To compare the graph of [tex]\( y = \sqrt[3]{27x - 54} + 5 \)[/tex] with the parent cube root function [tex]\( y = \sqrt[3]{x} \)[/tex], we need to identify the transformations involved:
### Horizontal Translation:
The argument inside the cube root function, [tex]\( 27x - 54 \)[/tex], can be written as [tex]\( 27(x - 2) \)[/tex]. This indicates a horizontal translation of the function. Specifically, the graph is translated 2 units to the right.
### Vertical Translation:
The "+5" outside the cube root function indicates a vertical translation. The graph is translated 5 units upward.
### Stretch/Compression:
The coefficient 27 inside the cube root function affects the horizontal stretch/compression. It indicates a horizontal compression by a factor of [tex]\(\frac{1}{27} \)[/tex]. This means the graph is compressed horizontally by this factor.
### Reflection:
There is no negative sign in front of the cube root function, nor inside it, so there is no reflection.
In summary:
- Horizontal translation: 2 units to the right
- Vertical translation: 5 units upward
- Stretch/compression: Compressed horizontally by a factor of 1/27
- Reflection: None
Therefore, the solution is:
Horizontal translation:
[tex]\[ 2 \][/tex]
Vertical translation:
[tex]\[ 5 \][/tex]
Stretch/compression:
[tex]\[ \frac{1}{27} \][/tex]
Reflection:
[tex]\[ 0 \][/tex]
