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.
