The given equation \((y + 2)^2 = 8(x - 5)\) is in the standard form of a horizontal parabola:

\[
(y - k)^2 = 4p(x - h)
\]

In this form:
- \((h, k)\) is the **vertex** of the parabola.
- \(p\) represents the distance from the vertex to the **focus** along the x-axis for a horizontal parabola.
- The directrix is a vertical line located at \(x = h - p\).
  
Now, comparing \((y + 2)^2 = 8(x - 5)\) with the standard form:
- \(h = 5\) and \(k = -2\), so the vertex is at \((5, -2)\).
- \(4p = 8\), so \(p = 2\), meaning the distance from the vertex to the focus is 2 units to the right (since \(p > 0\)).

### Important features:
- **Vertex**: \((5, -2)\)
- **Focus**: \((5 + 2, -2) = (7, -2)\)
- **Directrix**: \(x = 5 - 2 = 3\)
- **Axis of symmetry**: The line \(y = -2\), as the parabola opens horizontally.

### General Form:
To express this equation in general form, we can expand and rearrange it.

\[
(y + 2)^2 = 8(x - 5)
\]
Expanding both sides:
\[
y^2 + 4y + 4 = 8x - 40
\]
Rearrange to get:
\[
y^2 + 4y - 8x + 44 = 0
\]

This is the general form of the given parabola.

Would you like more details on any step? Here are some related questions you can explore:

1. What is the general form of a vertical parabola?
2. How do you find the focus and directrix of a parabola?
3. What are the key differences between horizontal and vertical parabolas?
4. How do you convert a parabola equation from general to standard form?
5. How can you graph a parabola given its equation in standard form?

**Tip:** The value of \(p\) determines the "width" of the parabola; larger \(p\) values make the parabola wider, while smaller \(p\) values make it narrower.

Discover how to express the parabola (y+2)²=8(x-5) in general form and find its key features such as the vertex, focus, and directrix. This tutorial covers essential conic section concepts and is suitable for students in Grades 10-12.

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