Precalculus Examples

$y = \csc (- \frac{1}{2} x)$

Step 1

Simplify $\csc (- \frac{1}{2} x)$ .

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Step 1.1

Combine $x$ and $\frac{1}{2}$ .

$y = \csc (- \frac{x}{2})$

Step 1.2

Since $\csc (- \frac{x}{2})$ is an odd function, rewrite $\csc (- \frac{x}{2})$ as $- \csc (\frac{x}{2})$ .

$y = - \csc (\frac{x}{2})$

Step 2

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 $y = - \csc (\frac{x}{2})$ . 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 $y = - \csc (\frac{x}{2})$ .

$\frac{x}{2} = 0$

Step 3

Set the numerator equal to zero.

$x = 0$

Step 4

Set the inside of the cosecant function $\frac{x}{2}$ equal to $2 π$ .

$\frac{x}{2} = 2 π$

Step 5

Solve for $x$ .

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Step 5.1

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (2 π)$

Step 5.2

Simplify both sides of the equation.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Cancel the common factor of $2$ .

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Step 5.2.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 (2 π)$

Step 5.2.1.1.2

Rewrite the expression.

$x = 2 (2 π)$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Multiply $2$ by $2$ .

$x = 4 π$

Step 6

The basic period for $y = - \csc (\frac{x}{2})$ will occur at $(0, 4 π)$ , where $0$ and $4 π$ are vertical asymptotes.

$(0, 4 π)$

Step 7

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

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Step 7.1

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Step 7.2

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 7.3

Multiply $2$ by $2$ .

$4 π$

Step 8

The vertical asymptotes for $y = - \csc (\frac{x}{2})$ occur at $0$ , $4 π$ , and every $2 π n$ , where $n$ is an integer. This is half of the period.

$x = 2 π n$

Step 9

Cosecant only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = 2 π n$ where $n$ is an integer

Step 10