To graph the function [tex]\( y = 0.5 \cot(0.5x) \)[/tex], let’s examine the transformations applied to the parent function [tex]\( y = \cot(x) \)[/tex].
1. Horizontal Transformation:
The parent function is [tex]\( y = \cot(x) \)[/tex], which has a period of [tex]\( \pi \)[/tex]. The function given for transformation is [tex]\( y = 0.5 \cot(0.5x) \)[/tex].
- The term [tex]\( 0.5x \)[/tex] inside the cotangent function affects the period. The period of [tex]\( \cot(bx) \)[/tex] is [tex]\( \pi / |b| \)[/tex].
- Here, [tex]\( b = 0.5 \)[/tex]. Therefore, the new period is [tex]\( \pi / 0.5 = 2\pi \)[/tex].
- This results in a horizontal stretch by a factor of 2, changing the period from [tex]\( \pi \)[/tex] to [tex]\( 2\pi \)[/tex].
2. Vertical Transformation:
- The coefficient [tex]\( 0.5 \)[/tex] in front of the cotangent function leads to a vertical transformation.
- This coefficient compresses the graph vertically by a factor of 0.5.
In summary, to graph [tex]\( y = 0.5 \cot(0.5x) \)[/tex], you would:
- Apply a horizontal stretch to produce a period of [tex]\( 2\pi \)[/tex].
- Apply a vertical compression by a factor of 0.5.
Thus, the appropriate choice is:
a horizontal stretch to produce a period of [tex]\( 2\pi \)[/tex] and a vertical compression
Hence, the correct answer is:
a horizontal stretch to produce a period of [tex]\( 2\pi \)[/tex] and a vertical compression.
