Why? What happens NEAR the asymptotes?
The best way to figure how to sketch a graph is to identify the critical x values. Those would be any zeros of g(x) and any vertical asymptotes. These cut the x-axis into intervals for you to check how g(x) behaves.
x value significance
-3 vertical asymptote
0 root or zero
3 vertical asymptote
Next look at the signs for g in each interval on x. Pick any x value in each interval.
INTERVAL pick x sign of g
(-infinity,-3) -4 -5(-4)/((-4-3)(-4+3))=(+)/((-)(-))=POSITIVE
(-3,0] -1 -k(-1)/((-)(+))=NEGATIVE
[0,3] 1 -k(+)/((-)(+))=POSITIVE
[3,infinity) 10 -k(+)/((+)(+))=NEGATIVE
Think about g in those four intervals.
Starting very far from the left, approaching -3, g is positive, changing from very small to very large, and large without bound toward this asymptote.
Immediately to the right of this x=-3 asymptote, suddenly g is very large BUT NEGATIVE..., Do you see how this works?
Also notice change from negative to positive as g crosses the x-axis at 0.
,...
The horizontal asymptote is y=0.