Problem 36 - Precalculus Essentials

Question

Graph two periods of the given cosecant or secant function. $$y=\sec \frac{x}{2}$$

Step-by-Step Solution

Answer

To graph the secant function \( y = \sec(\frac{x}{2}) \), start by graphing the cosine function \( y = \cos(\frac{x}{2}) \), identify where the cosine function equals 0 and then graph the secant function which intersects at the minima and maxima of the cosine function and goes towards infinity at the cosine's zeros.

Step 1: Plot the Parent Function

Since the secant is the reciprocal of the cosine we will begin by plotting two periods of the function \( y = \cos(\frac{x}{2}) \). The period of \( \cos(x) \) is \( 2\pi \) but since the function is \( \cos(\frac{x}{2}) \), then the period becomes \( 4\pi \) from \( -2\pi \) to \( 2\pi \). Sketch the minima and maxima and zeros of the cosine function.

Step 2: Identify the Asymptotes

The secant function has vertical asymptotes at points where the cosine function equals zero. This is because the secant function is undefined at those points. For every zero in the cosine function, draw a dashed line, representing the asymptote.

Step 3: Plot the Secant Function

Plot the secant function. At points where the cosine function has local maxima or minima, the secant function will intersect. Repeat this process for all maxima and minima.

Step 4: Sketch Two Periods

Run a curve through the intersections of the secant and the cosine functions and along the asymptotes. Then you will have a graph of two periods of the function \( y = \sec(\frac{x}{2}) \) from \( -2\pi \) to \( 2\pi \).