Answer
The graph of $r=f(\theta-\alpha)$ is the result of a rotation from the original $r=f(\theta)$ by an angle of $\alpha$ counterclockwise around the pole.
Work Step by Step
See graph, the original is the red curve (no shift), for the one with a $\frac{\pi}{6}$ shift, the new curve is a rotation of the original for a $\frac{\pi}{6}$ angle counterclockwise around the pole, and the same happens for the case of $\frac{\pi}{3}$ shift.
In general, the graph of $r=f(\theta-\alpha)$ is the result of a rotation from the original $r=f(\theta)$ by an angle of $\alpha$ counterclockwise around the pole.
