Precalculus Examples

$f (x) = \csc (x)$

For any $y = \csc (x)$ , vertical asymptotes occur at $x = n π$ , where $n$ is an integer. Use the basic period for $y = c s c (x)$ , $(0, 2 π)$ , to find the vertical asymptotes for $y = \csc (x)$ . Set the inside of the cosecant function, $b x + c$ , for $y = a \csc (b x + c) + d$ equal to $0$ to find where the vertical asymptote occurs for $y = \csc (x)$ .

$x = 0$

Set the inside of the cosecant function $x$ equal to $2 π$ .

$x = 2 π$

The basic period for $y = \csc (x)$ will occur at $(0, 2 π)$ , where $0$ and $2 π$ are vertical asymptotes.

$(0, 2 π)$

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

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The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

The vertical asymptotes for $y = \csc (x)$ occur at $0$ , $2 π$ , and every $π n$ , where $n$ is an integer. This is half of the period.

$π n$

There are only vertical asymptotes for secant and cosecant functions.

Vertical Asymptotes: $x = π n$ for any integer $n$

No Horizontal Asymptotes

No Oblique Asymptotes