1. **Problem statement:** Given the function $f(x) = \sqrt{x}$, find the functions for the following transformations: - Reflection across the x-axis - Shift right 3 units - Shift up 1 unit - Shift left 1 unit and up 1 unit 2. **Reflection across the x-axis:** The reflection of a function $f(x)$ across the x-axis is given by $y = -f(x)$. Since $f(x) = \sqrt{x}$, the reflected function is: $$y = -\sqrt{x}$$ 3. **Shift right 3 units:** Shifting a function right by $h$ units means replacing $x$ by $x - h$ in the function. So shifting right 3 units: $$y = \sqrt{x - 3}$$ 4. **Shift up 1 unit:** Shifting a function up by $k$ units means adding $k$ to the function. So shifting up 1 unit: $$y = \sqrt{x} + 1$$ 5. **Shift left 1 unit and up 1 unit:** Shifting left 1 unit means replacing $x$ by $x + 1$, and shifting up 1 unit means adding 1. So the function becomes: $$y = \sqrt{x + 1} + 1$$ **Summary of transformations:** - Reflected across x-axis: $y = -\sqrt{x}$ - Shift right 3 units: $y = \sqrt{x - 3}$ - Shift up 1 unit: $y = \sqrt{x} + 1$ - Shift left 1 unit and up 1 unit: $y = \sqrt{x + 1} + 1$