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The problem you're presenting is related to identifying the characteristics of a parabola based on its vertex and focus.

Given:
- Vertex at \( (-3, -2) \)
- Focus at \( (-2, -2) \)

Let's solve it step by step:

### 1. **General Equation**:
Since the vertex and focus have the same \( y \)-coordinate and different \( x \)-coordinates, the parabola opens **horizontally** (to the right or left).

The general form of a parabola that opens horizontally is:
\[
(y - k)^2 = 4p(x - h)
\]
Where:
- \( (h, k) \) is the vertex
- \( p \) is the distance from the vertex to the focus.

### 2. **Standard Equation**:
The vertex is \( (-3, -2) \), so \( h = -3 \) and \( k = -2 \).

Now, to find \( p \), we calculate the distance between the vertex and the focus. The distance along the x-axis is:
\[
p = -2 - (-3) = 1
\]

So, the equation becomes:
\[
(y + 2)^2 = 4(x + 3)
\]

### 3. **Length of Latus Rectum**:
The length of the latus rectum is given by \( |4p| \). Since \( p = 1 \), the length is:
\[
|4p| = 4
\]

### 4. **Opening**:
Since \( p > 0 \), the parabola opens **to the right**.

### 5. **Axis of Symmetry**:
The axis of symmetry is a horizontal line passing through the vertex. Since the \( y \)-coordinate of the vertex is \( -2 \), the axis of symmetry is:
\[
y = -2
\]

### 6. **Directrix**:
The directrix is a vertical line, and its distance from the vertex is equal to \( p \). The directrix is located \( p \) units to the left of the vertex, so:
\[
x = -3 - 1 = -4
\]

### 7. **End points of Latus Rectum**:
The endpoints of the latus rectum are located \( 2p \) units above and below the focus.

Since \( p = 1 \), the distance from the focus is \( 2p = 2 \).

- The coordinates of the endpoints are \( (-2, -2 + 2) = (-2, 0) \) and \( (-2, -2 - 2) = (-2, -4) \).

### Summary:
1. **General Equation**: \( (y + 2)^2 = 4(x + 3) \)
2. **Standard Equation**: \( (y + 2)^2 = 4(x + 3) \)
3. **Length of Latus Rectum**: 4
4. **Opening**: To the right
5. **Axis of Symmetry**: \( y = -2 \)
6. **Directrix**: \( x = -4 \)
7. **End points of Latus Rectum**: \( (-2, 0) \), \( (-2, -4) \)

Would you like further clarification on any step or more details? Here are some questions to deepen your understanding:

1. How would the equation change if the parabola opened vertically instead of horizontally?
2. Can you explain the relationship between the directrix, focus, and vertex in a parabola?
3. How is the length of the latus rectum derived in terms of \( p \)?
4. What is the significance of the axis of symmetry for the graph of a parabola?
5. How would the standard equation change if the parabola opened to the left instead of to the right?

**Tip**: Always check the distance between the vertex and the focus or directrix to determine the direction of the parabola's opening.

Learn how to find the equation of a parabola with a vertex at (-3, -2) and focus at (-2, -2). This problem includes key geometry concepts such as the equation of a parabola, axis of symmetry, and directrix. Suitable for students in Grades 10-12.

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