Definition

The vertex of a parabola is its turning point: the topmost point if the parabola opens downward, or the bottommost point if it opens upward, the single highest or lowest value on the graph. For a quadratic in standard form $y = ax^2 + bx + c$, the vertex's x-coordinate is $x = \frac{-b}{2a}$ (substitute back for the y-coordinate), while in vertex form $y = a(x-h)^2 + k$, the vertex is directly $(h, k)$; it is a minimum when $a > 0$ and a maximum when $a < 0$. Formally, the vertex is the unique critical point of $f(x) = ax^2 + bx + c$: setting $f'(x) = 2ax + b = 0$ gives $x = -b/(2a)$, and since $f$ is strictly convex or concave, this critical point is a global minimum or maximum, with $k$ in vertex form representing that optimal value.