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:

cf (x)
c + f (x)
f (cx)
f (c + x)

where c is a constant.

We have rules like this when dealing with polar functions too, but not as many.

• The graph of r = cf (θ) will be the same shape as the graph of r = f(θ), but stretched away from or squished toward the origin by a factor of c.

• The graph of r = f (θ - c) is the same as the graph of r = f(θ), but rotated by an angle of c.

As far as nice rules for graphing go, that's all we get.

• There's no nice rule that tells us how the function r = f(cθ) looks.
    
• There's no nice rule that tells us how the function r = c + f (θ) looks.

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.

Practice: