Parabolas

A parabola is the set of points equally distant from a fixed point, the focus, and a fixed line, the directrix. That equal-distance rule is exactly what makes a satellite dish work: every incoming signal that hits the curved surface parallel to the axis bounces to the same single point, the focus, no matter where it lands on the dish. A random curved bowl has no such guarantee. Only the parabola's shape sends every parallel ray to one spot.

You already know y - k = a(x - h)², where the vertex sits at (h, k) and the parabola opens up (a > 0) or down (a < 0) along a vertical axis. Swap the roles of x and y and you get x - h = a(y - k)²: same vertex (h, k), but now the axis of symmetry is horizontal, y = k, and the parabola opens right (a > 0) or left (a < 0). The variable that is squared tells you which direction the curve opens; the other variable sits alone and names the axis.

This swap connects to a bigger idea: exchanging x and y in any equation reflects its graph across the line y = x. Do that to a sideways parabola and it becomes a vertical one, and vice versa. That reflection is the same move you use later when thinking about inverse relations, so a parabola solved for x is really just the inverse-style twin of one solved for y.