1. **Problem Statement:** Graph the polar curve given by $$r = -3 - 5 \cos \theta$$. 2. **Understanding the Curve Type:** This is a limaçon curve of the form $$r = a + b \cos \theta$$ where $$a = -3$$ and $$b = -5$$. 3. **Key Properties:** - If $$|a| < |b|$$, the limaçon has an inner loop. - Here, $$|-3| = 3 < 5 = |-5|$$, so the curve has an inner loop. 4. **Calculate critical points:** - Maximum radius occurs when $$\cos \theta = -1$$: $$r_{max} = -3 - 5(-1) = -3 + 5 = 2$$ - Minimum radius occurs when $$\cos \theta = 1$$: $$r_{min} = -3 - 5(1) = -3 - 5 = -8$$ 5. **Plot points at key angles:** - At $$\theta = 0$$: $$r = -3 - 5 \times 1 = -8$$ - At $$\theta = \pi$$: $$r = -3 - 5 \times (-1) = 2$$ - At $$\theta = \frac{\pi}{2}$$ or $$\frac{3\pi}{2}$$: $$r = -3 - 5 \times 0 = -3$$ 6. **Interpretation:** - Negative radius means the point is plotted in the opposite direction of the angle. - The inner loop occurs because the radius becomes negative and large in magnitude. 7. **Summary:** - The curve is a limaçon with an inner loop. - It loops inside the origin due to negative radius values. **Final answer:** The graph of $$r = -3 - 5 \cos \theta$$ is a limaçon with an inner loop.