f(x)=(x-1)²(3x+1)/((x-2)²(x+3)) is an example.

The vertical asymptotes are at x=2 and x=-3, because the denominator becomes zero at these points. The turning point is caused by (x-1)² in the numerator. Note that the degrees of the numerator and denominator are the same (degree 3, because they each expand to x³). The horizontal asymptote is caused by the coefficient 3 in the numerator. As x gets larger in magnitude, the constants can be ignored and we end up with 3x³/x³=3.

To find another function, you could choose constants other than 2 and 3 in the denominator, but avoid 1, because that’s in the numerator.