Answer
$f(x)=(x-2)^2+1$.
Work Step by Step
If the vertex of a graph is at $(h, k)$ then the general formula for the quadratic function is $f(x)=a(x-h)^2+k$.
According to the picture the vertex of the graph is at $(2,1)$, hence the quadratic function becomes $f(x)=a(x-2)^2+1$.
The point $(0,5)$ is on the graph, hence if we plug in the values we get:
$5=a(0-2)^2+1\\
\\5=a(4)+1
\\5=4a+1\\
\\5-1=4a\\
4=4a
\\1=a$,
Thus, the equation of the function is $f(x)=(x-2)^2+1$.