To determine how the transformed function [tex]\( y = -(0.4) \sqrt[3]{x-2} \)[/tex] compares with the parent function [tex]\( y = \sqrt[3]{x} \)[/tex], we need to analyze each modification applied to the parent function:
1. Reflected over the x-axis:
The negative sign in front of the [tex]\( 0.4 \)[/tex] indicates that the function is reflected over the x-axis. This means that the graph will be inverted vertically.
2. Translated 2 units right:
The [tex]\( (x-2) \)[/tex] inside the cube root suggests that the graph is translated 2 units to the right. This horizontal shift moves every point on the original graph 2 units to the right.
3. Compressed by a factor of 0.4:
The coefficient [tex]\( 0.4 \)[/tex] in front of the cube root indicates a vertical compression. Since the coefficient is less than 1, the graph is scaled down vertically by a factor of 0.4.
Given these observations, the correct transformations applied to the parent function [tex]\( y = \sqrt[3]{x} \)[/tex] to obtain [tex]\( y = -(0.4) \sqrt[3]{x-2} \)[/tex] are:
- The function is reflected over the x-axis.
- The function is translated 2 units to the right.
- The function is compressed by a factor of 0.4.
Therefore, the correct responses are:
- Reflected over the x-axis
- Translated 2 units right
- Compressed by a factor of 0.4
