Westonci.ca makes finding answers easy, with a community of experts ready to provide you with the information you seek. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
To graph the function [tex]\(y = \frac{-3}{2} + \frac{1}{2} \cot(\pi x - 4i\pi)\)[/tex], we need to understand how the graph of the basic cotangent function, [tex]\(\cot(\pi x)\)[/tex], will be transformed. Here's the step-by-step analysis of these transformations:
### Step-by-Step Breakdown of the Transformations:
1. Base Function:
- Start with the basic cotangent function, [tex]\(y = \cot(\pi x)\)[/tex].
2. Horizontal Shift (Phase Shift):
- We have [tex]\(\cot(\pi x - 4i\pi)\)[/tex].
- This can be rewritten as [tex]\(\cot(\pi(x - 4i))\)[/tex], indicating a phase shift to the right by [tex]\(4i\)[/tex] units (though in standard real analysis we might consider it differently, we'll follow the given function).
3. Vertical Scaling (Stretch/Compress):
- The function [tex]\(\frac{1}{2} \cot(\pi x - 4i\pi)\)[/tex] represents a vertical compression by a factor of [tex]\(\frac{1}{2}\)[/tex].
4. Vertical Shift:
- Adding [tex]\(-\frac{3}{2}\)[/tex] results in a vertical shift downwards by [tex]\(\frac{3}{2}\)[/tex] units.
5. Reflection:
- There is no explicit flip across the x-axis in this transformation.
### Summary of Transformations:
- Reflection Across the x-axis: No
- Vertical Shift: Down
- Horizontal Shift: Right
- Vertical Compression: Yes
- Horizontal Compression: No
### Applying These Transformations to the Graph:
1. Start with the graph of [tex]\(y = \cot(\pi x)\)[/tex].
2. Shift the graph horizontally to the right by [tex]\(4i\)[/tex] units.
3. Compress the graph vertically by a factor of [tex]\(\frac{1}{2}\)[/tex].
4. Shift the graph vertically downwards by [tex]\(\frac{3}{2}\)[/tex] units.
The final graph will show the cotangent function that has been horizontally shifted, vertically compressed, and shifted downward.
### Summary:
The transformations on the function [tex]\(y = \frac{-3}{2} + \frac{1}{2} \cot(\pi x - 4i\pi)\)[/tex] result in:
- No reflection across the x-axis.
- A vertical shift down by [tex]\( \frac{3}{2} \)[/tex] units.
- A horizontal shift to the right by [tex]\( 4i \)[/tex] units.
- A vertical compression by a factor of [tex]\( \frac{1}{2} \)[/tex].
- No horizontal compression.
This step-by-step approach results in transforming and graphing the given function appropriately.
### Step-by-Step Breakdown of the Transformations:
1. Base Function:
- Start with the basic cotangent function, [tex]\(y = \cot(\pi x)\)[/tex].
2. Horizontal Shift (Phase Shift):
- We have [tex]\(\cot(\pi x - 4i\pi)\)[/tex].
- This can be rewritten as [tex]\(\cot(\pi(x - 4i))\)[/tex], indicating a phase shift to the right by [tex]\(4i\)[/tex] units (though in standard real analysis we might consider it differently, we'll follow the given function).
3. Vertical Scaling (Stretch/Compress):
- The function [tex]\(\frac{1}{2} \cot(\pi x - 4i\pi)\)[/tex] represents a vertical compression by a factor of [tex]\(\frac{1}{2}\)[/tex].
4. Vertical Shift:
- Adding [tex]\(-\frac{3}{2}\)[/tex] results in a vertical shift downwards by [tex]\(\frac{3}{2}\)[/tex] units.
5. Reflection:
- There is no explicit flip across the x-axis in this transformation.
### Summary of Transformations:
- Reflection Across the x-axis: No
- Vertical Shift: Down
- Horizontal Shift: Right
- Vertical Compression: Yes
- Horizontal Compression: No
### Applying These Transformations to the Graph:
1. Start with the graph of [tex]\(y = \cot(\pi x)\)[/tex].
2. Shift the graph horizontally to the right by [tex]\(4i\)[/tex] units.
3. Compress the graph vertically by a factor of [tex]\(\frac{1}{2}\)[/tex].
4. Shift the graph vertically downwards by [tex]\(\frac{3}{2}\)[/tex] units.
The final graph will show the cotangent function that has been horizontally shifted, vertically compressed, and shifted downward.
### Summary:
The transformations on the function [tex]\(y = \frac{-3}{2} + \frac{1}{2} \cot(\pi x - 4i\pi)\)[/tex] result in:
- No reflection across the x-axis.
- A vertical shift down by [tex]\( \frac{3}{2} \)[/tex] units.
- A horizontal shift to the right by [tex]\( 4i \)[/tex] units.
- A vertical compression by a factor of [tex]\( \frac{1}{2} \)[/tex].
- No horizontal compression.
This step-by-step approach results in transforming and graphing the given function appropriately.
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.
