Sure, let's analyze and describe the transformations in detail.
The parent function given is [tex]\( y = \sqrt[3]{x} \)[/tex].
The transformed function given is [tex]\( y = \sqrt[3]{8x} - 3 \)[/tex].
We need to determine the transformations applied to the parent function to obtain the transformed function.
1. Horizontal Transformation:
The term inside the cube root function, [tex]\( 8x \)[/tex], indicates a horizontal transformation. When a function is of the form [tex]\( \sqrt[3]{kx} \)[/tex], it represents a horizontal compression or stretch depending on the value of [tex]\( k \)[/tex].
- For [tex]\( y = \sqrt[3]{8x} \)[/tex]:
The factor [tex]\( 8 \)[/tex] suggests a horizontal compression by a factor of [tex]\( \frac{1}{8} \)[/tex].
2. Vertical Transformation:
The term outside of the cube root function, [tex]\( - 3 \)[/tex], indicates a vertical transformation. Specifically, it indicates a vertical translation.
- For [tex]\( y = \sqrt[3]{8x} - 3 \)[/tex]:
This implies a vertical translation downward by 3 units.
Putting it all together:
- The graph of [tex]\( y = \sqrt[3]{x} \)[/tex] is horizontally compressed by a factor of [tex]\( \frac{1}{8} \)[/tex].
- Then, the graph is translated 3 units downward.
Therefore, the answer is:
The graph is translated 3 units [tex]\(\textbf{downward}\)[/tex].
