Vertex Form

Vertex form gives the vertex without any calculation.

y = a(x − h)² + k has vertex (h, k).

Watch the sign inside. y = 2(x − 3)² + 4 has vertex (3, 4), but y = 2(x + 3)² − 1 has vertex (−3, −1), because x + 3 is x − (−3).

Read it as a transformation of y = x²: shifted h right, k up, stretched by a, and flipped if a is negative.

To convert from standard form, complete the square. To go the other way, expand and simplify.

Which form to use depends on the question. Standard form gives the y-intercept instantly; vertex form gives the maximum or minimum instantly.

Vertex form f(x) = a(x − h)² + k encodes the transformation of the parent y = x² directly, with axis of symmetry x = h and vertex (h, k).

Conversion from standard form is completing the square, and the resulting relationship h = −b/(2a), k = f(h) shows that the vertex formula is not independent information.

The three standard forms serve different purposes and convert freely: standard ax² + bx + c exposes the y-intercept c and feeds the quadratic formula; vertex form exposes the extremum and the transformations; factored form a(x − r₁)(x − r₂) exposes the roots, and the axis sits at their average (r₁ + r₂)/2.

The range follows immediately from vertex form: [k, ∞) for a > 0 and (−∞, k] for a < 0. Optimisation questions are therefore answered by conversion rather than by calculus at this level.