Answer
$y^2=\frac{9}{2}x$, $(\frac{9}{8},-\frac{9}{4}),(\frac{9}{8},\frac{9}{4})$. See graph.
Work Step by Step
1. Given vertex $(0,0)$ and x-axis as the symmetry axis, we have the general equation is $x=ky^2$.
2. Use point $(2,3)$, we have $2=3^2k$, thus $k=\frac{2}{9}$ and $x=\frac{2}{9}y^2$ or $y^2=\frac{9}{2}x$, thus $p=\frac{9}{8}$.
3. Let $x=\frac{9}{8}$, we have $y^2=\frac{81}{16}$ and $y=\pm\frac{9}{4}$, thus we have the two points that define the latus rectum $(\frac{9}{8},-\frac{9}{4}),(\frac{9}{8},\frac{9}{4})$
4. See graph.