The best way to graph polar functions is by using a graphing calculator or a computer program. We can wave our hands and pull a rabbit out of a hat. That's because there aren't as many rules about graphing polar functions. Those few rules that we do have can be much more complex.
With a rectangular function
y = f (x)
there are certain rules about how the function stretches or translates if we look at variations such as:
c + f (x)
f (c + x)
where c is a constant.
We have rules like this when dealing with polar functions too, but not as many.
As far as nice rules for graphing go, that's all we get.
We can verify that the function r = f (cθ) is weird by trying different values in the graphing calculator.
The function r = c + f(θ) is also weird. Adding a constant can change whether your r values are positive or negative, which can totally change the shape of the graph. It may also change the bounds we need for θ if we want to find the whole graph.