https://file.aiquickdraw.com/mathbot/file/17291718409806ogbexix.jpg

The image contains a graph of a parabola along with multiple-choice options representing quadratic functions. Let's break down the problem:

### Step-by-Step Analysis:

1. **Vertex of the Parabola:**
   - The vertex is located at \((-2, -2)\), as the lowest point of the parabola is at \((-2, -2)\).

2. **Direction and Shape:**
   - The parabola opens upwards, indicating a positive leading coefficient.
   - The graph is narrower than a standard parabola, which suggests the quadratic function is vertically stretched (meaning the coefficient of \((x + 2)^2\) is greater than 1).

3. **Equation Structure:**
   - Since the vertex is at \((-2, -2)\), the function should follow the form \( y = a(x + 2)^2 - 2 \), where \(a\) is a positive constant.
   - The stretch factor (how "narrow" the parabola is) corresponds to \(a = 3\), based on the graph.

### Matching with the Options:
The correct quadratic function, based on the vertex form, is:
\[
y = 3(x + 2)^2 - 2
\]
This matches the second option.

### Final Answer:
The correct equation is the **second option: \( y = 3(x - 2)^2 - 2 \)**.

Would you like more details or have any further questions?

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Here are 5 related questions to explore further:
1. How can we determine the vertex of a parabola from its equation?
2. What effect does the coefficient of \( (x - h)^2 \) have on the graph of a parabola?
3. How would the graph change if \( a = -3 \) instead of \( a = 3 \)?
4. What is the standard form of a quadratic equation, and how do you convert between standard and vertex form?
5. How can we find the axis of symmetry for a quadratic function from its equation?

**Tip:** In quadratic functions, the vertex form \( y = a(x-h)^2 + k \) makes it easy to identify the vertex at \( (h, k) \) and understand vertical shifts and stretches.

Which of the given quadratic equations represents the parabola in the graph?

In this problem, we determine the quadratic equation that represents a parabola based on its graph. The graph's vertex is identified, and the equation is written in vertex form y = a(x - h)^2 + k, where the values of a, h, and k are derived from the graph. Suitable for students in Grades 9-10.

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