Algebra 67 - Deriving the Vertex Form of a Quadratic Function

The graph of a quadratic function in a single variable is always a parabola, and when the function is written in vertex form, we can identify the coordinates of the parabola's vertex simply by looking at the function. But how is the vertex form derived and why does it work? The process explored here involves shifting or 'translating' the basic quadratic function "a x-squared".