For the equation y = a(x - h)^2 + k, the vertex is at (h, k), and the focus is at (h, k + 1/4a).
We first complete the square of the equation:
y = x^2 - 2x - 3
y = (x - 1)^2 - 4
Following this format, a = 1, h = 1, k = -4.
Then the vertex is at (1, -4), and the focus is at (1, -4 + 1/(4*1)) which is (1, -15/4).
