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The image contains a problem related to designing an obstacle in the form of a parabolic trap hole. The equation provided for the depth of the hole is:

\[
y = \frac{1}{2}(x - 2)(x - 6)
\]

This equation models the depth of the hole as a function of the horizontal distance \(x\).

### Step 1: Expand the Equation
First, let's expand the quadratic equation to express it in standard form:

\[
y = \frac{1}{2}(x - 2)(x - 6)
\]

Expanding the equation:

\[
y = \frac{1}{2}[(x \cdot x) - (6x) - (2x) + 12]
\]

\[
y = \frac{1}{2}(x^2 - 8x + 12)
\]

\[
y = \frac{1}{2}x^2 - 4x + 6
\]

So, the equation of the parabola is:

\[
y = \frac{1}{2}x^2 - 4x + 6
\]

### Step 2: Analyzing the Parabola
- **Vertex**: The vertex of a parabola in the form \(y = ax^2 + bx + c\) can be found using the formula \(x = -\frac{b}{2a}\).

Here \(a = \frac{1}{2}\), \(b = -4\):

\[
x = -\frac{-4}{2 \times \frac{1}{2}} = \frac{4}{1} = 4
\]

To find the depth \(y\) at the vertex \(x = 4\):

\[
y = \frac{1}{2}(4^2) - 4(4) + 6
\]
\[
y = \frac{1}{2}(16) - 16 + 6 = 8 - 16 + 6 = -2
\]

So, the vertex of the parabola is at \((4, -2)\), indicating that the maximum depth of the hole is 2 meters (since the problem specifies depth).

### Step 3: Confirming the Depth and Width
- **Depth of the hole**: The depth is 2 meters, which aligns with the problem constraints.
- **Width of the hole**: The roots of the equation can be found by setting \(y = 0\):

\[
\frac{1}{2}(x - 2)(x - 6) = 0
\]

So, the roots are \(x = 2\) and \(x = 6\), meaning the hole spans from \(x = 2\) to \(x = 6\), a total width of \(6 - 2 = 4\) meters.

### Summary
- The hole is 4 meters wide.
- The maximum depth is 2 meters, occurring at \(x = 4\).

Would you like more details on any step? Here are five related questions:

1. How do you find the axis of symmetry for a parabola?
2. What is the significance of the vertex in a parabolic curve?
3. How do you determine the roots of a quadratic equation?
4. How does the coefficient \(a\) in a quadratic equation affect the shape of the parabola?
5. What are other real-world applications of parabolic equations?

**Tip**: When analyzing quadratic functions, the vertex often provides critical information about maximum or minimum values in the context of real-world problems.

Explore how to analyze a parabolic trap hole using the quadratic equation. Learn about finding the vertex, determining the roots, and calculating depth and width. Suitable for high school students and enthusiasts of mathematical problem-solving.

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