For the equation y = x² - 2x - 3 the variables are a=1, b=-2, c=-3

y-intercept (where x = 0):
    y = 1(0)² - 2(0) - 3 = -3

vertex format (completing the square):
    y = x² - 2x + 1 - 3 - 1 => y = (x - 1)² - 4 (vertex is (1, -4))

x-intercept format ("factored"):

Find factors of -3 (c) whose sum is -2 (b)

Pairwise factors of -3

Candidate	Factor 1	Factor 2	Sum
1	-1	3	2
2	1	-3	-2

The candidate highlighted above gives us the factors.
    y = (x - 3)(x + 1) (using zero product property, the x-intercepts are 3 and -1)

Graph the three versions of the highlighted equations (using Desmos.com here) and confirm they are equivalent. ALSO, confirm that the y- and x-intercepts are correct, as well as the vertex: