Find the Asymptotes y=tan(x+pi/6)

$y = \tan (x + \frac{π}{6})$

For any $y = \tan (x)$ , vertical asymptotes occur at $x = \frac{π}{2} + n π$ , where $n$ is an integer. Use the basic period for $y = \tan (x)$ , $(- \frac{π}{2}, \frac{π}{2})$ , to find the vertical asymptotes for $y = \tan (x + \frac{π}{6})$ . Set the inside of the tangent function, $b x + c$ , for $y = a \tan (b x + c) + d$ equal to $- \frac{π}{2}$ to find where the vertical asymptote occurs for $y = \tan (x + \frac{π}{6})$ .

$x + \frac{π}{6} = - \frac{π}{2}$

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

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Subtract $\frac{π}{6}$ from both sides of the equation.

$x = - \frac{π}{2} - \frac{π}{6}$

Simplify the right side of the equation.

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To write $\frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = - \frac{π}{2} \cdot \frac{3}{3} - \frac{π}{6}$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$x = - \frac{π \cdot 3}{2 \cdot 3} - \frac{π}{6}$

Multiply $2$ by $3$ .

$x = - \frac{π \cdot 3}{6} - \frac{π}{6}$

Combine the numerators over the common denominator.

$x = \frac{- π \cdot 3 - π}{6}$

Simplify the numerator.

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Multiply $3$ by $- 1$ .

$x = \frac{- 3 π - π}{6}$

Subtract $π$ from $- 3 π$ .

$x = \frac{- 4 π}{6}$

Reduce the expression by cancelling the common factors.

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Cancel the common factor of $- 4$ and $6$ .

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Factor $2$ out of $- 4 π$ .

$x = \frac{2 (- 2 π)}{6}$

Cancel the common factors.

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Factor $2$ out of $6$ .

$x = \frac{2 (- 2 π)}{2 (3)}$

Cancel the common factor.

$x = \frac{2 (- 2 π)}{2 \cdot 3}$

Rewrite the expression.

$x = \frac{- 2 π}{3}$

Move the negative in front of the fraction.

$x = - \frac{2 π}{3}$

Set the inside of the tangent function $x + \frac{π}{6}$ equal to $\frac{π}{2}$ .

$x + \frac{π}{6} = \frac{π}{2}$

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

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Subtract $\frac{π}{6}$ from both sides of the equation.

$x = \frac{π}{2} - \frac{π}{6}$

Simplify the right side of the equation.

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To write $\frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = \frac{π}{2} \cdot \frac{3}{3} - \frac{π}{6}$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$x = \frac{π \cdot 3}{2 \cdot 3} - \frac{π}{6}$

Multiply $2$ by $3$ .

$x = \frac{π \cdot 3}{6} - \frac{π}{6}$

Combine the numerators over the common denominator.

$x = \frac{π \cdot 3 - π}{6}$

Simplify the numerator.

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Move $3$ to the left of $π$ .

$x = \frac{3 \cdot π - π}{6}$

Subtract $π$ from $3 π$ .

$x = \frac{2 π}{6}$

Cancel the common factor of $2$ and $6$ .

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Factor $2$ out of $2 π$ .

$x = \frac{2 (π)}{6}$

Cancel the common factors.

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Factor $2$ out of $6$ .

$x = \frac{2 π}{2 \cdot 3}$

Cancel the common factor.

$x = \frac{2 π}{2 \cdot 3}$

Rewrite the expression.

$x = \frac{π}{3}$

The basic period for $y = \tan (x + \frac{π}{6})$ will occur at $(- \frac{2 π}{3}, \frac{π}{3})$ , where $- \frac{2 π}{3}$ and $\frac{π}{3}$ are vertical asymptotes.

$(- \frac{2 π}{3}, \frac{π}{3})$

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

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

$\frac{π}{1}$

Divide $π$ by $1$ .

$π$

The vertical asymptotes for $y = \tan (x + \frac{π}{6})$ occur at $- \frac{2 π}{3}$ , $\frac{π}{3}$ , and every $x = - \frac{2 π}{3} + π n$ , where $n$ is an integer.

$x = - \frac{2 π}{3} + π n$

Tangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = - \frac{2 π}{3} + π n$ where $n$ is an integer

Find the Asymptotes y=tan(x+pi/6)

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