The vertex is basically the very top or very lowest, so you will notice how values around (-3,-2) are relatively the same in the y coordinate, but -2 is the "highest/lowest" so that's how you know it's a vertex.
If you try picturing the points in your head, you will see it make a U and at the bottom of the U is the vertex.
The function forms a parabola which has an axis of symmetry through the vertex. The y values are symmetric either side of x = -3 so the vertex is at (-3, -2) This is enough information to identify the function is y =(x + 3)^2 - 2
