Answer
The graphs were obtained by rotating $r= 1+\sin\theta.$
In general, the graph $r=f(\theta-\alpha)$ is obtained by rotating the graph of $r=f(\theta)$ by the angle $\alpha$, counterclockwise.
Work Step by Step
Graphing the three curves, also graph the lines
$\displaystyle \theta=\frac{\pi}{6}$ and $\displaystyle \theta=\frac{\pi}{3}$.
Note the point A on $ r=1+\sin\theta$ (black curve).
Rotating it by $\displaystyle \frac{\pi}{6}$, we have its corresponding point on $ r=1+\sin\theta$ (blue).
Also, we note that the red curve is obtained by rotating the black curve by $\displaystyle \frac{\pi}{3}$, counterclockwise.
In general, the graph $r=f(\theta-\alpha)$ is obtained by rotating the graph of $r=f(\theta)$ by the angle $\alpha$, counterclockwise.
