The Graph of the Secant Function

On $[0,2\pi]\setminus\left\{\tfrac{\pi}2,\tfrac{3\pi}2\right\}$: in Quadrants I and IV (i.e. $-\tfrac{\pi}2<x<\tfrac{\pi}2$), $\sec x$ is positive; in Quadrants II and III ($\tfrac{\pi}2<x<\tfrac{3\pi}2$), $\sec x$ is negative. Piece by piece: rises from $1$ to $+\infty$ over $\left[0,\tfrac{\pi}2\right)$; rises from $-\infty$ to $-1$ over $\left(\tfrac{\pi}2,\pi\right]$; falls from $-1$ to $-\infty$ over $\left[\pi,\tfrac{3\pi}2\right)$; and falls from $+\infty$ to $1$ over $\left(\tfrac{3\pi}2,2\pi\right]$. This is continuous except at $x=\tfrac{\pi}2,\tfrac{3\pi}2$, and produces Fig. 4.23: a U-shaped branch above $y=1$ centred near $x=0,2\pi$, and an inverted-U branch below $y=-1$ centred near $x=\pi$.

Being periodic with period $2\pi$, this same segment repeats over every subsequent interval of length $2\pi$, producing the entire graph Fig. 4.24 with a vertical asymptote at every odd multiple of $\tfrac{\pi}2$. …