Problem 35 - Precalculus Essentials

Question

Graph two periods of the given cosecant or secant function. $$y=\sec \frac{x}{3}$$

Step-by-Step Solution

Answer

The graph of \(y=\sec \frac{x}{3}\) is a series of U-shaped curves, each approaching vertical asymptotes at \(x=-5\pi\), \(x=-\pi\), \(x=\pi\), \(x=5\pi\), \(x=11\pi\), and \(x=17\pi\) for two periods. The curves pass through the maxima and minima of the cosine graph which are its x-intercepts. These are located halfway between the vertical asymptotes. The y-intercept is at the point \((0, 1)\)

Step 1: Define the Cosine function

Since the function given is \(y=\sec \frac{x}{3}\), we can define the corresponding cosine function as \(y=\cos \frac{x}{3}\). This will allow us to find the points at which the cosine function crosses the x-axis and hence determine the vertical asymptotes for the graph of the secant function.

Step 2: Identify the period of the function

Next, calculate the period of the function. The period of the standard secant function \(y = sec(x)\) is \(2\pi\). In this case, the input to the secant function is divided by 3, such that there are 3 cycles in where there was one before. Hence, the period of \(y = sec(\frac{x}{3})\) is \(6\pi\).

Step 3: Draw the graph of the cosine function

Graph the cosine function, \(y=\cos \frac{x}{3}\), for two periods, taking account of its period, which is \(6\pi\).

Step 4: Identify the vertical asymptotes

To find the vertical asymptotes of the secant graph, find the x-values for which the cosine function is equal to zero. There are vertical asymptotes at these x-values. The vertical asymptotes for one period of the secant function are at \(x=\pi\) and \(x=5\pi\). Therefore, the vertical asymptotes for two periods will also occur at \(x=-5\pi\), \(x=-\pi\), \(x=\pi\), \(x=5\pi\), \(x=11\pi\), and \(x=17\pi\). Any secant graph is undefined at these points.

Step 5: Draw the secant function

The secant graph will approach asymptotes, passing through maxima and minima of the cosine graph which are its x-intercepts. The x-intercepts are consistently located halfway between the vertical asymptotes. The y-intercept is at \((0, 1)\).