Sketch a scaled rational curve and verify its asymptote

At $x=0$, $at=0$ and $a>0$ give $t=0$, so the intercept is $(0,a)$. As $x\to\pm\infty$, also $t\to\pm\infty$, and $y\to0$. Thus $y=0$ is the horizontal asymptote. The denominator is positive for all real t, so there is no finite vertical asymptote.

Eliminating t gives $y=a^3/(a^2+x^2)$. It is positive and symmetric about the y-axis, with a smooth maximum at $(0,a)$ and both tails approaching but never reaching the x-axis. The source peak is drawn sharply schematically; the actual rational curve is smooth.