# Find information from basic root locus

I have to draw a root locus with respect to K and find for which values of K the system is stable, then draw conclusions on the principal shape of the step response.

$$G_{0}(s)=\frac{K(s+2)}{s(s+1)(s+3)}$$

1. Isn't the step response just $$\displaystyle \frac{1}{s}$$?

2. I can get the characteristic polynomial of $$G_c$$ which is $$s(s + 1)(s + 3) + K(s + 2)=0$$, and I know it is in the form of $$Q + KP$$ wher $$Q$$ stands for $$G_0$$ numerator and $$P$$ for $$G_0$$ denominator, so the function is supposed to move between the zeroes to the poles. I checked the solution and I still don't understand how to do the asymptotes, how do they come up with the intersection with the real axis, how do they draw the shape in form of a bow and 2 bars on the real axis. The intersection with the imaginary axis part makes much more sense.

What's the reasoning behind the asymptotes and the root locus?

The solution: