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

The given equation is:

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

Let's first simplify and analyze the equation.

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

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

### Identifying Parts of the Parabola:
1. **Vertex**: The vertex is \((h, k)\). From the equation, \((x + 3)\) corresponds to \((x - h)\), so \(h = -3\). \((y + 5.75)\) corresponds to \((y - k)\), so \(k = -5.75\).

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

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

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

4. **Focus**: The focus is located at \((h, k + p)\). So, 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
Here’s how the parabola looks:
- 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 sketch of this parabola for you if you'd like. Would you like me to generate a sketch? 

---

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

1. Find the equation of a parabola with vertex at \((2, -1)\) and opening upward.
2. Determine the focus and directrix of the parabola \(y^2 = 4x\).
3. Sketch the parabola \(x^2 = 12y\).
4. Convert the equation \(x^2 - 6x + 9 = -8y + 46\) to vertex form.
5. Find the vertex and focus of \(4(y + 2) = (x - 1)^2\).
6. Determine the vertex, focus, and directrix of \(x^2 + 8x + 16 = -12y\).
7. Write the equation of a parabola with a vertex at the origin and a focus at \((0, 3)\).
8. How does changing the value of \(p\) affect the shape of a parabola?

**Tip**: Always start by identifying the vertex of the parabola when analyzing its equation. This helps you determine its direction and position.

Explore how to sketch and identify the parts of the parabola given by the equation (x + 3)^2 = -8(y + 5.75). Learn about the vertex, focus, directrix, and axis of symmetry. Suitable for high school students studying quadratic equations and parabolas.

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