Downwardopening

Downward-opening describes a parabola that opens in the downward direction on the Cartesian plane. It is the case when the parabola’s axis of symmetry is vertical and the curvature bends downward, so the vertex represents the maximum point of the graph.

In algebra, a downward-opening parabola can be written as y = ax^2 + bx + c with a < 0. Its axis of symmetry is the vertical line x = -b/(2a), and the vertex is at (h, k) where h = -b/(2a) and k = f(h). In vertex form, the same curve is y = a(x − h)^2 + k with a < 0; the vertex is (h, k) and is the maximum value of the function.

Focus and directrix provide another description. For a parabola in the form y = a(x − h)^2 + k, the focal length p = 1/(4a) is negative when a < 0. The focus lies at (h, k + p) and the directrix is the line y = k − p. This reflects the geometric definition of a parabola as the set of points equidistant from the focus and the directrix.

Relation to concavity and applications: In calculus, a function is concave downward on an interval when its second derivative is negative, which corresponds to a locally downward-opening shape. Downward-opening parabolas appear in physics and engineering, such as in modeling projectile trajectories near their apex or in optical designs using parabolic reflectors that concentrate light toward a point along the axis below the vertex.

Example: y = −2x^2 + 3x + 1 opens downward, with vertex at (0.75, 2.125).