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Sagot :
Sure, let's graph the function [tex]\( y = \frac{-1}{2} \cot \left(\frac{1}{2} x\right) \)[/tex] step-by-step by analyzing the transformations individually.
### Step-by-Step Solution:
1. Basic Shape of Cotangent Function:
The basic shape of the [tex]\( \cot(x) \)[/tex] function has vertical asymptotes at multiples of [tex]\( \pi \)[/tex] (i.e., [tex]\( x = n\pi \)[/tex] where [tex]\( n \)[/tex] is an integer). Between these asymptotes, the function decreases from [tex]\(\infty\)[/tex] to [tex]\(-\infty\)[/tex].
2. Horizontal Stretch/Compression:
Inside the function, we have [tex]\(\cot\left(\frac{1}{2} x\right)\)[/tex]. The coefficient [tex]\(\frac{1}{2}\)[/tex] causes a horizontal stretch by a factor of 2. This means that the period of the function, which is originally [tex]\(\pi\)[/tex] for [tex]\(\cot(x)\)[/tex], will now be [tex]\(2\pi\)[/tex].
3. Vertical Stretch/Compression:
The function [tex]\(\frac{-1}{2} \cot\left(\frac{1}{2} x\right)\)[/tex] has a coefficient of [tex]\(\frac{-1}{2}\)[/tex] multiplying it. This will vertically stretch/compress the graph by a factor of [tex]\(\frac{1}{2}\)[/tex]. Each point on the basic [tex]\(\cot\)[/tex] function will be scaled down by a factor of [tex]\(\frac{1}{2}\)[/tex].
4. Reflection Across x-axis:
The negative sign in [tex]\(\frac{-1}{2} \cot\left(\frac{1}{2} x\right)\)[/tex] indicates a reflection across the x-axis. This will flip the graph upside down.
### Summary of Transformations:
1. Reflection Across x-axis:
- Reflect the graph across the x-axis.
2. Vertical Stretch/Compression:
- Compress the graph vertically by a factor of [tex]\(\frac{1}{2}\)[/tex].
3. Horizontal Stretch/Compression:
- Stretch the graph horizontally by a factor of 2. The period changes from [tex]\(\pi\)[/tex] to [tex]\(2\pi\)[/tex].
4. Shifts:
- There are no horizontal or vertical shifts.
### Final Analysis:
- Reflection Across x-axis: Yes, the graph will be reflected across the x-axis.
- Shift Graph Vertically:
- None.
- Shift Graph Horizontally (Phase Shift):
- None.
- Stretch/Compress Graph Vertically:
- Yes, the graph is compressed by a factor of [tex]\(\frac{1}{2}\)[/tex].
So, the final transformations to the [tex]\( y = \cot(x) \)[/tex] graph to obtain [tex]\( y = \frac{-1}{2} \cot\left(\frac{1}{2} x\right) \)[/tex] are:
1. Reflect across the x-axis.
2. Compress vertically by a factor of [tex]\(\frac{1}{2}\)[/tex].
3. Stretch horizontally by a factor of 2.
The graph of [tex]\( y = \frac{-1}{2} \cot\left(\frac{1}{2} x\right) \)[/tex] will look similar to the [tex]\(\cot(x)\)[/tex] graph, but reflected across the x-axis, compressed vertically by [tex]\(\frac{1}{2}\)[/tex], and stretched horizontally by a factor of 2.
### Step-by-Step Solution:
1. Basic Shape of Cotangent Function:
The basic shape of the [tex]\( \cot(x) \)[/tex] function has vertical asymptotes at multiples of [tex]\( \pi \)[/tex] (i.e., [tex]\( x = n\pi \)[/tex] where [tex]\( n \)[/tex] is an integer). Between these asymptotes, the function decreases from [tex]\(\infty\)[/tex] to [tex]\(-\infty\)[/tex].
2. Horizontal Stretch/Compression:
Inside the function, we have [tex]\(\cot\left(\frac{1}{2} x\right)\)[/tex]. The coefficient [tex]\(\frac{1}{2}\)[/tex] causes a horizontal stretch by a factor of 2. This means that the period of the function, which is originally [tex]\(\pi\)[/tex] for [tex]\(\cot(x)\)[/tex], will now be [tex]\(2\pi\)[/tex].
3. Vertical Stretch/Compression:
The function [tex]\(\frac{-1}{2} \cot\left(\frac{1}{2} x\right)\)[/tex] has a coefficient of [tex]\(\frac{-1}{2}\)[/tex] multiplying it. This will vertically stretch/compress the graph by a factor of [tex]\(\frac{1}{2}\)[/tex]. Each point on the basic [tex]\(\cot\)[/tex] function will be scaled down by a factor of [tex]\(\frac{1}{2}\)[/tex].
4. Reflection Across x-axis:
The negative sign in [tex]\(\frac{-1}{2} \cot\left(\frac{1}{2} x\right)\)[/tex] indicates a reflection across the x-axis. This will flip the graph upside down.
### Summary of Transformations:
1. Reflection Across x-axis:
- Reflect the graph across the x-axis.
2. Vertical Stretch/Compression:
- Compress the graph vertically by a factor of [tex]\(\frac{1}{2}\)[/tex].
3. Horizontal Stretch/Compression:
- Stretch the graph horizontally by a factor of 2. The period changes from [tex]\(\pi\)[/tex] to [tex]\(2\pi\)[/tex].
4. Shifts:
- There are no horizontal or vertical shifts.
### Final Analysis:
- Reflection Across x-axis: Yes, the graph will be reflected across the x-axis.
- Shift Graph Vertically:
- None.
- Shift Graph Horizontally (Phase Shift):
- None.
- Stretch/Compress Graph Vertically:
- Yes, the graph is compressed by a factor of [tex]\(\frac{1}{2}\)[/tex].
So, the final transformations to the [tex]\( y = \cot(x) \)[/tex] graph to obtain [tex]\( y = \frac{-1}{2} \cot\left(\frac{1}{2} x\right) \)[/tex] are:
1. Reflect across the x-axis.
2. Compress vertically by a factor of [tex]\(\frac{1}{2}\)[/tex].
3. Stretch horizontally by a factor of 2.
The graph of [tex]\( y = \frac{-1}{2} \cot\left(\frac{1}{2} x\right) \)[/tex] will look similar to the [tex]\(\cot(x)\)[/tex] graph, but reflected across the x-axis, compressed vertically by [tex]\(\frac{1}{2}\)[/tex], and stretched horizontally by a factor of 2.
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