$$ \leftarrow $$ 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 find the function that results from the given transformations applied to the graph of $$ y = \sqrt{x} $$, we will go through each transformation step by step.

1. **Shift down 8 units**: 
   - Shifting a function down by $$ k $$ units involves subtracting $$ k $$ from the function. 
   - For $$ y = \sqrt{x} $$, shifting down 8 units gives us:
   $$
   y = \sqrt{x} - 8
   $$

2. **Reflect about the $$ x $$-axis**: 
   - Reflecting a function about the $$ x $$-axis involves multiplying the entire function by -1.
   - Applying this to $$ y = \sqrt{x} - 8 $$ results in:
   $$
   y = -(\sqrt{x} - 8) = -\sqrt{x} + 8
   $$

3. **Reflect about the $$ y $$-axis**: 
   - Reflecting a function about the $$ y $$-axis involves replacing $$ x $$ with $$ -x $$ in the function.
   - Applying this to $$ y = -\sqrt{x} + 8 $$ gives us:
   $$
   y = -\sqrt{-x} + 8
   $$

Thus, the final function after applying all the transformations is:
$$
y = -\sqrt{-x} + 8
$$
The final function after the transformations is $$ y = -\sqrt{-x} + 8 $$.

\( \leftarrow \) 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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