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Sagot :
To graph the given function [tex]\( y = \frac{-1}{2} + 2 \cot \left(\frac{4}{3} x + 5 \pi \right) \)[/tex], let's follow these steps:
### Step-by-Step Solution
1. Identify the basic shape of [tex]\(y = \cot(x)\)[/tex]:
- The cotangent function, [tex]\(y = \cot(x)\)[/tex], has undefined points (vertical asymptotes) where [tex]\(x = k\pi \)[/tex] for [tex]\( k \in \mathbb{Z} \)[/tex].
- It is periodic with a period of [tex]\(\pi\)[/tex].
- The basic shape of the cotangent function decreases from positive to negative as [tex]\(x\)[/tex] increases between asymptotes.
2. Identify transformations:
- Start with [tex]\(y = \cot(x)\)[/tex] and apply transformations step-by-step.
1. Horizontal Compression and Phase Shift:
- The function is modified to [tex]\( y = \cot\left(\frac{4}{3} x + 5\pi\right) \)[/tex].
- This represents a horizontal compression with a horizontal scaling factor [tex]\(\frac{4}{3}\)[/tex], which reduces the period from [tex]\(\pi\)[/tex] to [tex]\(\frac{3\pi}{4}\)[/tex].
- The phase shift is given by solving [tex]\(\frac{4}{3} x + 5 \pi = 0\)[/tex] or when [tex]\(\frac{4}{3} x + 5 \pi \)[/tex] equals typical behavior points ([tex]\(0, \pi, 2\pi, \ldots\)[/tex]).
- Solving for [tex]\(x\)[/tex], [tex]\(x = -\frac{15\pi}{4}\)[/tex].
2. Vertical Stretch:
- The function is now [tex]\( y = 2 \cot\left(\frac{4}{3} x + 5\pi\right) \)[/tex].
- This represents a vertical stretch by a factor of 2. The graph will be stretched vertically by a factor of 2.
3. Vertical Shift and Reflection:
- Finally, we apply the vertical shift and add the constant term to get [tex]\( y = \frac{-1}{2} + 2 \cot\left(\frac{4}{3} x + 5\pi\right) \)[/tex].
- This includes a downward vertical shift by [tex]\(\frac{1}{2}\)[/tex] and also an x-axis reflection due to the combination of transformations.
### Summary of Transformations
- x-Axis Reflection: Yes, inherently due to the negative fraction.
- Shift Graph Vertically: Down by [tex]\(\frac{1}{2}\)[/tex].
- Shift Graph Horizontally (Phase Shift): Left by [tex]\(\frac{15\pi}{4}\)[/tex].
- Stretch/Compress Graph Vertically: Yes, by a factor of 2.
- Stretch/Compress Graph Horizontally (Period): Yes, period changes to [tex]\(\frac{3\pi}{4}\)[/tex].
### Graphing the Final Function
- Vertical Asymptotes:
They occur where [tex]\( \cot\left(\frac{4}{3} x + 5\pi\right) \)[/tex] is undefined.
- Solving for the asymptotes: [tex]\( \frac{4}{3} x + 5\pi = k\pi \Rightarrow \frac{4}{3} x = (k - 5)\pi \Rightarrow x = \frac{3}{4} (k - 5)\pi \)[/tex].
- Shape of the Curve:
Between the asymptotes, the curve has a standard cotangent shape but stretched by a factor of 2, reflected along x-axis and shifted vertically down by [tex]\(\frac{1}{2}\)[/tex].
- Graphing Transformations:
Reflect [tex]\( \cot(x) \)[/tex] initially.
Shift left by [tex]\( \frac{15\pi}{4} \)[/tex].
Compress horizontally to change period to [tex]\( \frac{3\pi}{4} \)[/tex].
Stretch vertically by 2.
Finally shift down by [tex]\(\frac{1}{2}\)[/tex].
The combined transformations will give you the graph required, showcasing all the shifts, stretches, and reflections as above.
### Step-by-Step Solution
1. Identify the basic shape of [tex]\(y = \cot(x)\)[/tex]:
- The cotangent function, [tex]\(y = \cot(x)\)[/tex], has undefined points (vertical asymptotes) where [tex]\(x = k\pi \)[/tex] for [tex]\( k \in \mathbb{Z} \)[/tex].
- It is periodic with a period of [tex]\(\pi\)[/tex].
- The basic shape of the cotangent function decreases from positive to negative as [tex]\(x\)[/tex] increases between asymptotes.
2. Identify transformations:
- Start with [tex]\(y = \cot(x)\)[/tex] and apply transformations step-by-step.
1. Horizontal Compression and Phase Shift:
- The function is modified to [tex]\( y = \cot\left(\frac{4}{3} x + 5\pi\right) \)[/tex].
- This represents a horizontal compression with a horizontal scaling factor [tex]\(\frac{4}{3}\)[/tex], which reduces the period from [tex]\(\pi\)[/tex] to [tex]\(\frac{3\pi}{4}\)[/tex].
- The phase shift is given by solving [tex]\(\frac{4}{3} x + 5 \pi = 0\)[/tex] or when [tex]\(\frac{4}{3} x + 5 \pi \)[/tex] equals typical behavior points ([tex]\(0, \pi, 2\pi, \ldots\)[/tex]).
- Solving for [tex]\(x\)[/tex], [tex]\(x = -\frac{15\pi}{4}\)[/tex].
2. Vertical Stretch:
- The function is now [tex]\( y = 2 \cot\left(\frac{4}{3} x + 5\pi\right) \)[/tex].
- This represents a vertical stretch by a factor of 2. The graph will be stretched vertically by a factor of 2.
3. Vertical Shift and Reflection:
- Finally, we apply the vertical shift and add the constant term to get [tex]\( y = \frac{-1}{2} + 2 \cot\left(\frac{4}{3} x + 5\pi\right) \)[/tex].
- This includes a downward vertical shift by [tex]\(\frac{1}{2}\)[/tex] and also an x-axis reflection due to the combination of transformations.
### Summary of Transformations
- x-Axis Reflection: Yes, inherently due to the negative fraction.
- Shift Graph Vertically: Down by [tex]\(\frac{1}{2}\)[/tex].
- Shift Graph Horizontally (Phase Shift): Left by [tex]\(\frac{15\pi}{4}\)[/tex].
- Stretch/Compress Graph Vertically: Yes, by a factor of 2.
- Stretch/Compress Graph Horizontally (Period): Yes, period changes to [tex]\(\frac{3\pi}{4}\)[/tex].
### Graphing the Final Function
- Vertical Asymptotes:
They occur where [tex]\( \cot\left(\frac{4}{3} x + 5\pi\right) \)[/tex] is undefined.
- Solving for the asymptotes: [tex]\( \frac{4}{3} x + 5\pi = k\pi \Rightarrow \frac{4}{3} x = (k - 5)\pi \Rightarrow x = \frac{3}{4} (k - 5)\pi \)[/tex].
- Shape of the Curve:
Between the asymptotes, the curve has a standard cotangent shape but stretched by a factor of 2, reflected along x-axis and shifted vertically down by [tex]\(\frac{1}{2}\)[/tex].
- Graphing Transformations:
Reflect [tex]\( \cot(x) \)[/tex] initially.
Shift left by [tex]\( \frac{15\pi}{4} \)[/tex].
Compress horizontally to change period to [tex]\( \frac{3\pi}{4} \)[/tex].
Stretch vertically by 2.
Finally shift down by [tex]\(\frac{1}{2}\)[/tex].
The combined transformations will give you the graph required, showcasing all the shifts, stretches, and reflections as above.
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