Reduced equation of the horizontal parabola
Choose a point $P(x_0,0)$ in the $x$-axis. Find the equation of the parabola whose focus coincides with point $P$ and the origin with the vertex. Find its generator line.
We choose $P(1,0)$.
First, the origin is the vertex, so $A(0,0)$ and it is a reduced equation. The point $P(1,0)$ is the focus $F(\dfrac{p}{2},0)$, so $\dfrac{p}{2}=1$ and then $p=2$.
It is possible now to find the equation by substituting $p$ in $y^2=2px$. The equation is $$y^2=4x$$
To obtain the generator function we substitute $p$ in $x=-\dfrac{p}{2}$ and find the straight line $$x=-1$$
For $P(1,0)$ the parabola is $y^2=4x$ and the generator line is $x=-1$.