Sketching the Graph of a Cardioid

First, testing the equation for symmetry, we find that the graph of this equation will be symmetric about the polar axis. Next, we find the zeros and maximums. Setting $r=0,$ we have $\theta =\pi +2k\pi .$ The zero of the equation is located at $\left(0,\pi \right).$ The graph passes through this point.

The maximum value of $r=2+2\cos \theta$ occurs when $\cos \theta$ is a maximum, which is when $\cos \theta =1$ or when $\theta =0.$ Substitute $\theta =0$ into the equation, and solve for $r.$

The point $(4,0)$ is the maximum value on the graph.

We found that the polar equation is symmetric with respect to the polar axis, but as it extends to all four quadrants, we need to plot values over the interval $[0,\pi ].$ The upper portion of the graph is then reflected over the polar axis. Next, we make a table of values, as in Table 4, and then we plot the points and draw the graph. See Figure 8.

Figure 8