To analyze the graph of [tex]\( y = \sqrt[3]{27x - 54} + 5 \)[/tex] compared with the parent cube root function [tex]\( y = \sqrt[3]{x} \)[/tex], we must identify the transformations step by step.
1. Horizontal Translation:
The expression inside the cube root, [tex]\( 27x - 54 \)[/tex], can be factored to [tex]\( 27(x - 2) \)[/tex]. This indicates a horizontal shift.
- The value that shifts the graph is [tex]\( x - 2 \)[/tex], which means the graph is translated 2 units to the right.
2. Vertical Translation:
The +5 outside the cube root function [tex]\( \sqrt[3]{27x - 54} + 5 \)[/tex] indicates a vertical shift.
- This means the graph is translated 5 units upward.
3. Stretch/Compression:
The coefficient 27 inside the cube root, [tex]\( \sqrt[3]{27(x - 2)} \)[/tex], affects the horizontal stretch or compression.
- Since 27 is equivalent to [tex]\( 3^3 \)[/tex], this implies a horizontal compression by a factor of 3.
4. Reflection:
To determine if there is a reflection, we look at signs in front of the function and inside the cubic root.
- There are no negative signs in front of the cube root or the entire function, indicating there is no reflection.
Summarizing all the transformations:
- Horizontal translation: 2 units to the right
- Vertical translation: 5 units upward
- Stretch/compression: Horizontal compression by a factor of 3
- Reflection: None
Thus, compared with the parent cube root function [tex]\( y = \sqrt[3]{x} \)[/tex], the graph of [tex]\( y = \sqrt[3]{27x - 54} + 5 \)[/tex] is shifted 2 units to the right, 5 units upward, horizontally compressed by a factor of 3, and not reflected.
