To graph \( g(x) = -3(x + 3)^2 + 1 \), we start with the basic graph of \( f(x) = x^2 \) and apply the transformations step by step.

### Steps to Graph:
1. **Start with \( f(x) = x^2 \)**: This is a basic parabola that opens upwards with the vertex at the origin (0, 0).

2. **Horizontal Shift**: Shift the graph 3 units to the left. The vertex, which was originally at (0, 0), will now be at (-3, 0).

3. **Reflection across the \( x \)-axis**: Reflect the graph across the \( x \)-axis. The parabola that originally opened upwards will now open downwards, so the vertex at (-3, 0) will now be the highest point.

4. **Vertical Stretch by a Factor of 3**: The parabola will become narrower due to the vertical stretch by a factor of 3. The arms of the parabola will be steeper compared to the original \( f(x) = x^2 \).

5. **Vertical Shift Up by 1 Unit**: Finally, shift the graph up by 1 unit. The vertex of the parabola will now be at (-3, 1).

### Final Graph:
- **Vertex**: (-3, 1)
- **Direction**: The parabola opens downwards.
- **Narrowness**: The parabola is steeper than \( f(x) = x^2 \) due to the vertical stretch.

I can generate a visual representation of this graph to show exactly how it looks. Would you like to see that?

Learn how to graph the quadratic function g(x) = -3(x + 3)^2 + 1 step-by-step. Understand the transformations applied to the basic parabola, including horizontal shifts, reflections, vertical stretches, and shifts. Suitable for students in Grades 9-12.

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