Parabola Explorer

Skill goal: connect a quadratic’s coefficients to its vertex, axis of symmetry, intercepts, and roots.

Observe

Move the sliders and watch the parabola change shape and position. The teal marks track the vertex and the x-intercepts.

Manipulate

a1 b0 c0

y = x²

a = 0, so this is a straight line — not a parabola. Set a to a nonzero value to explore quadratics.

Teal: vertex, axis of symmetry, and x-intercepts · Gray dot: y-intercept

Predict

Before you change it, what do you think will happen? If you raise c while a and b stay fixed, what will happen to the roots? Make a guess, then test it.

See

Vertex

(0, 0)

Axis of symmetry

x = 0

y-intercept

(0, 0)

Discriminant

Roots

x = 0 (double root)

Explain

Why it works: the vertex always sits at x = −b / (2a) — the axis of symmetry runs through it, and the two roots (when they exist) are mirror images across that line. The discriminant b² − 4ac predicts the roots before you solve: positive means two real x-intercepts, zero means the vertex just touches the x-axis (a double root), and negative means the parabola never touches the axis — the roots are a complex pair.

Challenge

1. In Standard form, find values of a, b, and c so the parabola crosses the x-axis at x = 2 and x = 3.

2. Make the vertex of the parabola land exactly on (3, 1). Either form works.

Parabola Explorer — Fibo Math Lab