If the axis of symmetry is x=-1, the vertex will have x-coordinate -1.
The point (0, 5) cannot be the vertex. The friend is not correct.
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If the point (1, 5) is on the graph so is the image of that point when it is reflected across the axis of symmetry. That image point is (-3, 5). A parabola that goes through points (1, 5) and (-3, 5) will not go through point (0, 5).