1. graph the function $f(x)=cot(x + \frac{pi}{6})$. be sure to clearly label the axes and graph at least one…

Explanation:

Step1: Recall the period of cotangent

The period of (y = \cot(x)) is (\pi). For the function (y=\cot\left(x +\frac{\pi}{6}\right)), the period remains (\pi) since the addition of (\frac{\pi}{6}) inside the function is a horizontal - shift and does not affect the period.

Step2: Find the vertical asymptotes

Set (x+\frac{\pi}{6}=k\pi), where (k\in\mathbb{Z}). Solving for (x) gives (x = k\pi-\frac{\pi}{6}). For (k = 0), (x=-\frac{\pi}{6}); for (k = 1), (x=\pi-\frac{\pi}{6}=\frac{5\pi}{6}). These are two consecutive vertical asymptotes for one - period of the function.

Step3: Find the x - intercept

Set (\cot\left(x+\frac{\pi}{6}\right)=0). Then (x+\frac{\pi}{6}=\frac{\pi}{2}+k\pi), (k\in\mathbb{Z}). Solving for (x) gives (x=\frac{\pi}{3}+k\pi). For (k = 0), (x=\frac{\pi}{3}) is an x - intercept within one period (\left(-\frac{\pi}{6},\frac{5\pi}{6}\right)).

Step4: Plot key points and graph

We know that (\cot(x)) is decreasing on each of its intervals between vertical asymptotes. Plot the vertical asymptotes (x =-\frac{\pi}{6}) and (x=\frac{5\pi}{6}), the x - intercept (x=\frac{\pi}{3}), and a few other points (e.g., when (x = 0), (y=\cot\left(\frac{\pi}{6}\right)=\sqrt{3})). Then draw a smooth curve that is decreasing from left - to - right between the vertical asymptotes.

To graph:

1. Draw the x - axis and y - axis. Label them as (x) and (y) respectively.
2. Mark the vertical asymptotes (x =-\frac{\pi}{6}) and (x=\frac{5\pi}{6}) with dashed lines.
3. Mark the x - intercept at (x=\frac{\pi}{3}).
4. Plot additional points such as ((0,\sqrt{3})) and draw a smooth curve that approaches the vertical asymptotes and passes through the plotted points, repeating the pattern every (\pi) units along the x - axis.

Answer:

The graph of (y = \cot\left(x+\frac{\pi}{6}\right)) has vertical asymptotes at (x=k\pi-\frac{\pi}{6},k\in\mathbb{Z}), x - intercepts at (x=\frac{\pi}{3}+k\pi,k\in\mathbb{Z}), is decreasing on each interval (\left(k\pi-\frac{\pi}{6},(k + 1)\pi-\frac{\pi}{6}\right)), and has a period of (\pi). The graph should be drawn with clearly labeled axes, vertical asymptotes, x - intercepts, and at least one full period shown.