You're correct to ask for clarification on how the \( y \)-coordinate of the vertex was determined. Let's go through the steps more carefully.

Given:
- **Focus:** \( \left(\frac{107}{12}, -7\right) \)
- **Directrix:** \( x = \frac{109}{12} \)

### Understanding the Problem

The problem involves a parabola that opens horizontally (since the directrix is a vertical line). The vertex of the parabola is always equidistant from both the focus and the directrix.

### Step-by-Step to Find the Vertex

1. **X-Coordinate of the Vertex:**
   The x-coordinate of the vertex, \( h \), is the midpoint between the x-coordinate of the focus \( \left(\frac{107}{12}\right) \) and the x-coordinate of the directrix \( \left(\frac{109}{12}\right) \).

   \[
   h = \frac{\frac{107}{12} + \frac{109}{12}}{2} = \frac{216}{24} \cdot \frac{1}{2} = \frac{108}{12} = 9
   \]

   So, the x-coordinate of the vertex is \( h = 9 \).

2. **Y-Coordinate of the Vertex:**
   Since the directrix is a vertical line and the parabola opens horizontally, the y-coordinate of the vertex remains the same as the y-coordinate of the focus, which is \( -7 \).

   Hence, the y-coordinate \( k \) of the vertex is \( -7 \).

### Conclusion
The vertex is \( (9, -7) \).

This correctly identifies both coordinates of the vertex. I hope this clears up any confusion!

### Would you like more details or have any questions?

Here are 5 related questions you might find interesting:

1. How would you find the vertex if the directrix was a horizontal line?
2. What are the steps to derive the equation of a parabola from its focus and directrix in a general case?
3. How does changing the focus position affect the shape and direction of the parabola?
4. What if the vertex was not exactly halfway between the focus and directrix—how would that affect the equation?
5. Can the vertex form equation be converted into standard form easily? How?

**Tip:** When dealing with parabolas, always check the orientation first (horizontal or vertical) because it affects how you calculate the vertex.

Learn how to find the vertex of a parabola with a detailed step-by-step explanation. This problem involves understanding horizontal parabolas and using the midpoint formula. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation