\( \leftarrow \) Find the function that is finally graphed after the following transformations are applied to the graph of \( y=\sqrt{x} \) in the order listed. (1) Shift down 8 units (2) Rellect about the \( x \)-axis (3) Reflect about the \( y \)-axis
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To start, we take the basic function \( y = \sqrt{x} \). 1. **Shift down 8 units:** The function after this transformation becomes \( y = \sqrt{x} - 8 \). 2. **Reflect about the \( x \)-axis:** This flips the function over the \( x \)-axis, resulting in \( y = -(\sqrt{x} - 8) \) or \( y = -\sqrt{x} + 8 \). 3. **Reflect about the \( y \)-axis:** This reflection transforms the function to \( y = -\sqrt{-x} + 8 \). In conclusion, after applying all transformations, the final function is: \[ y = -\sqrt{-x} + 8 \] So, the function graphed has been dramatically transformed into a reflection and shift resultant from your given steps!
