To understand how the transformed function [tex]\( y = -0.4 \sqrt{x-2} \)[/tex] relates to the parent function [tex]\( y = \sqrt{x} \)[/tex], we need to analyze the transformations applied to the parent function step-by-step.
1. Reflected Over the x-axis:
- The negative sign in front of the coefficient [tex]\(-0.4\)[/tex] indicates that the function is reflected over the x-axis. This turns all the y-values of the function into their opposites, flipping the graph upside down.
2. Translated 2 Units Right:
- The expression [tex]\((x-2)\)[/tex] under the square root indicates a horizontal translation. Specifically, [tex]\((x - 2)\)[/tex] means the graph is shifted 2 units to the right. A horizontal translation is always opposite in direction to the sign inside the parentheses.
3. Compressed by a Factor of 0.4:
- The coefficient [tex]\(0.4\)[/tex] (which is less than 1 but greater than 0) in front of the square root leads to a vertical compression of the graph. This means that all the y-values are multiplied by 0.4, making the graph of the function closer to the x-axis.
So, to summarize the transformations:
- The graph is reflected over the x-axis.
- The graph is translated 2 units right.
- The graph is compressed by a factor of 0.4.
Therefore, the correct descriptions for the transformed function [tex]\( y = -0.4 \sqrt{x-2} \)[/tex] compared to the parent function [tex]\( y = \sqrt{x} \)[/tex] are:
- Reflected over the x-axis.
- Translated 2 units right.
- Compressed by a factor of 0.4.
