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Sagot :
To graph the given function [tex]\( y = -2 + \frac{3}{2} \cot \left( \frac{\pi}{2} x + 2\pi \right) \)[/tex], we need to analyze how the general shape of the cotangent function [tex]\( y = \cot(x) \)[/tex] will shift, stretch, and reflect based on the transformation components in the given equation.
### Step-by-Step Transformation
#### 1. Cotangent Function Basics:
The basic cotangent function is [tex]\( y = \cot(x) \)[/tex]. It has the following characteristics:
- Vertical asymptotes at [tex]\( x = k\pi \)[/tex] (where [tex]\( k \)[/tex] is an integer).
- Period of [tex]\( \pi \)[/tex].
- Decreasing from positive infinity to negative infinity between consecutive vertical asymptotes.
#### 2. Horizontal Stretch/Compression (Period Change):
The given function contains [tex]\( \frac{\pi}{2} \)[/tex] inside the cotangent function, [tex]\( \cot \left( \frac{\pi}{2} x + 2\pi \right) \)[/tex]. This affects the period of the cotangent function. In general, the period of [tex]\( \cot(bx) \)[/tex] is [tex]\( \frac{\pi}{|b|} \)[/tex]. For [tex]\( \frac{\pi}{2} \)[/tex]:
[tex]\[ \text{Period} = \frac{\pi}{\frac{\pi}{2}} = 2. \][/tex]
So, the period of our function is 2 instead of [tex]\( \pi \)[/tex].
#### 3. Phase Shift (Horizontal Shift):
Next, consider the phase shift term [tex]\( 2\pi \)[/tex] inside the cotangent function. For [tex]\( \cot(bx + c) \)[/tex], the phase shift is given by [tex]\( \frac{-c}{b} \)[/tex]. Here:
[tex]\[ \text{Phase Shift} = \frac{-2\pi}{\frac{\pi}{2}} = -4. \][/tex]
Thus, the function is shifted left by 4 units.
#### 4. Vertical Stretch/Compression:
The function includes the [tex]\( \frac{3}{2} \)[/tex] multiplier. This affects the amplitude (vertical stretch). Since the amplitude (the distance from the centerline to a peak or trough) is scaled by [tex]\( \frac{3}{2} \)[/tex], the cotangent graph will stretch vertically by a factor of [tex]\( \frac{3}{2} \)[/tex].
#### 5. Vertical Shift:
The function has a [tex]\( -2 \)[/tex] horizontal component outside the cotangent function. This constant term shifts the entire graph down by 2 units.
#### 6. Reflection across [tex]\( x \)[/tex]-Axis:
The overall function is [tex]\( -2 + \frac{3}{2} \cot(\cdots) \)[/tex]. Note that the [tex]\( \cot(\cdots) \)[/tex] itself is not negated, thus there’s no reflection across the [tex]\( x \)[/tex]-axis coming from the cotangent function being negative. The graph is simply shifted down and scaled; there is no [tex]\( x \)[/tex]-axis reflection.
### Summary of Transformation Instructions:
- Reflect graph across [tex]\( x \)[/tex]-axis: None.
- Shift graph vertically:
- Down by 2 units.
- Shift graph horizontally (Phase Shift):
- Left by 4 units.
- Stretch/Compress graph vertically: Yes
- Stretch by a factor of [tex]\( \frac{3}{2} \)[/tex].
- Stretch/Compress graph horizontally (Period): Yes
- New period of 2 instead of [tex]\( \pi \)[/tex].
### Graph the Results:
To graph [tex]\( y = -2 + \frac{3}{2} \cot \left( \frac{\pi}{2} x + 2\pi \right) \)[/tex]:
1. Start with the basic cotangent shape.
2. Apply the period change, so know that one cycle spans 2 units.
3. Shift left by 4 units.
4. Vertically stretch by [tex]\( \frac{3}{2} \)[/tex].
5. Shift the entire graph down by 2 units.
Following these steps will give the transformed graph of the function.
### Step-by-Step Transformation
#### 1. Cotangent Function Basics:
The basic cotangent function is [tex]\( y = \cot(x) \)[/tex]. It has the following characteristics:
- Vertical asymptotes at [tex]\( x = k\pi \)[/tex] (where [tex]\( k \)[/tex] is an integer).
- Period of [tex]\( \pi \)[/tex].
- Decreasing from positive infinity to negative infinity between consecutive vertical asymptotes.
#### 2. Horizontal Stretch/Compression (Period Change):
The given function contains [tex]\( \frac{\pi}{2} \)[/tex] inside the cotangent function, [tex]\( \cot \left( \frac{\pi}{2} x + 2\pi \right) \)[/tex]. This affects the period of the cotangent function. In general, the period of [tex]\( \cot(bx) \)[/tex] is [tex]\( \frac{\pi}{|b|} \)[/tex]. For [tex]\( \frac{\pi}{2} \)[/tex]:
[tex]\[ \text{Period} = \frac{\pi}{\frac{\pi}{2}} = 2. \][/tex]
So, the period of our function is 2 instead of [tex]\( \pi \)[/tex].
#### 3. Phase Shift (Horizontal Shift):
Next, consider the phase shift term [tex]\( 2\pi \)[/tex] inside the cotangent function. For [tex]\( \cot(bx + c) \)[/tex], the phase shift is given by [tex]\( \frac{-c}{b} \)[/tex]. Here:
[tex]\[ \text{Phase Shift} = \frac{-2\pi}{\frac{\pi}{2}} = -4. \][/tex]
Thus, the function is shifted left by 4 units.
#### 4. Vertical Stretch/Compression:
The function includes the [tex]\( \frac{3}{2} \)[/tex] multiplier. This affects the amplitude (vertical stretch). Since the amplitude (the distance from the centerline to a peak or trough) is scaled by [tex]\( \frac{3}{2} \)[/tex], the cotangent graph will stretch vertically by a factor of [tex]\( \frac{3}{2} \)[/tex].
#### 5. Vertical Shift:
The function has a [tex]\( -2 \)[/tex] horizontal component outside the cotangent function. This constant term shifts the entire graph down by 2 units.
#### 6. Reflection across [tex]\( x \)[/tex]-Axis:
The overall function is [tex]\( -2 + \frac{3}{2} \cot(\cdots) \)[/tex]. Note that the [tex]\( \cot(\cdots) \)[/tex] itself is not negated, thus there’s no reflection across the [tex]\( x \)[/tex]-axis coming from the cotangent function being negative. The graph is simply shifted down and scaled; there is no [tex]\( x \)[/tex]-axis reflection.
### Summary of Transformation Instructions:
- Reflect graph across [tex]\( x \)[/tex]-axis: None.
- Shift graph vertically:
- Down by 2 units.
- Shift graph horizontally (Phase Shift):
- Left by 4 units.
- Stretch/Compress graph vertically: Yes
- Stretch by a factor of [tex]\( \frac{3}{2} \)[/tex].
- Stretch/Compress graph horizontally (Period): Yes
- New period of 2 instead of [tex]\( \pi \)[/tex].
### Graph the Results:
To graph [tex]\( y = -2 + \frac{3}{2} \cot \left( \frac{\pi}{2} x + 2\pi \right) \)[/tex]:
1. Start with the basic cotangent shape.
2. Apply the period change, so know that one cycle spans 2 units.
3. Shift left by 4 units.
4. Vertically stretch by [tex]\( \frac{3}{2} \)[/tex].
5. Shift the entire graph down by 2 units.
Following these steps will give the transformed graph of the function.
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