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Parabola ↔ Quadratic

Why does a kicked ball trace a curve — and how can one equation describe it?

1 The kick

A ball is kicked from the ground. Watch its path — then look at the equations that describe its height.

2 Explore: how a, h and k shape the curve

Every parabola can be written in vertex form: y = a(x − h)² + k. The vertex is (h, k), so you can read the turning point straight off the equation. Move the sliders to see exactly what each number does — and watch the standard form y = ax² + bx + c change with it.

a
h
k
your parabola vertex & axis of symmetry x-intercepts (roots) y-intercept

3 Challenge: match the kick

A ball is kicked from the ground. It reaches its maximum height of 4 m when it has travelled 4 m, and it lands 8 m away. The dashed grey curve is the real path. Use the sliders to find the equation that matches it exactly.
Tip: the vertex is (4, 4). In vertex form that means h = 4 and k = 4 — so you only need to find a.

a
h
k

4 Check your understanding

Five quick questions on vertex form y = a(x − h)² + k. The vertex is (h, k) — read it straight off the equation. Every quiz is different, so you can practice as many times as you like!

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🔑 The big ideas

Vertex form is the key. y = a(x − h)² + k shows the vertex (h, k) instantly. The turning point of the parabola is right there in the equation.
Watch the sign of h. In y = a(x + 4)² + k, the (x + 4) means h = −4. The vertex is (−4, k), not (4, k).
a controls the shape: positive → opens up (a valley), negative → opens down (a hill). Bigger |a| → narrower curve.
h shifts the parabola left or right. It is the x-coordinate of the vertex and the axis of symmetry is x = h.
k shifts the parabola up or down. It is the y-coordinate of the vertex — the maximum or minimum height in real life.
Two forms, one curve. y = a(x − h)² + k and y = ax² + bx + c describe the same parabola. Expand the vertex form to convert: b = −2ah and c = ah² + k.
Roots (x-intercepts) = where the height is zero. From vertex form, solve a(x − h)² + k = 0 → x = h ± √(−k/a). In real life: when the ball hits the ground.