To graph the function [tex]\( y = \frac{-1}{2} \cot \left(\frac{1}{2} x\right) \)[/tex], we first need to understand the shape of the more basic function before any transformations are applied.
### Step 1: Identify the Shape of the Basic Function
The given function is [tex]\( y = \frac{-1}{2} \cot \left(\frac{1}{2} x\right) \)[/tex].
#### Basic Function:
The fundamental function in this case is the cotangent function, [tex]\( y = \cot(x) \)[/tex].
- The cotangent function [tex]\( \cot(x) \)[/tex] has the following characteristics:
- It is undefined at [tex]\( x = n\pi \)[/tex] for integers [tex]\( n \)[/tex], where it has vertical asymptotes.
- It has a period of [tex]\( \pi \)[/tex] because [tex]\( \cot(x + \pi) = \cot(x) \)[/tex].
- It crosses the x-axis at [tex]\( x = \left(n + \frac{1}{2}\right)\pi \)[/tex].
Therefore, the shape of the more basic function that has been shifted, reflected, stretched, or compressed is the cotangent function.
### Summary
Thus, the shape of the basic function is identified as:
- Cotangent function.
