https://file.aiquickdraw.com/mathbot/file/17269334890258ff8o794.jpg

The graph in the image appears to be a parabola that opens downwards and is shifted horizontally and vertically. Based on visual analysis:

- The vertex of the parabola seems to be at \( (-6, 2) \), indicating that the horizontal shift is to \( x = -6 \), and the vertical shift is to \( y = 2 \).
- Since the parabola opens downward, the leading coefficient is negative.

Let's analyze the given answer choices:
- The equation of a parabola in vertex form is \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex.
- The correct vertex is \( (-6, 2) \), so the equation should involve \( (x + 6) \) and have a positive 2 as the vertical shift.

Only **Option D** fits this:  
\[ y = -3(x + 6)^2 + 2 \]

This is the correct equation. 

Would you like further clarification, or to explore more details?

Here are 5 follow-up questions for deeper understanding:
1. Why does the negative sign in the equation indicate that the parabola opens downwards?
2. How does changing the coefficient affect the width of the parabola?
3. What is the general form of a parabola's equation, and how does it relate to the vertex form?
4. How would the graph change if the coefficient in front of the squared term was positive?
5. Can you derive the vertex form of a parabola from its standard form?

**Tip:** The coefficient \( a \) in the equation \( y = a(x - h)^2 + k \) controls the direction and width of the parabola. A larger \( |a| \) value makes the parabola narrower, while a smaller \( |a| \) makes it wider.

Learn how to find the equation of a parabola from its graph by identifying key elements such as the vertex and the direction of opening. This problem explores the vertex form of a quadratic function and applies transformations for Grades 8-10.

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