To determine the equation of the limaçon based on your description, let's analyze the provided information:
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Vertical intercepts: You have two vertical intercepts at \( \left(5, \frac{\pi}{2}\right) \) and \( \left(1, \frac{3\pi}{2}\right) \). The first intercept at \( \left(5, \frac{\pi}{2}\right) \) indicates that when \( \theta = \frac{\pi}{2} \), \( r = 5 \), and the second intercept at \( \left(1, \frac{3\pi}{2}\right) \) indicates that when \( \theta = \frac{3\pi}{2} \), \( r = 1 \).
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Horizontal intercepts: The horizontal intercepts at \( (3, 0) \) and \( (-3, 0) \) indicate that when \( \theta = 0 \), \( r = 3 \), and when \( \theta = \pi \), \( r = -3 \).
Given this information, we see that the limaçon crosses the y-axis at values 5 and 1, and it reaches its maximum at \( \theta = 0 \) with a value of 3.
Since there is a dimple on the graph and no inner loop, the equation generally takes the form: \[ r = a + b \cos(\theta) \] or \[ r = a + b \sin(\theta) \] where \( a \) and \( b \) determine the shape of the limaçon.
Given the intercepts:
- The maximum \( r \) at \( \theta = 0 \) suggests that for \( r = 3 + 2 \cos(\theta) \), the points would yield the correct intercepts.
To summarize:
- \( r = 3 \) when \( \theta = 0 \).
- \( r = 5 \) when \( \theta = \frac{\pi}{2} \).
- \( r = 1 \) when \( \theta = \frac{3\pi}{2} \).
This aligns with: \[ r = 3 + 2 \cos(\theta) \]
Thus, the equation that matches the description of the limaçon with the given intercepts is:
Response: \( r = 3 + 2 \cos(\theta) \)