Here is an interactive worksheet that will allow you to plot polar functions of the form [math](r, \theta)[/math].[br][br]Try entering different functions into the input box in the top left hand corner. I would suggest looking at functions like [math]r = \sin(a \theta)[/math], [math]r = p - \cos(a\theta)[/math], [math]r = a\theta[/math] where [math]a,p \in \mathbb{R}[/math].[br][br]You may need to use the [math]\alpha[/math] symbol on the right of the input box to insert a [math]\theta[/math] in your function. When you have typed in your function press enter, you can then move the slider (in blue) to vary the size of [math]\theta[/math].

[color=#555][b]Now answer these questions[/b][/color][br][br]What generalisations can you make from what you have discovered?[br][br]Look specifically at curve with the equation [math]r = a(p + q\cos(\theta))[/math] where [math]p \geq q[/math][br][list][br][*]What if [math]p = q[/math]?[br][*]What if [math]p \geq 2q[/math]?[br][*]What if [math]q \leq p < 2q[/math]?[br][/list]