Reduced equation of the vertical parabola


Let's consider the vertical parabola, where the vertex is in the origin and the parabola lies on the $$x$$ axis.

The focus is now at point $$F(0,\dfrac{p}{2})$$, and the equation of the generator line $$D$$ is: $$y=-\dfrac{p}{2}$$.

The equation of the parabola is $$$x^2=2py$$$

Considering the equation $$x^2=12y$$, find its focus, its generator line and its vertex.

The vertex is, by definition, at $$A(0,0)$$.

Comparing $$x^2=12y$$ with $$x^2=2py$$ we can see that $$2p=12$$ and therefore $$p=6$$.

Substituting $$p$$, we can find the focus $$F(0,3)$$ and the generator line $$y=-3$$.