Parabola Studio
A quadratic can be written three ways, and exams love switching between them. Here they all move together — drag one parabola and watch every form, root and feature update at once. Works on any phone.
One curve, three outfits
Every parabola is the same shape written in whichever form is most useful for the question:
- Vertex form y = a(x − h)² + k — reads the turning point straight off as (h, k).
- Standard form y = ax² + bx + c — the one you usually start with; c is the y-intercept.
- Factored form y = a(x − r₁)(x − r₂) — reads the x-intercepts (roots) straight off — when they exist.
Drag the parabola 🟠🟣
Drag the orange dot to move the vertex; drag the purple dot to change the width and direction. All three forms, the roots, the axis of symmetry and the discriminant update live.
What to watch for
The vertex is the easy bit
Drag the orange dot and read (h, k) right off the vertex form. The dashed purple line is the axis of symmetry x = h — the parabola is a perfect mirror across it.
The discriminant predicts the roots
Watch Δ = b² − 4ac. When it's positive there are two x-intercepts; zero means the vertex sits on the x-axis (one root); negative means the curve never reaches the x-axis and factored form has no real version.
Try this: drag the vertex slowly up toward the x-axis while it opens upward. The two roots slide together, meet when Δ = 0, then vanish — and the factored form switches to "no real roots" at exactly that moment. That's the discriminant doing its job, made visible.
Things to try
- Make a negative (drag the purple dot below the vertex) — the parabola flips to open downward. A bigger |a| makes it narrower.
- Park the vertex exactly on the x-axis — see the two roots become one repeated root, and factored form become a perfect square.
- Read the y-intercept (the orange dot on the y-axis) and check it equals c in standard form.