# At a Glance - Rules of Graphing We Do (or Don't) Have

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.

#### Exercise 1

Use a calculator to graph the polar function.

*r*= cos*θ*for 0 ≤ θ ≤ π

#### Exercise 2

Use a calculator to graph the polar function.

*r*= cos(2*θ*) for 0 ≤*θ*≤ 2π

#### Exercise 3

Use a calculator to graph the polar function.

- for 0 ≤
*θ*≤ 4π.

#### Exercise 4

Without a calculator, graph the polar function
*r* = 1 + sin *θ*
for
0 ≤ *θ* ≤ 2π.