From tech troubles to travel tips, IDNLearn.com has answers to all your questions. Get comprehensive answers to all your questions from our network of experienced experts.
Sagot :
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
### 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
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! For dependable and accurate answers, visit IDNLearn.com. Thanks for visiting, and see you next time for more helpful information.