Problem 30 - Precalculus Essentials

Question

Graph two periods of the given cosecant or secant function. $$y=2 \csc x$$

Step-by-Step Solution

Answer

The graph of the function \(y = 2 \csc{x}\) for two periods consists of a series of 'U'-shaped curves, centered at \(y = 2\) and \(y = -2\), with vertical asymptotes at integer multiples of \(\pi\).

Step 1: Identify the Key Features

For \(y=csc(x)\), the period is \(2\pi\), the amplitude is undefined, and it has vertical asymptotes at \(x = 0\), \(x = \pi\), \(x = 2\pi\), etc. There are maximum and minimum values at \(x = \pi/2 + n\pi\), where \(n\) is any integer. The modifications due to the 2 coefficient will only affect the maximum and minimum values of the function.

Step 2: Draw the Vertical Asymptotes

Draw vertical dotted lines (representing the asymptotes) at \(x = 0\), \(x = \pi\), \(x = 2\pi\), \(x = -\pi\), and \(x = -2\pi\) to cover two periods of the function.

Step 3: Plot the Maximum and Minimum Values

Mark points at \(x = \pi/2\), \(x = 3\pi/2\), \(x = 5\pi/2\), \(x = -\pi/2\), and \(x = -3\pi/2\). Due to the 2 coefficient, these points will be at \(y = 2\) and \(y = -2\) instead of \(y = 1\) and \(y = -1\).

Step 4: Draw the Graph

Draw a curve passing through each of the plotted points and 'approaching' the vertical asymptotes, being careful to remember that the function is positive for \(0 < x < \pi\) and negative for \(-\pi < x < 0\). Repeat this for the second period of the function.