Graph y=sec(x)

For any ${\displaystyle y=\sec \left(x\right)}$, vertical asymptotes occur at ${\displaystyle x=\frac{\pi }{2}+n\pi }$, where ${\displaystyle n}$ is an integer. Use the basic period for ${\displaystyle y=sec\left(x\right)}$, ${\displaystyle (-\frac{\pi }{2},\frac{3\pi }{2})}$, to find the vertical asymptotes for ${\displaystyle y=\sec \left(x\right)}$. Set the inside of the secant function, ${\displaystyle bx+c}$, for ${\displaystyle y=a\sec (bx+c)+d}$ equal to ${\displaystyle -\frac{\pi }{2}}$ to find where the vertical asymptote occurs for ${\displaystyle y=\sec \left(x\right)}$.

Set the inside of the secant function ${\displaystyle x}$ equal to ${\displaystyle \frac{3\pi }{2}}$.

The basic period for ${\displaystyle y=\sec \left(x\right)}$ will occur at ${\displaystyle (-\frac{\pi }{2},\frac{3\pi }{2})}$, where ${\displaystyle -\frac{\pi }{2}}$ and ${\displaystyle \frac{3\pi }{2}}$ are vertical asymptotes.

Find the period ${\displaystyle \frac{2\pi }{\left|b\right|}}$ to find where the vertical asymptotes exist. Vertical asymptotes occur every half period.

The vertical asymptotes for ${\displaystyle y=\sec \left(x\right)}$ occur at ${\displaystyle -\frac{\pi }{2}}$, ${\displaystyle \frac{3\pi }{2}}$, and every ${\displaystyle \pi n}$, where ${\displaystyle n}$ is an integer. This is half of the period.

There are only vertical asymptotes for secant and cosecant functions.

Vertical Asymptotes: ${\displaystyle x=\frac{3\pi }{2}+\pi n}$ for any integer ${\displaystyle n}$

No Horizontal Asymptotes

No Oblique Asymptotes

The period of the function can be calculated using ${\displaystyle \frac{2\pi }{\left|b\right|}}$.

Replace ${\displaystyle b}$ with ${\displaystyle 1}$ in the formula for period.

The absolute value is the distance between a number and zero. The distance between ${\displaystyle 0}$ and ${\displaystyle 1}$ is ${\displaystyle 1}$.

Divide ${\displaystyle 2\pi }$ by ${\displaystyle 1}$.

The phase shift of the function can be calculated from ${\displaystyle \frac{c}{b}}$.

Phase Shift: ${\displaystyle \frac{c}{b}}$

Replace the values of ${\displaystyle c}$ and ${\displaystyle b}$ in the equation for phase shift.

Phase Shift: ${\displaystyle \frac{0}{1}}$

Divide ${\displaystyle 0}$ by ${\displaystyle 1}$.

Phase Shift: ${\displaystyle 0}$