The quadratic function f(x) = $ax^2$ + bx + c represents a parabola. Vertex form: f(x) = a(x + $\frac{b}{2a}$)^2 - $\frac{D}{4a}$, vertex at V(-$\frac{b}{2a}$, -$\frac{D}{4a}$). The parabola opens upward when a > 0 (minimum at vertex) and downward when a < 0 (maximum at vertex).

Key graphical properties: (1) Axis of symmetry x = -$\frac{b}{2a}$. (2) y-intercept is c (at x = 0). (3) x-intercepts exist only when D >= 0 and are at (-b +/- sqrt(D))/(2a). (4) The parabola is symmetric about the axis.

Range of f(x): For a > 0, range is [-$\frac{D}{4a}$, infinity). For a < 0, range is (-infinity, -$\frac{D}{4a}$]. This determines the set of values y = f(x) can take, critical for problems asking "for what values of k does f(x) = k have real solutions?"

Sign of quadratic expression: The wavy curve method applied to a(x - alpha)(x - beta) determines where the expression is positive, negative, or zero. For a > 0 with real roots alpha < beta: f(x) > 0 for x < alpha or x > beta, f(x) < 0 for alpha < x < beta.

Quadratic inequalities: Solve f(x) > 0, f(x) >= 0, f(x) < 0, or f(x) <= 0 by combining the sign chart with the discriminant. If D < 0 and a > 0: f(x) > 0 for all x. If D < 0 and a < 0: f(x) < 0 for all x.