Parabola y = ax² and quadratic equations

What it shows

The graph of y = ax² is a parabola with its vertex at the origin, symmetric about the y-axis; it opens upward when a > 0, downward when a < 0, and narrows as |a| grows. Intersecting it with the line y = bx + c leads to ax² − bx − c = 0, with discriminant Δ = b² + 4ac. Two intersection points mean two real roots (Δ > 0), a tangent line means a double root (Δ = 0), and no common point means no real roots (Δ < 0): a geometric meaning for the quadratic formula.

How to use

Drag point P on the parabola to change a, and the two points on the line to change b and c; the sliders do the same. The Example buttons 2 roots, Tangent (Δ = 0) and No roots give quick starting cases. Show/hide table of values lets students check symmetry about the y-axis. Ask them to compare the x-coordinates of the intersection points with the roots given by the formula in the info panel.

Parameters you can change

• Coefficient a in y = ax² -3–3
• Coefficient b in the line y = bx + c -5–5
• Coefficient c in the line y = bx + c -8–8
• Show table of values for y = ax²

Questions to explore

1. Keeping the line fixed, how must you change a to make the two intersection points disappear?
2. Why does ax² − bx − c = 0 always have the root x = 0 when c = 0?
3. How do the x-coordinates of the intersection points relate to the roots given by the quadratic formula?