Vertex Form and Transformations

Vertex form is y = a(x − h)² + k, and the vertex is (h, k). No calculation is needed to find it.

Watch the sign of h

In y = (x − 3)² the vertex is at x = 3, not −3. The form subtracts h, so the sign appears reversed.

As a transformation

Vertex form describes y = x² shifted h right, k up, and stretched by a.

Inside and outside again

The h sits inside the bracket and acts horizontally, backwards. The k sits outside and acts vertically, as expected.

Getting there

Completing the square converts standard form to vertex form. Take half the x coefficient, square it, and add and subtract it.

Why bother

The vertex is the maximum or minimum, so vertex form answers most optimisation questions in one line.

Vertex form shows the vertex

y = a(x − h)² + k has its vertex at (h, k), readable directly. The minus sign inside means the h is the x-coordinate as written, which catches people out at first.

The parameters are transformations

h shifts horizontally, k shifts vertically, and a stretches and possibly reflects. The same transformation rules apply to every function family, which is why they are worth learning generally.

Completing the square converts to vertex form

Standard form hides the vertex; completing the square reveals it. The technique adds and subtracts the square of half the x coefficient, which changes the form without changing the value.

Each form reveals something different

Standard form gives the y-intercept, factored form gives the roots, vertex form gives the turning point. Choosing which to convert into depends entirely on what you need to know.