Functions, Quadratic Equations, and Maximum Profit

A function assigns exactly one output to each allowed input. Several inputs may have the same output, but one input cannot have two different outputs in the same function. A table with the same input paired with conflicting outputs does not describe a function.

Function notation names the rule and its input. Read f(4) as 'f of four,' not f multiplied by four. To evaluate it, replace every occurrence of the input variable with 4 and follow the order of operations.

The domain is the collection of permitted inputs; the range is the collection of outputs actually produced. A formula alone does not settle a real-world domain. A workshop with room for 20 participants uses n = 0, 1, 2, ..., 20, not negative values or fractional people.

A graph represents a function when no vertical line intersects it more than once. Each horizontal location then has at most one output. Tables and formulas also help distinguish linear functions, which have constant first differences for equally spaced inputs, from nonlinear functions, whose changes may vary.

A quadratic has a nonzero squared term and can be written ax² + bx + c. Its graph is a parabola. Positive a produces an upward-opening curve, while negative a produces a downward-opening curve.

A root is an input making the function zero. To solve a quadratic equation, first move all terms to one side. When an expression factors, the zero-product property says at least one factor must equal zero.

Not every quadratic factors conveniently. The quadratic formula works for any quadratic equation with a ≠ 0. Substitute signed coefficients carefully, especially when b or c is negative, and keep exact values until the final rounding step.

The discriminant b² − 4ac tells how many real roots exist: positive gives two distinct roots, zero gives one repeated root, and negative gives no real roots. A model with no real zero need not be meaningless; it simply does not cross the horizontal axis.

Interpret each root in context. A negative quantity may be algebraically valid but unusable. For a downward-opening profit model with two valid roots, profit is positive between them and negative outside them, within the model's permitted domain.

When the selling price depends on quantity, revenue need not be linear. Multiplying p(q) = 30 − 0.20q by q gives R(q) = 30q − 0.20q². Subtracting C(q) = 200 + 10q produces P(q) = −0.20q² + 20q − 200.

A parabola's vertex is its turning point. For f(x) = ax² + bx + c, the vertex input is −b ÷ (2a). When a is negative, the parabola opens downward and the vertex gives its unrestricted maximum. When a is positive, the vertex is a minimum.

Calculate the output by substituting the vertex input into the original function. In a business problem, also recover the corresponding price, revenue, and cost so the final recommendation is understandable and can be checked.

Optimization must respect the domain. Here nonnegative ticket price requires q ≤ 150, and actual attendance uses whole numbers. Capacity may impose a smaller upper bound. For a continuous quadratic on a closed interval, compare an allowed vertex with the interval endpoints; for whole-number quantities, compare the allowed integers closest to the vertex and any relevant boundaries.

The result is only as dependable as the assumptions. Actual demand, refunds, staffing costs, and capacity may differ. Maximum modeled profit is neither investment advice nor a promise that a business or event will earn that amount.