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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