Problem 46 - Precalculus Essentials

Question

Graph two periods of each function. $$y=2 \cot \left(x+\frac{\pi}{6}\right)-1$$

Step-by-Step Solution

Answer

The graph of `y = 2 cot(x + \pi/6) - 1` will show a repeated cotangent curve, starting from the phase shift of \(-\pi/6\), going upwards through the mid-point between asymptotes at \( y = -1 \), and continuing to the next asymptote. This pattern repeats every \( \pi \) units along the x-axis, and will be shifted downwards by 1 unit.

Step 1: Identifying Key Elements

We start by identifying key elements from the given function. The function is y = 2 cot(x + \pi/6) - 1. The amplitude is not applicable here. The period can be found by dividing \( \pi \) by the absolute value of 'b'. From the function, 'b' is 1, so the period is \( \pi \ / \ |1| \ = \ \pi \). The phase shift can be calculated as '-c/b'. Here, 'c' is \(\pi/6\) and 'b' is 1, so the phase shift is \(-\pi/6\). The vertical shift 'd' is -1, which means the function is shifted downward by 1 unit.

Step 2: Plotting the Asymptotes

We know that cotangent function has vertical asymptotes wherever the function is undefined i.e., at every multiplier of \( \pi \ + \ phase \ shift \). So the asymptotes will be located at multiples of \( \pi \ + \ -\pi/6 \). Therefore asymptotes will be located at \( -\pi/6 \ , \ 5\pi/6 \ , \ 3\pi/2 \)... Plot these asymptotes on the graph first.

Step 3: Plotting the Key Points

Plot the mid-point between each set of asymptotes. These represent the 'zeroes' of the cotangent function, where the function intersects the x-axis. It will be at the middle points between the asymptotes i.e., at \( \pi/6 \ , \ 4\pi/6 \ , \ 7\pi/6 \)..., and the corresponding 'y' values will be equal to the value of vertical shift, which is '-1'. Plot these on the graph. Next, plot more points by taking the middle of each half period. The corresponding 'y' values will vary depending on whether it is a maximum or minimum. In our case, all maxima points will take the value 1, and minima will take -3.

Step 4: Graphing the Function

Start at the point representing the phase shift, and make sure your function approaches asymptotes as x approaches the x-values of the asymptotes. The function will stay in-between two vertical asymptotes for every period. Make sure the function is periodic over the distance of the period calculated in step 1, and that it covers two full periods.