To understand how the graph of the function [tex]\( y = \sqrt[3]{8x - 64} - 5 \)[/tex] compares to the parent cube root function [tex]\( y = \sqrt[3]{x} \)[/tex], we will break down the transformations step-by-step.
### Step-by-Step Solution
1. Identify the parent function:
The parent function here is [tex]\( y = \sqrt[3]{x} \)[/tex].
2. Analyze the transformation inside the cube root:
The given function has [tex]\( 8x - 64 \)[/tex] under the cube root. We can factor 8 out:
[tex]\[
y = \sqrt[3]{8(x - 8)} - 5
\][/tex]
3. Horizontal translation:
The term inside the cube root, [tex]\( (x - 8) \)[/tex], indicates a horizontal shift. Specifically, the function is translated 8 units to the right.
4. Vertical stretch:
The 8 inside the cube root affects the vertical stretching. Because the cube root of 8 is 2, the function undergoes a vertical stretch by a factor of 2 compared to the parent function [tex]\( y = \sqrt[3]{x} \)[/tex].
5. Vertical translation:
The term -5 outside the cube root indicates a vertical shift. Specifically, the function is translated 5 units down.
### Conclusion
Based on the transformations we identified:
- The graph is stretched by a factor of 2,
- Translated 8 units to the right,
- And translated 5 units down.
Among the given choices, the correct description is:
"stretched by a factor of 2 and translated 8 units right and 5 units down"
Hence, the correct choice is:
stretched by a factor of 2 and translated 8 units right and 5 units down
