Find answers to your questions and expand your knowledge with IDNLearn.com. Discover the reliable solutions you need with help from our comprehensive and accurate Q&A platform.
Sagot :
To determine the transformations applied to the function [tex]\( y = \cot(x) \)[/tex] to obtain the graph of [tex]\( y = 6 \cot \left(3x - \frac{\pi}{2}\right) + 4 \)[/tex], we need to analyze each component in the given transformed function step-by-step:
1. Vertical Stretch:
- The coefficient 6 in [tex]\( y = 6 \cot \left(3x - \frac{\pi}{2}\right) + 4 \)[/tex] indicates a vertical stretch. Specifically, the function [tex]\( y = \cot(x) \)[/tex] is vertically stretched by a factor of 6. This stretches the range of [tex]\( y = \cot(x) \)[/tex].
2. Horizontal Compression:
- The factor 3 inside the cotangent function [tex]\( \cot(3x - \frac{\pi}{2}) \)[/tex] affects the period of the cotangent function. The period of the original cotangent function [tex]\( y = \cot(x) \)[/tex] is [tex]\( \pi \)[/tex]. When multiplied by 3, the new period becomes [tex]\( \frac{\pi}{3} \)[/tex], indicating a horizontal compression by a factor of 3.
3. Horizontal Shift:
- The term [tex]\(-\frac{\pi}{2}\)[/tex] inside the argument [tex]\( 3x - \frac{\pi}{2} \)[/tex] shifts the graph horizontally. To find the actual shift, observe that [tex]\( 3x = \frac{\pi}{2} \)[/tex] at [tex]\( x = \frac{\pi}{6} \)[/tex]. This means the graph is shifted [tex]\(\frac{\pi}{6}\)[/tex] units to the right.
4. Vertical Shift:
- The constant term +4 at the end indicates a vertical shift upward by 4 units.
Combining all these transformations, the graph of [tex]\( y = 6 \cot \left(3x - \frac{\pi}{2}\right) + 4 \)[/tex] is obtained from [tex]\( y = \cot(x) \)[/tex] by the following steps:
- A vertical stretch by a factor of 6.
- A horizontal compression resulting in the period being [tex]\( \frac{\pi}{3} \)[/tex].
- A horizontal shift [tex]\(\frac{\pi}{6}\)[/tex] units to the right.
- A vertical shift upward by 4 units.
Therefore, the correct transformation is:
- a vertical stretch,
- a horizontal compression to make the period [tex]\( \frac{\pi}{3} \)[/tex],
- a horizontal shift of [tex]\( \frac{\pi}{6} \)[/tex] units to the right,
- and a vertical shift of 4 units up.
Given these details, the correct transformation matches the first option provided:
- a vertical stretch, a horizontal compression to make the period [tex]\( \frac{\pi}{3} \)[/tex], a horizontal shift of [tex]\( \frac{\pi}{6} \)[/tex] units to the right, and a vertical shift of 4 units up.
1. Vertical Stretch:
- The coefficient 6 in [tex]\( y = 6 \cot \left(3x - \frac{\pi}{2}\right) + 4 \)[/tex] indicates a vertical stretch. Specifically, the function [tex]\( y = \cot(x) \)[/tex] is vertically stretched by a factor of 6. This stretches the range of [tex]\( y = \cot(x) \)[/tex].
2. Horizontal Compression:
- The factor 3 inside the cotangent function [tex]\( \cot(3x - \frac{\pi}{2}) \)[/tex] affects the period of the cotangent function. The period of the original cotangent function [tex]\( y = \cot(x) \)[/tex] is [tex]\( \pi \)[/tex]. When multiplied by 3, the new period becomes [tex]\( \frac{\pi}{3} \)[/tex], indicating a horizontal compression by a factor of 3.
3. Horizontal Shift:
- The term [tex]\(-\frac{\pi}{2}\)[/tex] inside the argument [tex]\( 3x - \frac{\pi}{2} \)[/tex] shifts the graph horizontally. To find the actual shift, observe that [tex]\( 3x = \frac{\pi}{2} \)[/tex] at [tex]\( x = \frac{\pi}{6} \)[/tex]. This means the graph is shifted [tex]\(\frac{\pi}{6}\)[/tex] units to the right.
4. Vertical Shift:
- The constant term +4 at the end indicates a vertical shift upward by 4 units.
Combining all these transformations, the graph of [tex]\( y = 6 \cot \left(3x - \frac{\pi}{2}\right) + 4 \)[/tex] is obtained from [tex]\( y = \cot(x) \)[/tex] by the following steps:
- A vertical stretch by a factor of 6.
- A horizontal compression resulting in the period being [tex]\( \frac{\pi}{3} \)[/tex].
- A horizontal shift [tex]\(\frac{\pi}{6}\)[/tex] units to the right.
- A vertical shift upward by 4 units.
Therefore, the correct transformation is:
- a vertical stretch,
- a horizontal compression to make the period [tex]\( \frac{\pi}{3} \)[/tex],
- a horizontal shift of [tex]\( \frac{\pi}{6} \)[/tex] units to the right,
- and a vertical shift of 4 units up.
Given these details, the correct transformation matches the first option provided:
- a vertical stretch, a horizontal compression to make the period [tex]\( \frac{\pi}{3} \)[/tex], a horizontal shift of [tex]\( \frac{\pi}{6} \)[/tex] units to the right, and a vertical shift of 4 units up.
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! IDNLearn.com provides the answers you need. Thank you for visiting, and see you next time for more valuable insights.