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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 = (x)

there are certain rules about how the function stretches or translates if we look at variations such as:

c(x)
c + (x)
(cx)
(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(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 = (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 + (θ) looks.

We can verify that the function r = (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.

Example 1

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


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.

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