vertex

The vertex is the single turning point of a parabola — the spot where the curve stops going one way and starts going the other. For an upward parabola it is the bottom of the valley (the lowest point); for a downward parabola it is the top of the hill (the highest point).

If the quadratic is written in vertex form a(x - h)^2 + k, the vertex sits exactly at the point (h, k) — that is the whole point of vertex form. From the standard form ax^2 + bx + c, the x-coordinate of the vertex is h = -b/(2a), and you find k by plugging that x back into the function.

The vertex matters because it pins down the parabola's most important feature: its extreme value. The y-coordinate of the vertex is the maximum or minimum output of the function, which is why vertices show up constantly in optimization problems — the most area, the least cost, the highest profit.