Reduced equation of the horizontal parabola

Let's consider the parabola which vertex coincides with the origin and which axis coincides with the $x$ -axis.

In this case, the focus is at point $F (\frac{p}{2}, 0)$ , and the equation of the generator line $D$ is: $x = - \frac{p}{2}$ .

The equation of the parabola is $y^{2} = 2 p x$

Example

Considering the equation $y^{2} = - 6 x$ , find its vertex, its focus and its generator line.

By definition, in this type of equations the vertex is $A (0, 0)$ .

We can identify $y^{2} = - 6 x$ with $y^{2} = 2 p x$ and obtain $2 p = - 6$ and $p = - 3$ .

Therefore, the focus is at $F (\frac{p}{2}, 0)$ , which is at $F (- \frac{3}{2}, 0)$ .

To substitute $p$ in $x = - \frac{p}{2}$ .

The equation of the generator line is $x = - \frac{3}{2}$ .