to produce the graph of the function (y = 0.5cot(0.5x)), what transformations should be applied to the graph…

Answer:

A. a horizontal compression to produce a period of $\frac{\pi}{2}$ and a vertical compression

Explanation:

Step1: Recall period formula for cotangent

The period of $y = A\cot(Bx)$ is $T=\frac{\pi}{|B|}$, for $y = \cot(x)$ the period is $\pi$. For $y=0.5\cot(0.5x)$, $B = 0.5$, and $T=\frac{\pi}{0.5}=2\pi$. But if we consider the general form and the change from $y = \cot(x)$ to $y=\cot(Bx)$ with $B> 1$ is a horizontal - compression. Here, for the standard $y = \cot(x)$ to $y=\cot(2x)$ (to get period $\frac{\pi}{2}$), it's a horizontal compression.

Step2: Analyze vertical transformation

The coefficient $A$ in $y = A\cot(Bx)$ represents a vertical transformation. For $y = 0.5\cot(0.5x)$, $A = 0.5<1$, so it is a vertical compression.