## 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
$y=(x-2)^2+3=x^2-4x+4+3=x^2-4x+7$
.