−√(−𝑥)−2, use transformation to graph the function

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.

The query submitted by the user: −√(−𝑥)−2, use transformation to graph the function

Learn how to graph the transformed function f(x) = -√(-x) - 2. This problem covers key concepts such as reflections, square root functions, and vertical shifts. Suitable for students in Grades 9-12, with step-by-step instructions and transformations explained.

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