The Full Quadratic Workup

The full quadratic workup reads every feature from the formula: opening direction from a, vertex and axis from the vertex formula, y-intercept from c, x-intercepts from the zeros, extreme value from the vertex.

Work up f(x) = x² − 4x + 3. Direction: a = 1, opens up. Vertex: h = 2, k = f(2) = −1 — so (2, −1), axis x = 2, and the extreme is a minimum of −1. The y-intercept is (0, 3), read straight from c. The x-intercepts come from the zeros: the discriminant 16 − 12 = 4 promises two, and factoring delivers 1 and 3 — points (1, 0) and (3, 0). Five features, one formula, and every one checkable: the intercepts sit symmetric about the axis, 1 and 3 straddling x = 2 evenly, which is the geometry auditing the algebra.