Quadratic functions in vertex form

Mathematical statement

Let q:R→R be q(x)=a(x−h)²+k with real a,h,k and a≠0. Its vertex is (h,k) and its axis is x=h. When a>0, the global minimum is k; when a<0, the global maximum is k, both attained at x=h. The point (2h−x,q(x)) mirrors (x,q(x)) across the axis.

Representative formula (exact TeX)

q(x)

Assumptions

• The input ranges over all real numbers. Restricting the domain to an interval requires a separate extremum check.
• Require a≠0; otherwise the function is not quadratic in this entry. Exercise parameters are integers and substitution is exact.
• The representative formula q(x) denotes function evaluation only; the full vertex form and its conditions are in the statement.

Symbol scope

• q — real quadratic function in vertex form
• x — real input

Intuition, definition, and misconceptions

The square (x−h)² is nonnegative. A positive a makes the parabola open upward, and a negative a makes it open downward. The vertex is an ordered pair, whereas the extremum value is the scalar k; the function is unbounded in the other direction.

Evaluate by subtracting h, squaring, multiplying by a, and adding k. A negative a is not replaced by its absolute value, and signs in the vertex form must be retained.

The reflected input is 2h−x, whose squared displacement from h agrees with that of x. When x=h, the point is its own reflection. A common translation of h and the evaluation input does not create a substantively new evaluation instance.

Common errors include confusing the vertex coordinates with the scalar extremum, discarding the coefficient sign, or applying the full-real-domain extremum to a restricted interval without checking.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Function mapping — Declare the inputs and outputs of a total real function.
• Quadratic equations over the reals — Fix the meaning of the quadratic term and its nonzero coefficient.

Verification boundary

Finite exact evaluations can be recomputed; global extrema and symmetry have no newly attached formal Evidence.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Bind nonnegativity of squares and vertex-form identities over all real inputs, without inheriting a quadratic-equation proof.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.