The Parabola Dossier

Unit 6 turns two function families into working models — the line for steady change, the parabola for change that turns — and every feature of both reads straight off the coefficients.

A linear model is two facts wearing algebra: a constant rate and a starting value. The tank filling at 10 gallons per minute from 50 is V(t) = 10t + 50 before any formula is quoted, and the model's slope keeps its units — 10 gallons per minute, not a bare 10. Two data points rebuild the whole line, the unused point runs the audit, and the zero of the model is a fact with a name in the world: break-even. The parabola is the first graph that turns, and the unit's work is a dossier on the turn. The quadratic wears two forms, each with a superpower: general form surrenders the y-intercept on sight and feeds the vertex formula, h = −b/(2a); standard form surrenders the vertex on sight, once the inside's sign-flip is respected. Completing the square converts one costume to the other — on a function, add and subtract in the same line, since there is no second side. The vertex is the whole story's hinge: axis of symmetry through it, minimum or maximum at it by the sign of a, and the optimization problems of the world — the largest area forty meters of fence can hold — are vertex hunts wearing overalls. The discriminant counts the x-intercepts before anyone factors, and 333 closes a promise made back when the course fixed its names: zero of the function, x-intercept of the graph, solution of the equation — one object, three roles, no drift. The dossier method is the takeaway: a formula is not a problem to be survived but a document to be read, and every quadratic files the same five reports.