Answer :
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.
