Consider the polar equation $r=1+\cos (\theta )$. Identify this equation as a cardioid and sketch its graph.

Step 1 ：Step 1: Identify the type of polar graph. The general form of a cardioid is either $r=a+a\cos (\theta )$ or $r=a+a\sin (\theta )$, where a is the length of the cardioid. Comparing this with the given equation, we can see that $a=1$ and the graph is a cardioid that opens to the right.

Step 2 ：Step 2: Sketch the graph. Start by sketching the cardioid in the standard position (opening to the right). The length from the pole (origin) to the cusp (pointed end) of the cardioid is 1 (the value of a), and the length from the pole to the far side of the cardioid is 2 (the maximum value of r).

Step 3 ：Step 3: Label important points. The cusp is at $\theta =0$, and the far side of the cardioid is at $\theta =\pi$.