1. **Problem Statement:** Graph the polar curve given by the equation $$r = -4 - 3 \sin \theta$$. 2. **Understanding the curve type:** The general form of a limaçon is $$r = a + b \sin \theta$$ or $$r = a + b \cos \theta$$. 3. **Identify parameters:** Here, $$a = -4$$ and $$b = -3$$ (since the equation is $$r = -4 - 3 \sin \theta$$). 4. **Determine the type of limaçon:** The shape depends on the ratio $$\left| \frac{b}{a} \right|$$. - If $$\left| \frac{b}{a} \right| > 1$$, the limaçon has an inner loop. - If $$\left| \frac{b}{a} \right| = 1$$, it is a cardioid. - If $$0 < \left| \frac{b}{a} \right| < 1$$, it is a limaçon without an inner loop. Calculate: $$\left| \frac{b}{a} \right| = \left| \frac{-3}{-4} \right| = \frac{3}{4} = 0.75$$ Since $$0 < 0.75 < 1$$, the curve is a limaçon without an inner loop. 5. **Plotting points:** To sketch the curve, evaluate $$r$$ at key angles: - At $$\theta = 0$$: $$r = -4 - 3 \sin 0 = -4 - 0 = -4$$ - At $$\theta = \frac{\pi}{2}$$: $$r = -4 - 3 \sin \frac{\pi}{2} = -4 - 3 = -7$$ - At $$\theta = \pi$$: $$r = -4 - 3 \sin \pi = -4 - 0 = -4$$ - At $$\theta = \frac{3\pi}{2}$$: $$r = -4 - 3 \sin \frac{3\pi}{2} = -4 - (-3) = -4 + 3 = -1$$ 6. **Interpret negative $$r$$ values:** Negative $$r$$ means the point is plotted in the opposite direction of the angle. 7. **Summary:** The curve is a limaçon without an inner loop, shifted and reflected due to negative $$a$$ and $$b$$ values. **Final answer:** The polar curve $$r = -4 - 3 \sin \theta$$ is a limaçon without an inner loop.