## 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

\[y=(x-2)^2+3=x^2-4x+4+3=x^2-4x+7\]

.