the vertex of a conic
On any conic there are a few special points where the curve makes its sharpest turn — the 'nose' of the parabola, the ends of an ellipse, the tips of a hyperbola's branches. These are the vertices, and they are the natural landmarks you reach for when sketching or measuring the curve.
Formally, a vertex is a point where the conic crosses its axis of symmetry. The parabola has exactly one vertex: the point where it turns around, sitting halfway between the focus and the directrix; for y = x^2 it is the origin. An ellipse has four points where it meets its two axes — but the two on the major (long) axis, at distance a from the center, are usually called the (principal) vertices, and the two on the minor axis, at distance b, the co-vertices. A hyperbola has two vertices, one on each branch, at distance a from the center along the transverse axis, at (-a, 0) and (a, 0) for x^2/a^2 - y^2/b^2 = 1; the hyperbola never crosses its conjugate axis, so it has no vertices there.
Vertices matter because they pin down the curve's size and position with the least fuss: the distance between an ellipse's or hyperbola's two main vertices is 2a, and the vertex of a parabola is the reference point for its standard equation (x - h)^2 = 4p(y - k). When you translate a conic away from the origin, tracking the vertex tells you exactly how far it moved.
The parabola y = (x - 2)^2 + 3 has its vertex at (2, 3) — the lowest point, where the curve turns. The ellipse x^2/9 + y^2/4 = 1 has principal vertices (-3, 0), (3, 0) and co-vertices (0, -2), (0, 2). The hyperbola x^2/9 - y^2/4 = 1 has vertices (-3, 0) and (3, 0), one on each branch.
Vertices sit where a conic crosses its axis of symmetry — the curve's sharpest turns.
The 'vertex' of a conic is not the same as the vertex of the cone it was cut from, nor the same as a polygon's corner. Here it means a turning point on the curve where it meets an axis of symmetry.