Problem 18 - Precalculus Essentials

Question

Graph two periods of the given cotangent function. $$y=\frac{1}{2} \cot x$$

Step-by-Step Solution

Answer

The graph of \( y = \frac{1}{2}\cot x \) will have the same shape as the basic cotangent function, but less steep due to the factor of \( \frac{1}{2} \). Vertical asymptotes are located at multiples of \( \pi \). It will start high, pass through a point in the middle and end low within each period.

Step 1: Identify the Amplitude

The cotangent function doesn't actually have an amplitude in the same way as a sine or cosine function, as it goes to infinity at lift-off points. The function as given is \( y=\frac{1}{2} cot x \), so the \( \frac{1}{2} \) factor will reduce the steepness of the cotangent curve.

Step 2: Identify the Period

For a cotangent function, the period is normally \( \pi \), but the \(\frac{1}{2}\) in this case, which is in front of the cotangent, doesn't change the period. So, the period is still \( \pi \).

Step 3: Identify Lift-off Points

The lift-off points of the cotangent function occur at multiples of \( \pi \). These points are where the function is undefined, and will create vertical asymptotes in the graph.

Step 4: Plot the Graph

First, plot the lift-off points at multiples of \( \pi \) as vertical dashed lines. These denote where the function is undefined. For two periods, these points would be at \(x = 0, \pi, 2\pi, 3\pi \). Between each pair of asymptotes, the cotangent function starts high (you can imagine it coming from positive infinity), goes through a point in middle and ends low (going to negative infinity).