Quadratic functions
The vertex of y = a(x - p)^2 + q is (p, q), and the sign of a says which way it opens.
A worked example
For y = x^2 - 10x + 21 the vertex x is -b/(2a) = 5, and y = 25 - 50 + 21 = -4, so the vertex is (5, -4).
The mistake people actually make
The vertex x-coordinate is p, not -p: y = (x + 4)^2 - 5 has its vertex at x = -4. A discriminant of zero means one root, and a negative one means no x-intercepts.