Precalculus Examples

$f (x) = 2 \cot (x - \frac{π}{4})$

Step 1

For any $y = \cot (x)$ , vertical asymptotes occur at $x = n π$ , where $n$ is an integer. Use the basic period for $y = \cot (x)$ , $(0, π)$ , to find the vertical asymptotes for $y = 2 \cot (x - \frac{π}{4})$ . Set the inside of the cotangent function, $b x + c$ , for $y = a \cot (b x + c) + d$ equal to $0$ to find where the vertical asymptote occurs for $y = 2 \cot (x - \frac{π}{4})$ .

$x - \frac{π}{4} = 0$

Step 2

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

$x = \frac{π}{4}$

Step 3

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

$x - \frac{π}{4} = π$

Step 4

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

Tap for more steps...

Step 4.1

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

$x = π + \frac{π}{4}$

Step 4.2

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

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

Step 4.3

Combine $π$ and $\frac{4}{4}$ .

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

Step 4.4

Combine the numerators over the common denominator.

$x = \frac{π \cdot 4 + π}{4}$

Step 4.5

Simplify the numerator.

Tap for more steps...

Step 4.5.1

Move $4$ to the left of $π$ .

$x = \frac{4 \cdot π + π}{4}$

Step 4.5.2

Add $4 π$ and $π$ .

$x = \frac{5 π}{4}$

Step 5

The basic period for $y = 2 \cot (x - \frac{π}{4})$ will occur at $(\frac{π}{4}, \frac{5 π}{4})$ , where $\frac{π}{4}$ and $\frac{5 π}{4}$ are vertical asymptotes.

$(\frac{π}{4}, \frac{5 π}{4})$

Step 6

Find the period $\frac{π}{| b |}$ to find where the vertical asymptotes exist.

Tap for more steps...

Step 6.1

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

$\frac{π}{1}$

Step 6.2

Divide $π$ by $1$ .

$π$

Step 7

The vertical asymptotes for $y = 2 \cot (x - \frac{π}{4})$ occur at $\frac{π}{4}$ , $\frac{5 π}{4}$ , and every $x = \frac{π}{4} + π n$ , where $n$ is an integer.

$x = \frac{π}{4} + π n$

Step 8

Cotangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

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

Step 9