maximum or minimum value

The maximum or minimum value of a quadratic is the single greatest or least output the function ever reaches. Because a parabola is a smooth arc that bends only once, it has exactly one such extreme value — a peak if it opens downward, a trough if it opens upward — and it occurs at the vertex.

Which one you get depends on the leading coefficient a. If a > 0 the parabola opens upward and has a minimum (the lowest point); if a < 0 it opens downward and has a maximum (the highest point). Either way, the extreme value is the y-coordinate of the vertex, k, found at x = -b/(2a).

This is why quadratics are the workhorse of basic optimization: maximize the area of a rectangular pen with fixed fencing, minimize the cost of production, find the height at which a ball peaks. You set up a quadratic model and read off the vertex. Note the extreme is a value (an output), while the vertex is the whole point (input and output together).