a parabola
/ puh-RAB-uh-luh /
Throw a ball and watch its path; aim a satellite dish at the sky; trace the cross-section of a car headlight reflector. Each is a parabola — the gentle U-shaped curve that opens forever in one direction. It is the simplest of the conic sections and the one a child meets first, because the graph of y = x^2 is exactly this curve.
Precisely, a parabola is the set of all points that are equally far from a fixed point (the focus) and a fixed line (the directrix). Pick the focus F and the directrix line d; a point P lies on the parabola exactly when its distance to F equals its perpendicular distance to d. Putting the focus at (0, p) and the directrix at y = -p, this balance gives the clean equation x^2 = 4py, where p is the distance from the vertex to the focus. The lowest point, halfway between focus and directrix, is the vertex; the line through the focus perpendicular to the directrix is the axis of symmetry, and the curve is a perfect mirror image across it.
Why care? A parabola is the only conic with eccentricity exactly e = 1, sitting on the boundary between the closed ellipse and the open hyperbola. Its reflective property — every ray coming in parallel to the axis bounces off to the focus, and a source at the focus throws out a parallel beam — is why headlights, flashlights, telescopes, and dish antennas are parabolic. And under gravity alone (ignoring air), a thrown object really does follow a parabola, a fact Galileo first proved.
For x^2 = 8y, match 4p = 8 so p = 2: the vertex is (0, 0), the focus is (0, 2), and the directrix is the line y = -2. Check the point (4, 2) on the curve: its distance to the focus is sqrt((4-0)^2 + (2-2)^2) = 4, and its distance down to the directrix y = -2 is |2 - (-2)| = 4. They match, as they must.
Equal distances to focus and directrix define every point of the parabola.
A parabola is not just any U-shape: a hanging chain forms a catenary, not a parabola, and the two curves look similar but obey different equations. 'Parabola' specifically means the e = 1 conic, the graph of a quadratic.