91Ó°ÊÓ

To test for symmetry, check for three different conditions. For symmetry about the x-axis, plug -\(\theta\) into the equation and compare the resultant with the original equation. For the given equation, replacing \(\theta\) with -\(\theta\), you get \(r = 2 \cos 2(-\theta)\). Cosine is an even function, so \( \cos(-\theta) = \cos(\theta)\), hence the equation remains \(r = 2 \cos 2 \theta \), which is the original equation. Therefore, this equation is symmetric with respect to x-axis. Similarly, test for symmetry about the y-axis by plugging \(\pi - \theta\). However, replace \(\theta\) with \(\pi - \theta\), gets \(r = 2 \cos[2(\pi - \theta)] = 2 \cos(2 \pi - 2 \theta) = 2 \cos(2 \theta)\). Cosine function has a period of \(2 \pi\) hence \(\cos(2\pi - x) = \cos(x)\). Therefore, this equation is symmetric with respect to y-axis. Lastly, to test for symmetry about the origin, replace r and \(\theta\) with -r and \(\pi + \theta\) respectively, which doesn't hold, and equation is not symmetric about the origin.

To graph a polar equation, first construct a table with \(\theta\) and r. Then plot the polar points on the polar grid. Since this function is a cosine function, it varies between -1 and 1. Therefore, the radius r, will vary between -2 and 2. The graph has maximum points at \(\theta = 0\), \(\pi/2\), \(\pi\), \(\3 \pi / 2\)..., and minimum points at \(\theta = \pi / 4\), \(\3 \pi / 4\), \(\5 \pi / 4\), \(\7 \pi / 4\)... where the graph intersects the x-axis. Hence, plot the points accordingly.