identify the graph of r = 2 + cos(θ).

Explanation:

Step1: Analyze the polar - equation form

The general form of a limacon in polar coordinates is $r = a\pm b\cos\theta$ or $r = a\pm b\sin\theta$. When $a = 2$ and $b = 1$ (since $r=2 + \cos\theta$), and $a>b>0$, the graph of the polar equation $r=a + b\cos\theta$ is a limacon without an inner - loop.

Step2: Check key points

When $\theta = 0$, $r=2 + \cos(0)=3$. When $\theta=\frac{\pi}{2}$, $r = 2+\cos(\frac{\pi}{2})=2$. When $\theta=\pi$, $r=2+\cos(\pi)=1$. When $\theta=\frac{3\pi}{2}$, $r=2+\cos(\frac{3\pi}{2})=2$. The graph is symmetric about the polar axis (because the equation has $\cos\theta$).

Answer:

The bottom - right graph (assuming the bottom - right graph is a non - looped limacon symmetric about the horizontal axis with maximum value of $r = 3$ and minimum value of $r = 1$)