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.