#### Answer

$f(x)=2(x -1)^2-3$

#### Work Step by Step

The vertex form of the quadratic function $ax^2+bx+c=0$ can be expressed as $f(x)=a(x-h)^2+k$ and its vertex is at $(h, k)$.
As depicted in the picture, the vertex of the graph is at $(h, k)=(1,-3)$, and thus, the quadratic function becomes $f(x)=a(x -1)^2-3$.
Plug in the values $(3, 5)$ to obtain:
$5=a(3-1)^2 - 3 \\ 4a=8 \implies a=2$
Therefore, the equation of the function can be expressed as: $f(x)=2(x -1)^2-3$