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To solve problems 47 and 48, we need to sketch the corresponding polar curves based on the given graphs of \( r \) as a function of \( \theta \). Let me guide you through the process.

### Problem 47:
In the first graph (Problem 47), the function \( r \) oscillates between approximately 1 and 2, with a periodic behavior over the interval \( [0, 2\pi] \).

#### Steps to convert to a polar graph:
1. **Identify the periodic function**: The graph shows a sinusoidal pattern, likely of the form \( r = 1 + \sin(\theta) \).
2. **Determine key points**:
   - At \( \theta = 0 \), \( r = 1 \).
   - At \( \theta = \frac{\pi}{2} \), \( r \approx 2 \).
   - At \( \theta = \pi \), \( r \approx 1 \).
   - At \( \theta = \frac{3\pi}{2} \), \( r \approx 0 \).
   - At \( \theta = 2\pi \), \( r \approx 1 \).
3. **Sketch the polar curve**: Use the points above to sketch the corresponding polar plot. Since the

Learn how to graph polar equations such as r = 1 + sin(θ) and r = 1 + sin(2θ) by interpreting Cartesian graphs. This problem covers key concepts in polar coordinates and trigonometric functions, suitable for high school students in Grades 10-12.

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