# Points, Vectors, and Functions

### Topics

## Introduction to Points, Vectors, And Functions - At A Glance:

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*)

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 = c
*f*(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 each polar function. Notice that we're changing the bounds on θ each time.

- r = cos θ for 0≤ θ≤ π

#### Exercise 2

Use a calculator to graph each polar function. Notice that we're changing the bounds on θ each time.

- r = cos(2θ) for 0≤ θ ≤ 2π

#### Exercise 3

Use a calculator to graph each polar function. Notice that we're changing the bounds on θ each time.

- r = cos(\frac{1}{2}θ) for 0≤ θ ≤ 4π.

#### Exercise 4

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