Answer
$f(x)=\displaystyle \frac{3}{5}(x-1)^{2}+3$
Work Step by Step
$f(x)=\displaystyle \frac{3}{5}x^{2}$
In order to have a vertex in the point $(1,3)$, this function needs to be moved $1$ unit to the right and $3$
units up.
$f(x)=a(x-h)^{2}+k$
Substitute $a=\displaystyle \frac{3}{5},h=1,k=3$
$f(x)=\displaystyle \frac{3}{5}(x-1)^{2}+3$