Quadratic explorer

Slide a, b and c, or grab the vertex and drag it around. The three formulas under the graph (standard, vertex and factored form) are the same parabola written three ways. Watch which numbers change and which stay put, and watch the two roots meet and vanish as the discriminant crosses zero.

Mission. Keep a = 1 and b = −2. Before you touch c, predict the value of c at which the two roots merge into one. Then slide c and test. Which feature of the picture records the discriminant reaching zero?

Answer

c = 1. With a = 1 and b = −2, Δ = b² − 4ac = 4 − 4c, which is zero when c = 1. In the picture the two black root dots slide together and the orange vertex touches the x-axis: k = 0 and Δ = 0 are the same statement, because k = −Δ/(4a). Push c past 1 and the roots are gone. Extension: predict the merging value of c for a = 2, b = −4 (it is c = 2), then try a negative a.

a 1 b -2 c -3

What's going on

Standard form y = ax² + bx + c tells you c at a glance (the y-intercept) and not much else. Vertex form y = a(x − h)² + k tells you the turning point (h, k) and the axis of symmetry x = h. Completing the square converts one into the other, and the conversion gives h = −b/(2a) and k = c − b²/(4a). Factored form y = a(x − r₁)(x − r₂) tells you the roots, when there are real ones.

The discriminant Δ = b² − 4ac decides how many real roots there are. Geometrically, Δ/(4a²) is how far the vertex sits below the axis in units that make the parabola look like x², so Δ > 0 means the vertex is on the far side of the x-axis from the arms (two roots), Δ = 0 means the vertex touches the axis (one repeated root), Δ < 0 means the parabola never reaches the axis.

Drag the vertex straight up and down: a stays fixed, h stays fixed, so b stays fixed, and only c moves. Drag it sideways: now b and c both change but a does not, because the shape of a parabola is set by a alone. Every parabola with the same a is the same shape, just moved.

Sheets: The parabola's mirror line (grade 9), Why the quadratic formula works and What the roots are telling you.