Answer
slant: $y=-\frac{1}{2}x+1$
vertical: $x=1$
Work Step by Step
Rewrite the function as $r(x)=\frac{1}{2}\frac{-x^2+3x}{x-1}$
and use synthetic division to get
$r(x)=-\frac{1}{2}x+1+\frac{1}{x-1}$
identify the asymptotes as
slant: $y=-\frac{1}{2}x+1$
vertical: $x=1$