The given equation is \((x - 2)^2 = 8(y - 1)\). This equation resembles the standard form of a parabola, which is:

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

Where:
- \((h, k)\) is the vertex of the parabola.
- \(p\) is the distance from the vertex to the focus or directrix.

### Step-by-Step Breakdown:

1. **Identify the vertex:**
   - Comparing \((x - 2)^2 = 8(y - 1)\) to \((x - h)^2 = 4p(y - k)\), we see that \(h = 2\) and \(k = 1\).
   - Therefore, the vertex of the parabola is \((2, 1)\).

2. **Find the value of \(p\):**
   - The equation can be rewritten as \((x - 2)^2 = 4p(y - 1)\), where \(4p = 8\).
   - Solving for \(p\): \(p = \frac{8}{4} = 2\).

3. **Interpret the parabola:**
   - Since the equation involves \((x - 2)^2\), the parabola opens vertically (either up or down).
   - Because \(p = 2 > 0\), the parabola opens **upward**.

### Summary:
- The vertex of the parabola is at \((2, 1)\).
- The parabola opens upward.
- The distance from the vertex to the focus is 2 units.

Would you like more details or have any questions about this?

Here are 5 related questions:
1. What is the focus of the parabola \((x - 2)^2 = 8(y - 1)\)?
2. How do you find the directrix of a parabola?
3. Can you transform the given equation into general form?
4. What is the axis of symmetry for this parabola?
5. How do you find the length of the latus rectum in a parabola?

**Tip:** For parabolas, \(p\) helps determine how "wide" or "narrow" the parabola opens!

Explore how to solve the parabola equation (x-2)^2 = 8(y-1), including identifying the vertex, finding the value of p, and understanding the direction of the parabola. This detailed step-by-step guide is ideal for high school students learning conic sections in Grades 9-12.

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