To graph the function $f(x) = -\sqrt{-x} - 2$ using transformations, we'll break down the function step by step and apply transformations to the basic square root function $\sqrt{x}$. Here's how to approach it:

1. Start with the base function: $\sqrt{x}$

The base function is $y = \sqrt{x}$, which is the standard square root graph. It starts at the point $(0, 0)$, moves upwards, and only exists for $x \geq 0$.

2. Apply a horizontal reflection (over the y-axis): $\sqrt{-x}$

Next, replace $x$ with $-x$, which reflects the graph across the y-axis. The graph now exists only for $x \leq 0$, and the square root function opens to the left.

3. Apply a vertical reflection (over the x-axis): $-\sqrt{-x}$

By multiplying by -1, the graph flips vertically, reflecting it across the x-axis. Now, the graph will point downward.

4. Apply a vertical shift: $-\sqrt{-x} - 2$

Finally, subtract 2 from the function to shift the graph downward by 2 units. This lowers the entire graph by 2 units along the y-axis.

Final Function: $f(x) = -\sqrt{-x} - 2$

• The graph is defined only for $x \leq 0$.
• It starts at the point $(0, -2)$ and moves downward to the left.

Graph Transformations Summary:

1. Start with the square root function $y = \sqrt{x}$.
2. Reflect across the y-axis to get $y = \sqrt{-x}$.
3. Reflect across the x-axis to get $y = -\sqrt{-x}$.
4. Shift the graph down 2 units to get $y = -\sqrt{-x} - 2$.

Key Points:

• The domain is $(-\infty, 0]$ because the function involves $\sqrt{-x}$.
• The range is $(-\infty, -2]$ due to the downward shift.

Let me know if you need the graph itself, and feel free to ask for further clarifications!

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Would you like a detailed step-by-step illustration of the graph? Here are some related questions for further exploration:

1. How do you find the domain of more complex square root functions?
2. What happens when you reflect a function across both axes?
3. How do vertical and horizontal shifts affect the graph of a function?
4. Can you describe transformations involving cube root functions?
5. How do transformations change the intercepts of a function?

Tip: When applying transformations, always follow the order: horizontal shifts, stretches, reflections, and finally vertical shifts for accuracy.