the relabelling  
\[f(x) \rightarrow f(x)+2\]
.
Hence the equation of the translated curve is  
\[y=(x-2)^2+3=x^2-4x+4+3=x^2-4x+7\]
.