The root locus shows how the closed-loop poles move as a scalar loop gain changes.

For a loop transfer function

$$L(s)=KG(s),$$

the closed-loop poles satisfy

$$1+KG(s)=0.$$

Equivalently,

$$KG(s)=-1.$$

A point $s$ lies on the root locus when the phase of $G(s)$ is an odd multiple of $\pi$ and the gain can be chosen positive to satisfy the magnitude condition.

As $K$ varies from zero upward, branches begin at the open-loop poles and move toward open-loop zeros or toward infinity when there are more poles than zeros.

The plot makes controller-gain trade-offs visible geometrically. Pole motion toward the imaginary axis usually means slower decay or greater oscillation; crossing into the right half-plane means instability.

Root locus is therefore a design map: it connects one adjustable gain directly to the closed-loop pole locations that determine transient behavior and stability.