The four standard forms of a parabola

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

• 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: 

`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

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:

• 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

`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

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:

• 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

`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

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

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

`x^2 = 2y` and its characteristics

`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

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

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

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

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Four Standard Forms of a ParabolaNext:
Convert Equation of a Parabola to Standard Form