https://file.aiquickdraw.com/mathbot/file/1727976126819ramnd14d.jpg

The problem is asking for the transformations required to graph \( g(x) = -2(x + 3)^2 + 1 \) using the base function \( f(x) = x^2 \).

Let's break down the transformations:

1. **Shift the graph horizontally to the left** by 3 units: 
   - The term \( (x + 3) \) inside the square indicates a horizontal shift 3 units to the left.

2. **Vertical reflection**:
   - The negative sign in front of the 2, \( -2(x + 3)^2 \), reflects the graph about the x-axis.

3. **Vertical stretch by a factor of 2**:
   - The factor of 2 in front of \( (x + 3)^2 \) stretches the graph vertically by a factor of 2.

4. **Shift the graph vertically upward** by 1 unit:
   - The constant \( +1 \) at the end shifts the graph up by 1 unit.

Given this breakdown, the correct transformations are:
- **A**: Shift the graph 3 units to the left.
- **B**: Shift the graph 1 unit up (this option is correct but labeled incorrectly in the image; "down" should be "up").
- **E**: Reflect the graph about the x-axis.
- **F**: Stretch the graph vertically by a factor of 2.

Let me know if you'd like more details or explanations!

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Here are 5 related questions for deeper understanding:
1. What happens if the graph is shifted 3 units to the right instead of left?
2. How does a horizontal stretch affect the graph of a quadratic function?
3. What is the effect of a vertical compression on the graph of \( f(x) = x^2 \)?
4. How would the graph change if the constant was \( -1 \) instead of \( +1 \)?
5. How does a horizontal reflection differ from a vertical reflection?

**Tip**: Always check the signs inside and outside the parentheses to determine whether a graph shifts left or right, and whether it stretches or compresses.

Use transformations of f(x) = x^2 to graph the function g(x) = -2(x + 3)^2 + 1.

Learn how to apply function transformations to graph g(x) = -2(x + 3)^2 + 1, using the base function f(x) = x^2. This problem covers horizontal and vertical shifts, reflections, and vertical stretches. Suitable for high school students.

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