Let the rational function is .

The numerator of a rational function in lowest terms determines the -intercepts of its graph.

Observe the graph:

The -intercepts of the graph are and (odd multiplicity; graph crosses the -axis).

At the graph crosses the -axis has odd multiplicity.

At the graph touches the -axis even multiplicity.

The horizontal asymptote is .

The numerator is multiplied with .

So one possibility for the numerator is .

The denominator of a rational function in lowest terms determines the vertical asymptotes of its graph.

Observe the graph:

The vertical asymptotes of the graph are and .

Since approaches to the left of and approaches to the right of , is a factor of odd multiplicity in .

Since approaches to the left of and approaches to the right of , is a factor of even multiplicity in . \ \

A possibility for the denominator is .

Therefore, one possibility for the rational function .

One possibility; .