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Graph the following function:

[tex]\[ y = \frac{-3}{2} + \frac{1}{2} \cot (\pi x - 4i\pi) \][/tex]

Determine how the general shape of the graph would be shifted, stretched, and reflected for the given function. Graph the results on the provided axes.

1. Reflection across the x-axis?
- Yes / No

2. Shift vertically?
- Up / Down / None

3. Shift horizontally (phase shift)?
- Left / Right / None

4. Stretch/compress vertically?
- Yes / No

5. Stretch/compress horizontally (period)?
- Yes / No

Sagot :

GinnyAnswer
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.
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