To graph the function $y = -\sqrt{-x} - 2$ using transformations, we break it down step by step, starting from the basic square root function $y = \sqrt{x}$, and apply transformations accordingly:

1. Start with the basic function:

• The parent function is $y = \sqrt{x}$, which is the graph of a square root function. It starts at the origin (0,0) and increases slowly as $x$ increases.

2. Reflect across the y-axis ($y = \sqrt{-x}$):

• The term $-x$ inside the square root reflects the graph across the y-axis. This means the graph will now be on the left side of the y-axis instead of the right. It is only defined for $x \leq 0$, because the square root of a negative number inside the function is not defined in the real number system.

3. Reflect across the x-axis ($y = -\sqrt{-x}$):

• The negative sign outside the square root reflects the graph across the x-axis, flipping it downwards. So, the part that was above the x-axis is now below it.

4. Vertical shift down by 2 units ($y = -\sqrt{-x} - 2$):

• The $-2$ subtracts 2 from every y-value, shifting the entire graph down by 2 units.

Final description of the graph:

• The final graph starts at the point (0, -2) and extends leftward, going downwards. The function is only defined for $x \leq 0$. The overall shape is an upside-down, horizontally reflected square root function shifted downward by 2 units.

Summary of transformations:

1. Start with $y = \sqrt{x}$.
2. Reflect across the y-axis: $y = \sqrt{-x}$.
3. Reflect across the x-axis: $y = -\sqrt{-x}$.
4. Shift the graph down by 2: $y = -\sqrt{-x} - 2$.

Would you like a more detailed explanation of any part of the transformation, or have any questions?

Here are 5 questions related to this explanation:

1. How does the domain of the function change after reflecting across the y-axis?
2. What happens to the range after reflecting across the x-axis?
3. Why is the function only defined for $x \leq 0$?
4. How do vertical and horizontal shifts affect the graph of a function?
5. What real-world scenarios could this transformed function model?

Tip: Reflecting across an axis flips the graph over that axis, but it does not affect the shape of the graph—only its orientation changes.