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Let's analyze the function \( f(x) = -3(-x - 2)^2 + 3 \) in terms of translation and scaling.

### Step-by-step breakdown:

1. **Parent Function:**
   The basic form of this function is derived from the standard quadratic function \( y = x^2 \), which is a parabola.

2. **Translation:**
   The function \( f(x) = -3(-x - 2)^2 + 3 \) can be interpreted by rewriting the terms inside the parentheses:
   \[
   f(x) = -3(-(x + 2))^2 + 3 = -3(x + 2)^2 + 3
   \]
   - The term \( (x + 2)^2 \) represents a **horizontal shift** to the left by 2 units (because of the \( x + 2 \)).
   - The \( + 3 \) outside the squared term represents a **vertical shift** upward by 3 units.

3. **Scaling:**
   The coefficient \( -3 \) in front of \( (x + 2)^2 \) does two things:
   - The negative sign indicates a **reflection** across the x-axis (i.e., the parabola opens downward instead of upward).
   - The factor 3 causes a **vertical stretch** by a factor of 3. This means the parabola becomes narrower because it grows faster vertically.

### Summary of transformations:
- **Horizontal shift**: 2 units to the left.
- **Vertical shift**: 3 units upward.
- **Vertical stretch**: By a factor of 3.
- **Reflection**: Across the x-axis (the parabola opens downward).

Would you like further details or have any questions?

### Related questions:
1. How would the graph of \( f(x) = 2(x - 1)^2 - 4 \) be transformed compared to \( y = x^2 \)?
2. What happens if the coefficient of the squared term is less than 1 (e.g., \( f(x) = \frac{1}{2}(x - 3)^2 + 5 \))?
3. How do you determine the vertex of a function in vertex form?
4. What is the effect of changing the sign inside the parentheses (e.g., \( (x - h)^2 \) vs. \( (x + h)^2 \))?
5. What role does the horizontal shift play in the axis of symmetry of a parabola?

### Tip:
For quadratic functions, the vertex form \( f(x) = a(x - h)^2 + k \) is useful to quickly determine the vertex \((h, k)\), as well as the direction of the parabola and any stretching or shrinking.

Determine how the function f(x) = -3(-x - 2)^2 + 3 has been translated and scaled.

Analyze how the quadratic function f(x) = -3(-x - 2)^2 + 3 is translated and scaled. Learn about horizontal and vertical shifts, reflections, and vertical stretching, with a focus on transformation rules for quadratic functions. Suitable for students in Grades 9-11.

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