General and Vertex Form of a Quadratic Function

The equation of a parabola or a quadratic function can be given in various forms.

One of these forms is the so-called general form:

However, if you want to read the parabola directly from a coordinate system or draw it in a coordinate system, then an alternative form of representation is useful, the so-called vertex form:

From this form you can directly read off the vertex $\mathrm{S}$ of the parabola: It has the coordinates $\mathrm{S}\left( d\,\left|\, e\right.\right)$. The influence of the parameters in the vertex form is explained in more detail in the linked article.

Conversion general form - vertex form and vice versa

• The transformation from the general form to the vertex form is done by completing the square.

• The conversion from the vertex form to the general form is done by multiplying out the brackets using the binomial formulae and concluding.

The value of $a$ remains the same, but the values for $d$ and $e$ are generally not identical with those of $b$ and $c$.

Using the general and vertex form

General form:

• Sometimes the problem setting explicitly asks you to determine the function term in the general form. Here, if you start with the equation in the vertex form and transform it.

• For calculations, the general form is often better suited because there are no more brackets in this form. This is the case, for example, when calculating the intersection of a parabola with a straight line or another parabola. When calculating zeros, you can apply the quadratic solution formula directly to the general form, which is not possible for the vertex form.

Vertex form:

• From the vertex shape you can read off the vertex without further calculation.

• You can easily describe how to get the parabola by shifting and stretching the normal parabola in the vertex form.

• If the equation of a parabola is to be set up and the vertex of the parabola is given, one should work with the vertex form, since one can then plug in the vertex immediately.

Exercises

State in vertex form and determine the vertex:

Video on the conversion