Precalculus Examples

$y = \frac{1}{4} \cdot \csc (x + \frac{π}{4})$

Step 1

Combine $\frac{1}{4}$ and $\csc (x + \frac{π}{4})$ .

$y = \frac{\csc (x + \frac{π}{4})}{4}$

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 $y = \frac{\csc (x + \frac{π}{4})}{4}$ . 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 $y = \frac{\csc (x + \frac{π}{4})}{4}$ .

$x + \frac{π}{4} = 0$

Step 3

Subtract $\frac{π}{4}$ from both sides of the equation.

$x = - \frac{π}{4}$

Step 4

Set the inside of the cosecant function $x + \frac{π}{4}$ equal to $2 π$ .

$x + \frac{π}{4} = 2 π$

Step 5

Move all terms not containing $x$ to the right side of the equation.

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Step 5.1

Subtract $\frac{π}{4}$ from both sides of the equation.

$x = 2 π - \frac{π}{4}$

Step 5.2

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = 2 π \cdot \frac{4}{4} - \frac{π}{4}$

Step 5.3

Combine $2 π$ and $\frac{4}{4}$ .

$x = \frac{2 π \cdot 4}{4} - \frac{π}{4}$

Step 5.4

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - π}{4}$

Step 5.5

Simplify the numerator.

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Step 5.5.1

Multiply $4$ by $2$ .

$x = \frac{8 π - π}{4}$

Step 5.5.2

Subtract $π$ from $8 π$ .

$x = \frac{7 π}{4}$

Step 6

The basic period for $y = \frac{\csc (x + \frac{π}{4})}{4}$ will occur at $(- \frac{π}{4}, \frac{7 π}{4})$ , where $- \frac{π}{4}$ and $\frac{7 π}{4}$ are vertical asymptotes.

$(- \frac{π}{4}, \frac{7 π}{4})$

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

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

$\frac{2 π}{1}$

Step 7.2

Divide $2 π$ by $1$ .

$2 π$

Step 8

The vertical asymptotes for $y = \frac{\csc (x + \frac{π}{4})}{4}$ occur at $- \frac{π}{4}$ , $\frac{7 π}{4}$ , and every $x = - \frac{π}{4} + π n$ , where $n$ is an integer. This is half of the period.

$x = - \frac{π}{4} + π n$

Step 9

Cosecant only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = - \frac{π}{4} + π n$ where $n$ is an integer

Step 10