Graph two periods of the given cosecant or secant function.


y = sec x/2

Verified step by step guidance

1. Identify the given function: \(y = \sec\left(\frac{x}{2}\right)\). This is a secant function with the argument \(\frac{x}{2}\) inside the secant.
2. Recall the period of the basic secant function \(y = \sec x\) is \(2\pi\). For \(y = \sec(bx)\), the period is given by \(\frac{2\pi}{|b|}\). Here, \(b = \frac{1}{2}\), so the period is \(\frac{2\pi}{\frac{1}{2}} = 4\pi\).
3. Since the period is \(4\pi\), two periods will span \(2 \times 4\pi = 8\pi\). So, you will graph the function from \(x = 0\) to \(x = 8\pi\) (or any interval of length \(8\pi\)) to show two full periods.
4. Identify the vertical asymptotes of \(y = \sec\left(\frac{x}{2}\right)\), which occur where the cosine in the denominator is zero: \(\cos\left(\frac{x}{2}\right) = 0\). Solve for \(x\) to find these asymptotes within the interval of two periods.
5. Plot key points by evaluating \(y = \sec\left(\frac{x}{2}\right)\) at values where \(\cos\left(\frac{x}{2}\right)\) is \(\pm 1\) (maxima and minima of secant), and sketch the graph between the asymptotes to complete two periods.

Key Concepts

Understanding the Secant Function

The secant function, sec(x), is the reciprocal of the cosine function, defined as sec(x) = 1/cos(x). It is undefined where cosine equals zero, leading to vertical asymptotes in its graph. Recognizing these properties helps in sketching the secant curve accurately.

Effect of Horizontal Scaling on Trigonometric Functions

In the function y = sec(x/2), the input to secant is scaled by a factor of 1/2, which stretches the period horizontally. Since the standard period of sec(x) is 2π, dividing x by 2 doubles the period to 4π. This affects how many cycles appear over a given interval.

Graphing Periodic Functions with Asymptotes

Graphing secant involves plotting its periodic behavior along with vertical asymptotes where the function is undefined. Identifying these asymptotes, typically where cosine is zero, and marking key points such as maxima and minima, is essential for accurately sketching two full periods.