Polar graph question

Question

Can you only graph periodic functions using polar graphing? I'm not really understanding this I guess. It you are to get all of the x and y values on a finite graph, then the original must be periodic, no?

Answer 1

Not necessarily. Recall that:

$$r^2=x^2+y^2$$

$$x=r\cos \theta$$

$$y=r\sin \theta$$

You can manipulate that to graph non-periodic functions. Take for instance $y=2x^2$.

$$r^2=x^2+(2x^2)^2=x^2+4x^4$$ $$r=x\sqrt{1+4x^2}$$

Plug in for $x$ what we said earlier, $r\cos \theta$:

$$r=r\cos \theta\sqrt{1+4(r\cos \theta)^2}$$ $$1=\cos \theta \sqrt{1+4r^2\cos^2\theta}$$ $$\left(\frac1{\cos \theta}\right)^2-1=4r^2\cos^2 \theta$$ Too lazy to simplify:

$$r=\sqrt{\frac{\left(\frac1{\cos \theta}\right)^2-1}{4\cos^2 \theta}}$$

When you graph it, you'll get the graph of $y=2x^2$.