Question

What type of limaçon is graphed by the polar equation lower r equals 4 plus 2 sine theta? Identify the axis of symmetry and horizontal and vertical intercepts.
(4 points)

Answer 1

To analyze the polar equation \( r = 4 + 2 \sin \theta \), we can identify the type of limaçon it represents and the properties related to symmetry and intercepts.

### Type of Limaçon

The general form of a limaçon is given by:

\[
r = a + b \sin \theta \quad \text{or} \quad r = a + b \cos \theta
\]

In this case, we have \( r = 4 + 2 \sin \theta \), where \( a = 4 \) and \( b = 2 \).

Since \( a > 0 \) and \( b > 0 \) and \( a > b \) (i.e., \( 4 > 2 \)), this limaçon will not have an inner loop and is classified as a **limaçon with an outer loop**.

### Axis of Symmetry

The symmetry of the limaçon can be determined based on the trigonometric function used. Since we are using \( \sin \theta \), the graph of the limaçon will be symmetric about the vertical line \( \theta = \frac{\pi}{2} \) (the y-axis).

### Horizontal and Vertical Intercepts

1. **Vertical Intercepts:**
Vertical intercepts occur when \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \).

- At \( \theta = \frac{\pi}{2} \):
\[
r = 4 + 2 \sin\left(\frac{\pi}{2}\right) = 4 + 2(1) = 6
\]
Thus, the point is \( (0, 6) \) in Cartesian coordinates.

- At \( \theta = \frac{3\pi}{2} \):
\[
r = 4 + 2 \sin\left(\frac{3\pi}{2}\right) = 4 + 2(-1) = 2
\]
The point is \( (0, -2) \) in Cartesian coordinates (which corresponds to a point at \( (0, -2) \)).

2. **Horizontal Intercepts:**
Horizontal intercepts occur when \( r = 0 \). Therefore, we set the equation to zero:

\[
0 = 4 + 2 \sin \theta
\]
Rearranging gives us:

\[
2 \sin \theta = -4 \implies \sin \theta = -2
\]

However, the sine function cannot equal -2; hence there are no horizontal intercepts for this limaçon.

### Summary

- **Type of Limaçon:** Limaçon with an outer loop
- **Axis of Symmetry:** Vertical axis (y-axis)
- **Vertical Intercepts:** \( (0, 6) \) and \( (0, -2) \)
- **Horizontal Intercepts:** None

This completes the analysis of the given polar equation.