Precalculus Examples

$f (x) = \csc (\frac{1}{2} x - \frac{π}{4})$

Step 1

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

$y = \csc (\frac{x}{2} - \frac{π}{4})$

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 $y = \csc (\frac{x}{2} - \frac{π}{4})$ . 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 $y = \csc (\frac{x}{2} - \frac{π}{4})$ .

$\frac{x}{2} - \frac{π}{4} = 0$

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

Add $\frac{π}{4}$ to both sides of the equation.

$\frac{x}{2} = \frac{π}{4}$

Step 3.2

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \frac{π}{4}$

Step 3.3

Simplify both sides of the equation.

Tap for more steps...

Step 3.3.1

Simplify the left side.

Tap for more steps...

Step 3.3.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \frac{π}{4}$

Step 3.3.1.1.2

Rewrite the expression.

$x = 2 \frac{π}{4}$

Step 3.3.2

Simplify the right side.

Tap for more steps...

Step 3.3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.2.1.1

Factor $2$ out of $4$ .

$x = 2 \frac{π}{2 (2)}$

Step 3.3.2.1.2

Cancel the common factor.

$x = 2 \frac{π}{2 \cdot 2}$

Step 3.3.2.1.3

Rewrite the expression.

$x = \frac{π}{2}$

Step 4

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

$\frac{x}{2} - \frac{π}{4} = 2 π$

Step 5

Solve for $x$ .

Tap for more steps...

Step 5.1

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

Tap for more steps...

Step 5.1.1

Add $\frac{π}{4}$ to both sides of the equation.

$\frac{x}{2} = 2 π + \frac{π}{4}$

Step 5.1.2

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

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

Step 5.1.3

Combine $2 π$ and $\frac{4}{4}$ .

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

Step 5.1.4

Combine the numerators over the common denominator.

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

Step 5.1.5

Simplify the numerator.

Tap for more steps...

Step 5.1.5.1

Multiply $4$ by $2$ .

$\frac{x}{2} = \frac{8 π + π}{4}$

Step 5.1.5.2

Add $8 π$ and $π$ .

$\frac{x}{2} = \frac{9 π}{4}$

Step 5.2

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \frac{9 π}{4}$

Step 5.3

Simplify both sides of the equation.

Tap for more steps...

Step 5.3.1

Simplify the left side.

Tap for more steps...

Step 5.3.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.3.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \frac{9 π}{4}$

Step 5.3.1.1.2

Rewrite the expression.

$x = 2 \frac{9 π}{4}$

Step 5.3.2

Simplify the right side.

Tap for more steps...

Step 5.3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.3.2.1.1

Factor $2$ out of $4$ .

$x = 2 \frac{9 π}{2 (2)}$

Step 5.3.2.1.2

Cancel the common factor.

$x = 2 \frac{9 π}{2 \cdot 2}$

Step 5.3.2.1.3

Rewrite the expression.

$x = \frac{9 π}{2}$

Step 6

The basic period for $y = \csc (\frac{x}{2} - \frac{π}{4})$ will occur at $(\frac{π}{2}, \frac{9 π}{2})$ , where $\frac{π}{2}$ and $\frac{9 π}{2}$ are vertical asymptotes.

$(\frac{π}{2}, \frac{9 π}{2})$

Step 7

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

Tap for more steps...

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 $y = \csc (\frac{x}{2} - \frac{π}{4})$ occur at $\frac{π}{2}$ , $\frac{9 π}{2}$ , and every $x = \frac{π}{2} + 2 π n$ , where $n$ is an integer. This is half of the period.

$x = \frac{π}{2} + 2 π n$

Step 9

Cosecant only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = \frac{π}{2} + 2 π n$ where $n$ is an integer

Step 10