Graph the polar curve \( r=4+4 \cos \theta \)

To graph the polar curve \( r=4+4 \cos \theta \), start by recognizing that this is a type of limacon. At \( \theta = 0 \), \( r = 8 \) since \( \cos(0) = 1 \). As \( \theta \) reaches \( \pi \), \( r = 0 \) because \( \cos(\pi) = -1 \). The curve will have symmetry about the polar axis, and you can find several key points by plugging in values of \( \theta \) (like \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \)). Sketch the curve smoothly connecting these points, and you'll see that it narrows at the origin, forming a "heart shape" as it loops around. For a fun twist, think of it as a dance—each point on the graph is like a step synchronized to how far you are from the pole, out toward the rhythm of cosine waves. And remember, if you ever need to plot one, be sure to draw those angles carefully; they make all the difference in the "dance"!