## Pages

### Standards form of a parabola

There are four simplest equations of a parabola which are called its standard forms. They are:
• y^2 = 4ax
• y^2 = -4ax
• x^2 = 4ay
• x^2 = -4ay
These are called the standard forms because they represent the simplest possible graphs of parabolas. In each of these equations, x and y represents a variable and a can be any number.

For example, y^2 = 4ax represents a right handed parabola as follows:
 y^2 = 4ax
y^2 = -4ax represents a left handed parabola:
 y^2 = -4ax
x^2 = 4ay represents a upward facing parabola:
 x^2 = 4ay
x^2 = -4ay represents a upside-down or inverted parabola:
 x^2 = -4ay
As you can see these are the simplest kinds of graphs of a parabola possible. So they are termed as standard forms.

These standard forms help determine the various properties of a parabola. For example, for y = 2x^2 + 3x + 1, though it does not match with any of the standard forms above, can be easily converted to match any one of them (It can be rewritten as y = 2(x + 3/4)^2 - 1/8, which is comparable with the standard form x^2 = 4ay - you can learn more about converting a parabola to the above four standard forms here). Its characteristics can be determined by comparing it to the standard form.

Each standard form of a parabola has its own set of properties or characteristics, such as the coordinates of focus, vertex, the equation of directrix, etc. These are discussed in this post.