If you want to draw a graph by hand, you usually want to show the significant features. This takes some judgement. For your function, the places it goes off to infinity ($-\frac 13$ and $1$) is a good start. Finding local maxima and minima is another. You could look at the numerator and see it is always positive, then at the denominator and see that it is negative on $(-\frac 13,1)$. This gives you the sign changes, which here are at the asymptotes. There is a root at $0$, you want to show that. Looking at the numbers, you expect the interesting things to be within $(-10,10)$ or so.
You can probably do better than Wolfram Alpha, which has extraneous vertical lines at the asymptotes and plotted less in the negative $y$ direction than I would like. This is even after I gave it the range $(-5,5)$-it started with $(0,20)$