Answer :
To determine the transformations applied to produce the graph of the function [tex]\( y = 0.5 \cot(0.5x) \)[/tex] from the parent function [tex]\( y = \cot(x) \)[/tex], we need to analyze the components of the function.
### Step-by-Step Solution
1. Understanding the Parent Function:
- The parent function is [tex]\( y = \cot(x) \)[/tex].
- The period of the parent function [tex]\( y = \cot(x) \)[/tex] is [tex]\( \pi \)[/tex].
2. Examining the New Function Components:
- The given function is [tex]\( y = 0.5 \cot(0.5x) \)[/tex].
3. Horizontal Transformation:
- The input to the cotangent function has been modified from [tex]\( x \)[/tex] to [tex]\( 0.5x \)[/tex].
- To find the new period, we set up the relationship: [tex]\( \text{Period of } \cot(0.5x) = \frac{\pi}{0.5} = 2\pi \)[/tex].
- This indicates a horizontal stretch because the period has increased from [tex]\( \pi \)[/tex] to [tex]\( 2\pi \)[/tex].
4. Vertical Transformation:
- The output of the cotangent function is multiplied by 0.5.
- This multiplication by a factor less than 1 (0.5) indicates a vertical compression.
### Final Conclusion
- We have a horizontal stretch that changes the period from [tex]\( \pi \)[/tex] to [tex]\( 2\pi \)[/tex].
- We also have a vertical compression.
Hence, the transformations applied to the graph of [tex]\( y = \cot(x) \)[/tex] to produce the graph of [tex]\( y = 0.5 \cot(0.5x) \)[/tex] are:
- A horizontal stretch to produce a period of [tex]\( 2\pi \)[/tex]
- A vertical compression
Thus, the correct answer is:
- A horizontal stretch to produce a period of [tex]\( 2\pi \)[/tex] and a vertical compression
### Step-by-Step Solution
1. Understanding the Parent Function:
- The parent function is [tex]\( y = \cot(x) \)[/tex].
- The period of the parent function [tex]\( y = \cot(x) \)[/tex] is [tex]\( \pi \)[/tex].
2. Examining the New Function Components:
- The given function is [tex]\( y = 0.5 \cot(0.5x) \)[/tex].
3. Horizontal Transformation:
- The input to the cotangent function has been modified from [tex]\( x \)[/tex] to [tex]\( 0.5x \)[/tex].
- To find the new period, we set up the relationship: [tex]\( \text{Period of } \cot(0.5x) = \frac{\pi}{0.5} = 2\pi \)[/tex].
- This indicates a horizontal stretch because the period has increased from [tex]\( \pi \)[/tex] to [tex]\( 2\pi \)[/tex].
4. Vertical Transformation:
- The output of the cotangent function is multiplied by 0.5.
- This multiplication by a factor less than 1 (0.5) indicates a vertical compression.
### Final Conclusion
- We have a horizontal stretch that changes the period from [tex]\( \pi \)[/tex] to [tex]\( 2\pi \)[/tex].
- We also have a vertical compression.
Hence, the transformations applied to the graph of [tex]\( y = \cot(x) \)[/tex] to produce the graph of [tex]\( y = 0.5 \cot(0.5x) \)[/tex] are:
- A horizontal stretch to produce a period of [tex]\( 2\pi \)[/tex]
- A vertical compression
Thus, the correct answer is:
- A horizontal stretch to produce a period of [tex]\( 2\pi \)[/tex] and a vertical compression
