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The problem is asking which equation represents the parabola shown in the \(xy\)-plane. Here's how we can solve it step by step:

### 1. **Identifying the Vertex of the Parabola:**

The vertex of the parabola is clearly labeled as \((3, -8)\) in the graph. The general form for a parabola is:

\[
y = a(x - h)^2 + k
\]

where \((h, k)\) is the vertex of the parabola. From the graph, we see that the vertex is \((3, -8)\), so we substitute \(h = 3\) and \(k = -8\) into the general form:

\[
y = a(x - 3)^2 - 8
\]

### 2. **Determine the Value of \(a\):**

Next, we need to determine the value of \(a\). Looking at the shape of the parabola, we see that it opens upwards, indicating \(a > 0\). Also, the parabola appears to be relatively narrow, suggesting that \(a\) could be greater than 1. 

Now, let's analyze the answer choices:

- **Choice A:** \(y = (x - 3)^2 - 8\) matches the form of the equation with \(a = 1\).
- **Choice C:** \(y = 2(x - 3)^2 - 8\) matches the form of the equation with \(a = 2\).

Since the parabola looks narrower than a standard parabola, it is likely that \(a = 2\), meaning the correct answer is **Choice C**.

### Answer: 
The equation that represents the parabola is:

\[
\boxed{C: y = 2(x - 3)^2 - 8}
\]

Would you like further clarification or details on any step?

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Here are 5 related questions to expand on the topic:
1. How do you determine whether a parabola opens upwards or downwards?
2. How would you find the equation of a parabola given its focus and directrix?
3. What role does the coefficient \(a\) play in the width of a parabola?
4. How can you derive the equation of a parabola if you are given three points on it?
5. How does the vertex form of a parabola relate to its standard form \(y = ax^2 + bx + c\)?

**Tip:** To identify the direction of a parabola, check the sign of the coefficient \(a\). If \(a > 0\), it opens upwards, and if \(a < 0\), it opens downwards.

Which of the following equations represents the parabola shown in the xy-plane above?

Solve a problem involving identifying the equation of a parabola from a graph, where the vertex is at (3, -8). This problem involves using the vertex form of a quadratic equation and analyzing the effect of the coefficient 'a' on the width of the parabola. Suitable for Grades 9-12.

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