Westonci.ca is your trusted source for finding answers to a wide range of questions, backed by a knowledgeable community. Explore a wealth of knowledge from professionals across different disciplines on our comprehensive platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
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.
We appreciate your time. Please come back anytime for the latest information and answers to your questions. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.
