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The problem asks to find the vertex and axis of symmetry of a parabola, given the focus and the directrix.

- **Focus**: \( (3, -2) \)
- **Directrix**: \( x = -5 \)

### Solution:

1. **Vertex**: The vertex of a parabola lies exactly halfway between the focus and the directrix. Since the directrix is a vertical line (\(x = -5\)) and the focus has an x-coordinate of \(3\), we can calculate the x-coordinate of the vertex as the midpoint between \(3\) and \(-5\):

   \[
   \text{x-coordinate of vertex} = \frac{3 + (-5)}{2} = \frac{-2}{2} = -1
   \]

   The y-coordinate of the vertex is the same as the y-coordinate of the focus, because the parabola opens sideways (horizontal parabola). Therefore, the y-coordinate of the vertex is \(-2\).

   So, the vertex is \( (-1, -2) \).

2. **Axis of Symmetry**: The axis of symmetry is the vertical line that passes through the vertex. Since the x-coordinate of the vertex is \(-1\), the axis of symmetry is the line:

   \[
   x = -1
   \]

### Final Answer:
- Vertex: \( (-1, -2) \)
- Axis of symmetry: \( x = -1 \)

Would you like further details or explanations on any part of this solution?

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Here are 5 additional questions related to parabolas:

1. How do you determine whether a parabola opens horizontally or vertically?
2. What is the general formula for a parabola given its focus and directrix?
3. Can the vertex formula be used if the focus and directrix are not parallel?
4. What are the steps to graph a parabola from its focus and directrix?
5. How does the distance between the focus and the directrix affect the shape of the parabola?

**Tip**: Remember that in horizontal parabolas, the vertex formula is still valid, but the standard form of the equation is \( (y - k)^2 = 4p(x - h) \), where \(p\) is the distance between the vertex and the focus.

The focus of a parabola is (3, -2), and its directrix is the line x = -5. What are the vertex and the axis of symmetry of the parabola?

This problem demonstrates how to find the vertex and axis of symmetry of a parabola with a focus at (3, -2) and a directrix at x = -5. Using the midpoint formula and properties of parabolas, the solution explains the step-by-step process to determine the vertex and symmetry. Suitable for students in Grades 10-12.

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