Sure! Let's break down the transformations applied to the parent function [tex]\( y = \sqrt{x} \)[/tex] to obtain the transformed function [tex]\( y = -3 \sqrt{x-6} \)[/tex].
1. Translation:
- The term inside the square root, [tex]\( x - 6 \)[/tex], indicates a horizontal shift. Specifically, [tex]\( x \)[/tex] is replaced by [tex]\( x - 6 \)[/tex], which translates the graph 6 units to the right.
- Hence, the graph of [tex]\( y = \sqrt{x} \)[/tex] is translated 6 units to the right.
2. Reflection:
- The negative sign in front of the expression [tex]\(-3 \sqrt{x-6}\)[/tex] indicates a reflection over the x-axis.
- Therefore, the graph of [tex]\( y = \sqrt{x} \)[/tex] is reflected over the x-axis, changing all positive y-values to their negative counterparts.
3. Vertical Stretch/Compression:
- The coefficient 3 in front of the square root represents a vertical stretch or compression factor.
- Since the coefficient is 3, which is greater than 1, it indicates a vertical stretch by a factor of 3.
Putting it all together:
- The graph is translated 6 units to the right.
- The graph is reflected over the x-axis.
- The graph has a vertical stretch by a factor of 3.
So the complete explanation of the transformations is as follows:
1. The graph is translated 6 units to the right.
2. The graph is reflected over the x-axis.
3. The graph has a vertical stretch by a factor of 3.
Therefore, the correct results are:
- Translation: The graph is translated 6 units right.
- Reflection: The graph is reflected over the x-axis.
- Vertical Stretch: The graph has a vertical stretch by a factor of 3.
