As discussed earlier, there are four standard forms of a parabola. In short, they represent

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`y^2 = 4ax` or

For example, for the equation `y^2 = 2x`, comparing it with standard form `y^2 = 4ax`, we get a 4a = 2, that is, a = 1/2. Therefore the focus, directrix, etc. for `y^2 = 2x` are as follows:

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`y^2 = -4ax` or

For example, for the equation `y^2 = -2x`, comparing it with standard form `y^2 = -4ax`, we get a 4a = 2, that is, a = 1/2. Therefore the focus, directrix, etc. for `y^2 = -2x` are as follows:

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`x^2 = 4ay` or

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`x^2 = -4ay` or

- A right handed parabola: `y^2 = 4ax`
- A left handed parabola: `y^2 = -4ax`
- An up facing parabola: `x^2 = 4ay`
- A down facing parabola: `x^2 = -4ay`

In each of these equations, `x` and `y` are variables and `a` is any number.

Now as you may know, a parabola has some characteristic properties, namely the vertex, focus, directrix, axis, latus rectum, etc. Each of the four standard forms have their own set of properties. These are discussed below:

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`y^2 = 4ax` or *Right Handed Parabola*

- Focus: (a, 0)
- Directrix: x = -a
- Vertex: (0, 0)
- Axis: y = 0
*(that is, the x-axis)* - Length of latus rectum: 4a

- Focus: (1/2, 0)
- Directrix: x = -1/2
- Vertex: (0, 0)
- Axis: y = 0
*(that is, the x-axis)* - Length of latus rectum: 4(1/2) = 1/2

`y^2 = 2x` and its characteristics |

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`y^2 = -4ax` or *Left Handed Parabola*

- Focus: (-a, 0)
- Directrix: (a, 0)
- Vertex: (0, 0)
- Axis: y = 0
*(that is, the x-axis)* - Length of latus rectum: 4a

- Focus: (-1/2, 0)
- Directrix: x = 1/2
- Vertex: (0, 0)
- Axis: y = 0
*(that is, the x-axis)* - Length of latus rectum: 4(1/2) = 1/2

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`x^2 = 4ay` or *Up Facing Parabola*

- Focus: (0, a)
- Directrix: (0, -a)
- Vertex: (0, 0)
- Axis: x = 0
*(that is, the y-axis)* - Length of latus rectum: 4a

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`x^2 = -4ay` or *Down Facing Parabola*

- Focus: (0, -a)
- Directrix: (0, a)
- Vertex: (0, 0)
- Axis: x = 0
*(that is, the y-axis)* - Length of latus rectum: 4a