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Let's go through the given parabolas one by one and solve for their vertex, focus, directrix, and latus rectum.

### 1) \( (y - 3)^2 = 48(x - 5) \)
This is a **horizontal parabola** since it is in the form \( (y - k)^2 = 4p(x - h) \).

- **Vertex**: \( (h, k) = (5, 3) \)
- **4p**: Here, \( 4p = 48 \), so \( p = 12 \).
- **Focus**: The focus is located \( p \) units to the right of the vertex for a horizontal parabola. Hence, the focus is \( (5 + 12, 3) = (17, 3) \).
- **Directrix**: The directrix is \( p \) units to the left of the vertex. So, the equation of the directrix is \( x = 5 - 12 = -7 \).
- **Latus Rectum**: The length of the latus rectum is \( |4p| = 48 \).

### 2) \( (y + 4)^2 = -16(x + 3) \)
This is another **horizontal parabola**, but it opens to the left because of the negative sign.

- **Vertex**: \( (h, k) = (-3, -4) \)
- **4p**: \( 4p = -16 \), so \( p = -4 \).
- **Focus**: Since \( p = -4 \), the focus is to the left of the vertex: \( (-3 - 4, -4) = (-7, -4) \).
- **Directrix**: The directrix is \( p \) units to the right of the vertex: \( x = -3 + 4 = 1 \).
- **Latus Rectum**: The length of the latus rectum is \( |4p| = 16 \).

### 3) \( (x + 5)^2 = 8(y - 6) \)
This is a **vertical parabola** since it is in the form \( (x - h)^2 = 4p(y - k) \).

- **Vertex**: \( (h, k) = (-5, 6) \)
- **4p**: \( 4p = 8 \), so \( p = 2 \).
- **Focus**: For vertical parabolas, the focus is \( p \) units above the vertex. So, the focus is \( (-5, 6 + 2) = (-5, 8) \).
- **Directrix**: The directrix is \( p \) units below the vertex: \( y = 6 - 2 = 4 \).
- **Latus Rectum**: The length of the latus rectum is \( |4p| = 8 \).

### 4) \( (x - 7)^2 = -36(y - 7) \)
This is a **vertical parabola** that opens downward because of the negative sign.

- **Vertex**: \( (h, k) = (7, 7) \)
- **4p**: \( 4p = -36 \), so \( p = -9 \).
- **Focus**: Since \( p = -9 \), the focus is \( 9 \) units below the vertex: \( (7, 7 - 9) = (7, -2) \).
- **Directrix**: The directrix is \( p \) units above the vertex: \( y = 7 + 9 = 16 \).
- **Latus Rectum**: The length of the latus rectum is \( |4p| = 36 \).

### 5) \( (x - 2)^2 = -12y(y - 3) \)
This equation is not clearly in standard form, so it needs rearrangement for proper interpretation.

For now, do you want to continue working on this or have any particular clarification? 

Here are some relative follow-up questions to further explore the topic:
1. How do we derive the formula for a parabola's focus and directrix?
2. How do you determine the orientation of a parabola based on its equation?
3. What role does the sign of \( p \) play in the direction of the parabola's opening?
4. Can you explain the geometrical significance of the latus rectum in a parabola?
5. How do transformations affect the position of a parabola's vertex?

**Tip**: When dealing with parabolas, always check the squared term to identify whether it opens horizontally or vertically!

Given the following equation of a parabola, determine the vertex, focus, directrix, and latus rectum. Graph it in a rectangular coordinate system.

Learn how to determine the vertex, focus, directrix, and latus rectum of a parabola based on its equation. This involves working with horizontal and vertical parabolas, applying the formula for conic sections, and graphing in a rectangular coordinate system. Suitable for high school students in Grades 10-12.

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