We start with the parent function:
[tex]y=\sqrt[]{x}[/tex]
The transformations are:
• Shift up 6 units
,
• Reflect about the x-axis
,
• Reflect about the y-axis
The first transformation is equal to add 6 units to y:
[tex]\begin{gathered} y_2=y+6 \\ y_2=\sqrt[]{x}+6 \end{gathered}[/tex]
Then, a reflection across the x-axis makes y change sign, so we will have:
[tex]\begin{gathered} y_3=-y_2 \\ y_3=-\sqrt[]{x}-6 \end{gathered}[/tex]
Finally, a reflection across the y-axis makes the argument x change sign, so we will have:
[tex]\begin{gathered} y_4=y_3(-x) \\ y_4=-\sqrt[]{-x}-6 \end{gathered}[/tex]
This is the final function.
We can graph this function, and the intermediate steps, as:
Answer: y = -√-x - 6
