# Sketch the polar graph $r=e^{-2\phi}$

How are you supposed to sketch this type of polar graph?

Are you supposed to somehow relate this to $\cos\phi+i\sin\phi$ but can polar graphs even have an imaginary axis?!

I am thinking that you relate it to $\cos\phi+i\sin\phi$ because $x=r\cos\phi$ and $y=r\sin\phi$.

I have also sketched the graph plotting each point individually $\phi= 0, \frac{\pi}{6}, \frac{\pi}{3}, \frac{\pi}{2}, \pi.$ $r=1,0.35,0.12,0.04...$ But seems to plot the Cartesian graph. If you want to see the answer on the mark scheme it is here Q4(i) page 24.

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If the angle is just a real value, I don't see how complex numbers relate to this problem. The graph is essentially a spiral, that converges to the Origin. (At a finite length, but that's just on the side) – imranfat May 31 '13 at 22:58

Hint: note that the distance from the point $P=(r,\phi)$ to the origin, that is, the value of $r$, varies with the angle $\phi$.