To graph a limacon, we start with the general equation of the form \( r = a \pm b \cos \theta \) or \( r = a \pm b \sin \theta \). The values of \( a \) and \( b \) determine the characteristics of the graph, specifically whether it has a dimple or an inner loop. If \( a > b \), the limacon will have a dimple; if \( a < b \), it will have an inner loop. For example, in the equation \( r = 3 - 2 \sin \theta \), we see that \( a = 3 \) and \( b = 2 \), indicating that the graph will have a dimple.

Next, we analyze the symmetry of the graph. Since the equation contains a sine function, the graph will be symmetric about the line \( \theta = \frac{\pi}{2} \). This symmetry helps in plotting points at the quadrantal angles: \( 0 \), \( \frac{\pi}{2} \), \( \pi \), and \( \frac{3\pi}{2} \).

Calculating the points involves substituting these angles into the equation:

• For \( \theta = 0 \): \[ r = 3 - 2 \sin(0) = 3 \] This gives the point \( (3, 0) \).
• For \( \theta = \frac{\pi}{2} \): \[ r = 3 - 2 \sin\left(\frac{\pi}{2}\right) = 1 \] This gives the point \( (1, \frac{\pi}{2}) \).
• For \( \theta = \pi \): Using symmetry, we reflect the point from \( \theta = \frac{\pi}{2} \) to get \( (3, \pi) \).
• For \( \theta = \frac{3\pi}{2} \): \[ r = 3 - 2 \sin\left(\frac{3\pi}{2}\right) = 5 \] This gives the point \( (5, \frac{3\pi}{2}) \).

After plotting these points, we connect them with a smooth curve, ensuring to reflect the dimple at the top of the graph. The resulting shape will clearly illustrate the characteristics of a limacon with a dimple. For greater accuracy, additional points can be plotted as needed.