Answer
Please see the graph.
Work Step by Step
$h(x) = (x-3)^2+2$
Axis of symmetry:
$h(x) = (x-3)^2+2$
$h(x) = (x- 3)^2+2$
$x=3$
Vertex:
$h(x) = (x-3)^2+2$
$h(3) = (3-3)^2+2$
$h(3)=0^2+2$
$h(3) = 0+2$
$h(3)=2$
The green line $x=3$ is the axis of symmetry, the black dot at $(3,2)$ is the vertex, and the red line is the function $h(x)=(x-3)^2+2$.