To me it looks like a reciprocal function mixed with a cubic function. I tried out x/((x+1)(x-4)) and it looks very close, but I can't figure out the correct transformations to apply. What is the best way to go about trying to find an equation for this?
Based on your answer, our function should take the following form:
$$f(x)=\frac{a(x-b)}{(x+1)(x-4)}$$
Since $f(2)=0$, then $b=2$.
$$\implies f(x)=\frac{a(x-2)}{(x+1)(x-4)}$$
Since $f(0)=2$, then $a=4$. Therefore,
$$f(x)=\frac{4(x-2)}{(x+1)(x-4)}$$