Maximum or Minimum

The vertex is the turning point, and which way the parabola opens decides what kind of turning point it is. When a is positive the curve opens up, like a cup: the vertex is its lowest point. When a is negative it opens down, like a hill: the vertex is its highest point.

So the vertex's height is a minimum or a maximum of the function. For f(x) = 2x^2 + 8x + 1, a is 2 and the vertex is (-2, -7), so -7 is the smallest output f ever gives, reached at x = -2. For g(x) = -x^2 + 4x + 1, a is -1, the axis is 2 and g(2) = 5, so 5 is the largest output g ever gives.

This is why the vertex matters outside the classroom. A thrown ball's highest point, the cheapest production level, the biggest area a fence can enclose: each is the vertex of some parabola.