(x-3)² = -4(2)(y+5.75). Sketch and determine the parts of the parabola

Let's analyze and sketch the given equation of the parabola:

\[
(x - 3)^2 = -4(2)(y + 5.75)
\]

### Simplification:
\[
(x - 3)^2 = -8(y + 5.75)
\]

This is in the standard form of a parabola equation:
\[
(x - h)^2 = 4p(y - k)
\]
where \((h, k)\) is the vertex, and \(p\) determines the distance from the vertex to the focus and the directrix.

### Identifying Parts of the Parabola:

1. **Vertex**: The vertex \((h, k)\) is derived from the equation. Here, \(h = 3\) and \(k = -5.75\).

   **Vertex**: \((3, -5.75)\)

2. **Direction**: Since the coefficient of \(y\) is negative, the parabola opens downward.

3. **Parameter \(p\)**: In the standard form, \(4p = -8\), so:
   \[
   p = -2
   \]
   Since \(p\) is negative, the parabola opens downward.

4. **Focus**: The focus is located at \((h, k + p)\). Therefore, the focus is:
   \[
   (3, -5.75 - 2) = (3, -7.75)
   \]

5. **Directrix**: The directrix is a horizontal line located at \(y = k - p\). Therefore, the directrix is:
   \[
   y = -5.75 + 2 = -3.75
   \]

6. **Axis of Symmetry**: The axis of symmetry is the vertical line that passes through the vertex, which is \(x = 3\).

### Sketch:

The sketch of the parabola would look like this:
- The vertex is at \((3, -5.75)\).
- The parabola opens downward.
- The focus is at \((3, -7.75)\).
- The directrix is the line \(y = -3.75\).
- The axis of symmetry is \(x = 3\).

I can create a visual sketch for you. Would you like me to generate it?

---

If you have any questions or need more details, feel free to ask. Here are some related problems you might find useful:

1. Find the vertex, focus, and directrix of the parabola \(x^2 = 8y\).
2. Determine the equation of a parabola with a vertex at \((0, 0)\) and a focus at \((0, -2)\).
3. Convert \(y^2 - 6y + 9 = 8x + 16\) to its standard form and identify its parts.
4. Sketch the parabola \(x^2 = -16y\).
5. Find the equation of a parabola with vertex at \((4, -3)\) and opening upward.
6. Determine the effect of changing the coefficient of \(y\) on the shape of the parabola.
7. Write the equation of a parabola with a vertex at \((-2, 1)\) and directrix \(y = 3\).
8. How does the sign of \(p\) affect the direction in which the parabola opens?

**Tip**: The vertex form of a parabola makes it easier to identify its direction, vertex, and the focus or directrix.

Learn how to sketch and determine the vertex, focus, directrix, and axis of symmetry of the parabola defined by the equation (x-3)^2 = -8(y+5.75). This problem covers key concepts in quadratic equations and parabola properties, suitable for advanced high school students.

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