The value of h is exactly the axis of symmetry. So when h changes, it completely changes the line of symmetry. This is because the vertex is (h, k) and the x value of the vertex is always equal to the line of symmetry in a quadratic.
Changing the k value does nothing to the line of symmetry. This is because it is just moving the graph up and down, which doesn't change the symmetry of a parabola.