Sketching the Graph of a Polar Equation for a Circle

First, testing the equation for symmetry, we find that the graph is symmetric about the polar axis. Next, we find the zeros and maximum $\left|r\right|$ for $r=4\cos \theta .$ First, set $r=0,$ and solve for $\theta$. Thus, a zero occurs at $\theta =\frac{\pi }{2}\pm k\pi .$ A key point to plot is $\left(0,\text{}\text{}\frac{\pi }{2}\right).$

To find the maximum value of $r,$ note that the maximum value of the cosine function is 1 when $\theta =0\pm 2k\pi .$ Substitute $\theta =0$ into the equation:

The maximum value of the equation is 4. A key point to plot is $(4,0).$

As $r=4\cos \theta$ is symmetric with respect to the polar axis, we only need to calculate r-values for $\theta$ over the interval $[0,$ $\pi ].$ Points in the upper quadrant can then be reflected to the lower quadrant. Make a table of values similar to Table 3. The graph is shown in Figure 6.

Figure 6