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The question asks to describe how the graph of the function \( f(x) = x^2 \) can be transformed into the graph of the function \( g(x) = 2(x - 6)^2 - 9 \).

### Step-by-step breakdown of the transformation:

1. **Horizontal shift**: The term \( (x - 6) \) indicates a shift to the **right by 6 units**. This happens because subtracting 6 from \( x \) shifts the graph right.

2. **Vertical shift**: The term \( -9 \) indicates a shift **down by 9 units**. This is because subtracting 9 from the function translates it downward.

3. **Vertical stretch**: The coefficient 2 outside of the squared term indicates a **vertical stretch** by a factor of 2. This makes the graph narrower as the y-values are scaled up by 2.

### Now, looking at the choices:

- **Option 1**: Shift the graph left 6 units, down 9 units, and stretch vertically. (Incorrect: It should be a right shift, not left).
  
- **Option 2**: Shift the graph up 6 units, right 9 units, and compress horizontally. (Incorrect: It should be down 9 units, right 6 units, and a vertical stretch, not a horizontal compression).
  
- **Option 3**: Shift the graph down 6 units, left 9 units, and stretch horizontally. (Incorrect: The shifts and stretch direction are incorrect).
  
- **Option 4**: Shift the graph right 6 units, down 9 units, and stretch vertically. (Correct: This accurately describes the transformations).

### Conclusion:
The correct answer is **Option 4**.

Would you like more details on this or have any further questions?

### Related questions:
1. What happens to the graph when the function is multiplied by a fraction?
2. How does changing the sign of the \( x \)-term inside a square affect the graph?
3. How can we describe a reflection over the x-axis or y-axis using transformations?
4. What is the impact of adding a constant inside and outside the square term in a quadratic function?
5. How would the transformation change if the coefficient outside the squared term were negative?

### Tip:
When analyzing transformations, pay close attention to signs and coefficients—they determine the direction and type of shifts or stretches.

Describe how the graph of function f(x) = x^2 can be transformed to the graph of the function g(x) = 2(x - 6)^2 - 9.

Learn how to transform the graph of f(x) = x^2 to g(x) = 2(x - 6)^2 - 9, involving horizontal shifts, vertical shifts, and vertical stretching. Understand the core concepts of graph transformations in quadratic functions. Suitable for high school math students (Grades 9-12).

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