Let's analyze the transformations applied to the parent function \( y = \sqrt{\sqrt{x}} \) to obtain the transformed function \( y = -0.4\sqrt{x} - 2 \):
1. **Reflected over the x-axis**: The negative sign in front of \( 0.4\sqrt{x} \) reflects the function over the x-axis. Thus, the graph is reflected over the x-axis.
2. **Translated 2 units left**: The "+2" inside the function shifts the graph 2 units to the right, not left. So, this statement is incorrect.
3. **Translated 2 units right**: The "-2" outside the function shifts the graph 2 units down, not right. So, this statement is incorrect.
4. **Compressed by a factor of 0.4**: The coefficient "0.4" in front of \( \sqrt{x} \) compresses the graph horizontally by a factor of 0.4. So, this statement is correct.
5. **Stretched by a factor of 0.4**: The coefficient "0.4" actually compresses, not stretches, the graph horizontally. So, this statement is incorrect.
6. **Translated 2 units up**: The "-2" outside the function shifts the graph 2 units down, not up. So, this statement is incorrect.
7. **Translated 2 units down**: The "-2" outside the function shifts the graph 2 units down. So, this statement is correct.
Therefore, the correct descriptions of the transformed function compared with the parent function are:
- Reflected over the x-axis
- Compressed by a factor of 0.4
- Translated 2 units down
The options "Reflected over the x-axis," "Compressed by a factor of 0.4," and "Translated 2 units down" are the correct ones.
#studyharder
