identify the period and equation of one asymptote of the function f(x)=-5 cot(1/2 x)+3. period: equation of…

Explanation:

Step1: Recall cotangent period formula

The period of $y = A\cot(Bx - C)+D$ is $\frac{\pi}{|B|}$. For $f(x)=-5\cot(\frac{1}{2}x)+3$, $B = \frac{1}{2}$.

Step2: Calculate the period

$T=\frac{\pi}{\left|\frac{1}{2}\right|}=2\pi$.

Step3: Recall cotangent asymptote formula

The asymptotes of $y = \cot(x)$ occur at $x = n\pi$, $n\in\mathbb{Z}$. For $y=\cot(Bx)$, the asymptotes are $Bx=n\pi$. For $f(x)=-5\cot(\frac{1}{2}x)+3$, when $n = 0$, we solve $\frac{1}{2}x=0$.

Step4: Find an asymptote

Solving $\frac{1}{2}x = 0$ gives $x = 0$.

Answer:

Period: $2\pi$ Equation of one asymptote: $x = 0$