To determine the transformations applied to the parent function [tex]\( y = \sqrt{x} \)[/tex] to obtain [tex]\( g(x) = -2 \sqrt{x-6} - 3 \)[/tex], we need to analyze each component of the function.
Let's break it down step-by-step:
1. Horizontal Shift:
- The function inside the square root is [tex]\( x - 6 \)[/tex].
- This indicates a horizontal shift to the right by 6 units.
- Result: Horizontal shift of 6 right.
2. Reflection Across the x-axis:
- The negative sign in front of the square root, [tex]\( -2 \sqrt{x - 6} \)[/tex], indicates a reflection across the x-axis.
- Result: Reflect across the x-axis.
3. Vertical Stretch/Compression:
- The coefficient in front of the square root, which is -2, indicates a vertical transformation.
- Since the absolute value of the coefficient (| -2 | = 2) is greater than 1, this results in a vertical stretch.
- Result: Vertical stretch.
4. Vertical Shift:
- The -3 at the end of the function, [tex]\( -2 \sqrt{x - 6} - 3 \)[/tex], indicates a vertical shift.
- Since it is negative, it means a vertical shift down by 3 units.
- Result: Vertical shift of 3 down.
Combining all these observations, the transformations applied to the parent function [tex]\( y = \sqrt{x} \)[/tex] to obtain [tex]\( g(x) = -2 \sqrt{x-6} - 3 \)[/tex] are:
- Horizontal shift of 6 right
- Reflect across the x-axis
- Vertical stretch
- Vertical shift of 3 down
Here is the complete list of identified transformations:
- Reflect across the x-axis: True
- Vertical stretch: True
- Vertical compression: False
- Horizontal shift of 6 right: True
- Horizontal shift of 6 left: False
- Vertical shift of 3 up: False
- Vertical shift of 3 down: True.
