Answer :
To describe how the graph of the parent function [tex]\( y = \sqrt{x} \)[/tex] is transformed when graphing [tex]\( y = -3 \sqrt{x - 6} \)[/tex], let’s break down each component of the transformation:
1. Horizontal Shift:
- The function has been shifted horizontally. The term inside the square root, [tex]\( \sqrt{x - 6} \)[/tex], indicates a horizontal shift.
- Specifically, [tex]\( y = \sqrt{x - 6} \)[/tex] means the graph is translated 6 units to the right.
2. Reflection:
- The negative sign in front of the coefficient, "−3", indicates a reflection.
- This reflection occurs over the x-axis.
3. Vertical Stretch:
- The coefficient of the square root term is -3, which affects the vertical stretch.
- A coefficient of 3 (in absolute value) means the graph is stretched vertically by a factor of 3.
Combining these transformations, the graph of [tex]\( y = -3 \sqrt{x - 6} \)[/tex] is obtained by following these steps:
1. Start with the parent function [tex]\( y = \sqrt{x} \)[/tex].
2. Translate the graph 6 units to the right.
3. Reflect the graph across the x-axis.
4. Stretch the graph vertically by a factor of 3.
Therefore, the question about the translation can be completed by filling in the blank correctly:
[tex]\[ \begin{tabular}{l} left \\ right \(\checkmark\)\\ up \\ down \\ \hline \end{tabular} \][/tex]
1. Horizontal Shift:
- The function has been shifted horizontally. The term inside the square root, [tex]\( \sqrt{x - 6} \)[/tex], indicates a horizontal shift.
- Specifically, [tex]\( y = \sqrt{x - 6} \)[/tex] means the graph is translated 6 units to the right.
2. Reflection:
- The negative sign in front of the coefficient, "−3", indicates a reflection.
- This reflection occurs over the x-axis.
3. Vertical Stretch:
- The coefficient of the square root term is -3, which affects the vertical stretch.
- A coefficient of 3 (in absolute value) means the graph is stretched vertically by a factor of 3.
Combining these transformations, the graph of [tex]\( y = -3 \sqrt{x - 6} \)[/tex] is obtained by following these steps:
1. Start with the parent function [tex]\( y = \sqrt{x} \)[/tex].
2. Translate the graph 6 units to the right.
3. Reflect the graph across the x-axis.
4. Stretch the graph vertically by a factor of 3.
Therefore, the question about the translation can be completed by filling in the blank correctly:
[tex]\[ \begin{tabular}{l} left \\ right \(\checkmark\)\\ up \\ down \\ \hline \end{tabular} \][/tex]
