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

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.

Explore how to graph the function y = -√(-x) - 2 by using reflections and translations. Learn the key transformations applied to the square root function, including reflections across the x-axis and y-axis, and vertical shifts. Suitable for students in Grades 9-12.

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