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.