Problem 17 - Precalculus Essentials

Question

Graph two periods of the given cotangent function. $$y=2 \cot x$$

Step-by-Step Solution

Answer

The graph of the function \(y=2 \cot x\) shows two periods, stretching from \(-2\pi\) to \(2\pi\), with vertical asymptotes at \(x = n\pi\), (where \(n\) is an integer), and parabolic arcs which are stretched vertically by a factor of 2 transitioning from positive infinity to negative infinity at each period.

Step 1: Identify the Period, Phase Shift, and Vertical Shift

The general formula of cotangent function is \(y=A \cot (B(x - C)) + D\), where \(A\) is the amplitude (stretch/compress), \(B\) determines the period, \(C\) is the phase shift, and \(D\) is the vertical shift. Here, \(A=2\), \(B=1\), \(C=0\), and \(D=0\), which means the function is stretched vertically by a factor of 2, has a period of \(\pi\), no phase shift, and no vertical shift. The vertical asymptotes occurs at \(x = n\pi\), where \(n\) is an integer.

Step 2: Plot the Asymptotes and Key Points for Two Periods

Mark points at intervals of \(\pi\) from \(0\) on the x-axis for two periods, for example \(x=-2\pi, -\pi, 0, \pi, 2\pi\). These points will represent the vertical asymptotes of the function. Since cotangent is positive in the first and third quadrants, plot a point halfway between the asymptotes in those intervals showing that cotangent is positive.

Step 3: Sketch the Graph

Connect the points drawn with a smooth curve that approaches but never reaches the vertical asymptotes. The cotangent function decreases from positive infinity to negative infinity as it progresses from left to right. So, the graph trends downwards from each asymptote making parabolic arcs. Repeat this sketch for two periods to complete the graph.