In Grade 11 a parabola was a shape to describe. This year it is an objective function: something a business, an engineer or a physicist wants pushed as high or as low as it will go. The uniform has not changed — $y = ax^2 + bx + c$, read aloud y equals a x-squared plus b x plus c — but every letter now has a job title. a is the coefficient of the squared term, and its SIGN alone decides the whole question: negative a folds the graph downward, so the turn is a maximum; positive a cups it upward, so the turn is a minimum. b is the coefficient of the plain first power, and it is the only reason the optimum is not sitting at zero. c is the constant — the value the model returns before a single unit is made.

The turning point has two coordinates and they answer two different exam questions. $h$ is the x-coordinate of the vertex: an INPUT — how many units to build, what price to charge. $k$ is the y-coordinate: the OUTPUT — the profit, the cost, the height. Marks are awarded separately for each, and quoting one when the paper asked for the other is the single most expensive habit in this chapter. No computing today: today is about reading, and a sign is never decoration. In $P(n) = -2n^2 + 48n - 180$, b is 48 and c is −180.