select the graph of y = sec x.

Explanation:

Step1: Recall secant - cosine relationship

Since $\sec x=\frac{1}{\cos x}$, the secant function has vertical asymptotes where $\cos x = 0$. The values of $x$ for which $\cos x=0$ are $x=\frac{\pi}{2}+n\pi$, where $n\in\mathbb{Z}$.

Step2: Analyze the sign and range

When $\cos x$ is positive and close to 0, $\sec x$ approaches positive infinity, and when $\cos x$ is negative and close to 0, $\sec x$ approaches negative infinity. Also, $|\sec x|\geq1$ for all $x$ in its domain.

Step3: Evaluate the graphs

For $y = \sec x$, at $x = 0$, $\sec(0)=\frac{1}{\cos(0)} = 1$, at $x=\pi$, $\sec(\pi)=\frac{1}{\cos(\pi)}=- 1$. Graph A has the correct behavior with vertical asymptotes at $x=\frac{\pi}{2},\frac{3\pi}{2}$ and the correct values at $x = 0$ and $x=\pi$.

Answer:

A.