What is the completed square form of a parabola?

The completed square form of a parabola is expressed as ( y = a(x - h)^2 + k ). This form is advantageous because it clearly identifies the vertex of the parabola, which is located at the point ( (h, k) ). The parameter ( a ) indicates the direction and width of the parabola: if ( a ) is positive, the parabola opens upwards, while if ( a ) is negative, it opens downwards.

In the completed square form, the expression ( (x - h)^2 ) represents the transformation of the standard quadratic function ( y = x^2 ) horizontally by ( h ) units and vertically by ( k ) units. This makes it easier to graph the parabola or analyze its properties, such as vertex, axis of symmetry, and direction of opening.

Other forms listed don't specifically highlight the vertex derived from the completed square method. The linear representation of parabolas like ( y = ax^2 + bx + c ) or ( y = ax^2 + bx ) lack direct information about the vertex. The roots form ( y = a(x - r_1)(x - r_2) ) indicates the x