https://file.aiquickdraw.com/mathbot/file/1725672970245papi1ish.jpg

From the image you uploaded, the coordinates of the vertex and focus of a parabola are given as follows:

- Vertex: \((1, 9)\)
- Focus: \((-3, -9)\)

We can analyze this information to determine the equation of the parabola.

### Steps to find the equation of the parabola:
1. **Identify the direction of the parabola**:
   Since the x-coordinates of the vertex and the focus differ, the parabola is horizontal (it opens left or right).

2. **Calculate the distance between the vertex and the focus**:
   The distance between the vertex and the focus is the focal length \( p \). This will help in writing the equation of the parabola.

   The focal length \( p \) can be found by the formula:
   \[
   p = \text{distance between vertex and focus} = \frac{\text{difference in x-coordinates}}{2} = \frac{1 - (-3)}{2} = 2
   \]

3. **Equation of the parabola**:
   For a horizontally oriented parabola, the equation in standard form is:
   \[
   (y - k)^2 = 4p(x - h)
   \]
   where \((h, k)\) is the vertex and \( p \) is the focal length. Substituting \( h = 1 \), \( k = 9 \), and \( p = -2 \) (since the parabola opens to the left):

   \[
   (y - 9)^2 = -8(x - 1)
   \]

This is the equation of the parabola.

Would you like more details on the steps, or any clarifications?

### 5 Relative Questions:
1. How would the equation change if the parabola opened upwards instead of sideways?
2. Can you find the directrix of this parabola using the vertex and the focus?
3. What is the length of the latus rectum of the parabola?
4. How would you graph this parabola based on the equation and the vertex/focus points?
5. How does changing the distance between the vertex and the focus affect the shape of the parabola?

### Tip:
Always pay attention to the relationship between the vertex and the focus to determine the correct orientation of the parabola.

Given the vertex (1,9) and focus (-3,-9) of a parabola, determine the equation of the parabola.

Learn how to find the equation of a horizontally oriented parabola with vertex (1,9) and focus (-3,-9). This solution involves calculating the focal length and applying the standard form of a parabola's equation. Ideal for Grades 10-12.

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