What is a sufficient condition such that the graph of $f(\theta)$ in polar coordinates is symmetric relative to the $y$ axis?

You graph is symmetrical about the y-axis when $(x,y)\in G\implies (-x,y)\in G$.

Expressed in polar coordinates this means $\begin{cases}f(\theta_1)\cos(\theta_1)=-f(\theta_2)\cos(\theta_2)\\f(\theta_1)\sin(\theta_1)=f(\theta_2)\sin(\theta_2)\end{cases}$

Since we are not interested in $f=0$ which is automatically symmetric (since origin belongs to y-axis) we can simplify by $f$ to get the necessary condition:

$$\tan(\theta_1)=-\tan(\theta_2)\iff \theta_1\equiv-\theta_2\pmod\pi$$

And you get your conditions on $f$:

• $\theta_2=(2n)\pi-\theta_1\implies f(\theta_1)=-f(\theta_2)$

• $\theta_2=(2n+1)\pi-\theta_1\implies f(\theta_1)=f(\theta_2)$