Different Types of Parabolas

We see that for the equation $${y^2} = 4ax$$ the parabola opens to the right if $$a > 0$$ and to the left if $$a < 0$$. The point midway between the focus and the directrix on the parabola is called the vertex. The vertex of the parabola is the origin. The line through the vertex and the focus is called the axis of the parabola.

If the X-axis and the Y-axis are interchanged, then the focus is at the point $$F\left( {0,a} \right)$$, and the directrix is the line having the equation $$y = – a$$. The equation of this parabola is $${x^2} = 4ay$$. If $$a > 0$$ the parabola opens upwards, and if $$a < 0$$ the parabola opens downwards. In each case the vertex is at the origin, and the Y-axis is the axis of the parabola.

When one draws a sketch of the graph of a parabola, it is helpful to draw the chord through the focus, perpendicular to the axis of the parabola. This chord is called the latus rectum of the parabola. The length of the latus rectum is $$|4a|$$.

Standard Parabolas

Equation	$${y^2} = 4ax$$	$${y^2} = – 4ax$$	$${x^2} = 4ay$$	$${x^2} = – 4ay$$
Focus	$$\left( {a,0} \right)$$	$$\left( { – a,0} \right)$$	$$\left( {0,a} \right)$$	$$\left( {0, – a} \right)$$
Vertex	$$\left( {0,0} \right)$$	$$\left( {0,0} \right)$$	$$\left( {0,0} \right)$$	$$\left( {0,0} \right)$$
Axis	$$y = 0$$	$$y = 0$$	$$x = 0$$	$$x = 0$$
Directrix	$$x = – a$$	$$x = a$$	$$y = – a$$	$$y = a$$
Latus Rectum	$$x = a$$	$$x = – a$$	$$y = a$$	$$y = – a$$
Length of L.R	$$|4a|$$	$$|4a|$$	$$|4a|$$	$$|4a|$$
Graph