The Parabola Shape

Algebra 1 · Axiom Academy

A fountain throws water into the air, and the water carves out one particular curve — the same curve a thrown ball, a bridge cable, and the graph of y = ax² all share. Water knows a shape you'll meet over and over Stand by a fountain and watch a single stream. It leaves the nozzle, climbs, slows, tips over at the top, and falls — and it never wobbles. Every drop carves out the exact same smooth, symmetric arch. That arch has a name: a parabola . Before you ever write its equation, it's worth seeing one drawn in front of you. Press play. A jet of water sprays from the nozzle and the path it traces fills in across the screen. Notice where it peaks, and notice that the way up and the way down are mirror images. The arch is a parabola: one highest point — the vertex — with two identical halves folding around the line straight up through it. A gentle fountain makes a wide, lazy arch; a high-pressure jet makes a tall, narrow one. On the graph that's the single number a in y = a·x² . Drag the slider — or grab the handle on the curve — and watch the parabola open up and pinch in. The vertex never moves off the bottom, and the two halves stay perfect mirror images. Big a, narrow and steep; small a, wide and shallow — but it's always the same parabola shape, just stretched. The vertex stays put and the mirror symmetry never breaks. Fold it in half and the sides match

This is the written version of the interactive lesson above. See the full Algebra 1 course.