Domain of R: x<>-8 (x not equal to -8) because x+8 would be zero and division by zero is not permissible.
x=-8 is equation of vertical asymptote.
When x is very large and positive or negative, R(x) approaches x^3/x^3=1, so y=1 is the horizontal asymptote.
Draw the asymptotes. On the left of x=-8 R(x) is large and positive and drops steeply to continue closer to the horizontal asymptote as x becomes more negative. On the right of x=-8 R(x) rises steeply from a large negative value passing through the origin and rising a little above the x axis but levelling out to approach the horizontal asymptote. At x=2.5 the curve rises to its local flattish maximum at R(x)=0.66 approx., dropping to zero at x=20, then it rises towards the asymptote gradually as x gets larger.