To analyze the transformation of the function [tex]\( y = \sqrt[3]{27x - 54} + 5 \)[/tex] compared with the parent function [tex]\( y = \sqrt[3]{x} \)[/tex], we break it down step by step:
1. Horizontal Translation:
- The expression inside the cube root is [tex]\( 27x - 54 \)[/tex].
- Factoring this term, we get [tex]\( 27(x - 2) \)[/tex].
- The term [tex]\( (x - 2) \)[/tex] indicates a horizontal shift to the right by 2 units.
So, the horizontal translation is 2 units to the right.
2. Vertical Translation:
- The function [tex]\( y = \sqrt[3]{27x - 54} \)[/tex] becomes [tex]\( y = \sqrt[3]{27x - 54} + 5 \)[/tex].
- The "+5" outside the cube root indicates a vertical shift upwards by 5 units.
So, the vertical translation is 5 units up.
3. Stretch/Compression:
- The factor inside the cube root is 27.
- The cube root function [tex]\( y = \sqrt[3]{ax} \)[/tex] is horizontally compressed by a factor of [tex]\( \frac{1}{a} \)[/tex] when [tex]\( a > 0 \)[/tex].
- Here, [tex]\( a = 27 \)[/tex], so the function is horizontally compressed by a factor of [tex]\( \frac{1}{27} \)[/tex].
So, the stretch/compression is a horizontal compression by a factor of [tex]\( \frac{1}{27} \)[/tex]. Numerically, this factor is approximately [tex]\( 0.037037 \)[/tex].
4. Reflection:
- There is no negative sign either inside or outside the cube root.
- Therefore, no reflection occurs.
So, the reflection is absent.
Summarizing these transformations:
- Horizontal translation: 2 units to the right.
- Vertical translation: 5 units up.
- Stretch/compression: Horizontal compression by a factor of [tex]\( 0.037037 \)[/tex].
- Reflection: None.
