Physics

projectile motion height = ut + ½gt²

Throwing a ball: height(t) = ut − ½gt². The path is a parabola, the time-of-flight is the quadratic's root.

Engineering

parabolic mirrors, satellite dishes

Designing parabolic satellite dishes, mirror telescopes, and bridge cables — every parabolic shape is a quadratic.

Economics

profit-maximising quantities

Profit = Revenue − Cost; both can be quadratic in price. Maximum profit is the vertex of the resulting parabola.

Architecture

parabolic arches

St. Louis Gateway Arch is a (modified) parabola — 192 m tall, 192 m wide. Quadratic in stone.

Sports

optimal angle for max distance

Basketball trajectory, football kick, golf-ball flight — all parabolic. Pros optimise launch angle to maximise the quadratic.

Cricket

ball trajectory analysis

A six is a parabola from bat to stand. Hawk-Eye fits a quadratic to predicted ball-flight for every LBW decision.