Precalculus Examples

$f (x) = \tan (\frac{1}{2} \cdot (x + \frac{π}{6}))$

Step 1

Simplify $\tan (\frac{1}{2} \cdot (x + \frac{π}{6}))$ .

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Step 1.1

Apply the distributive property.

$y = \tan (\frac{1}{2} x + \frac{1}{2} \cdot \frac{π}{6})$

Step 1.2

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

$y = \tan (\frac{x}{2} + \frac{1}{2} \cdot \frac{π}{6})$

Step 1.3

Multiply $\frac{1}{2} \cdot \frac{π}{6}$ .

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Step 1.3.1

Multiply $\frac{1}{2}$ by $\frac{π}{6}$ .

$y = \tan (\frac{x}{2} + \frac{π}{2 \cdot 6})$

Step 1.3.2

Multiply $2$ by $6$ .

$y = \tan (\frac{x}{2} + \frac{π}{12})$

Step 2

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 $y = \tan (\frac{x}{2} + \frac{π}{12})$ . 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 $y = \tan (\frac{x}{2} + \frac{π}{12})$ .

$\frac{x}{2} + \frac{π}{12} = - \frac{π}{2}$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $\frac{π}{12}$ from both sides of the equation.

$\frac{x}{2} = - \frac{π}{2} - \frac{π}{12}$

Step 3.1.2

To write $- \frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{x}{2} = - \frac{π}{2} \cdot \frac{6}{6} - \frac{π}{12}$

Step 3.1.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1.3.1

Multiply $\frac{π}{2}$ by $\frac{6}{6}$ .

$\frac{x}{2} = - \frac{π \cdot 6}{2 \cdot 6} - \frac{π}{12}$

Step 3.1.3.2

Multiply $2$ by $6$ .

$\frac{x}{2} = - \frac{π \cdot 6}{12} - \frac{π}{12}$

Step 3.1.4

Combine the numerators over the common denominator.

$\frac{x}{2} = \frac{- π \cdot 6 - π}{12}$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Multiply $6$ by $- 1$ .

$\frac{x}{2} = \frac{- 6 π - π}{12}$

Step 3.1.5.2

Subtract $π$ from $- 6 π$ .

$\frac{x}{2} = \frac{- 7 π}{12}$

Step 3.1.6

Move the negative in front of the fraction.

$\frac{x}{2} = - \frac{7 π}{12}$

Step 3.2

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (- \frac{7 π}{12})$

Step 3.3

Simplify both sides of the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Cancel the common factor of $2$ .

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Step 3.3.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 (- \frac{7 π}{12})$

Step 3.3.1.1.2

Rewrite the expression.

$x = 2 (- \frac{7 π}{12})$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $2 (- \frac{7 π}{12})$ .

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Step 3.3.2.1.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1.1

Move the leading negative in $- \frac{7 π}{12}$ into the numerator.

$x = 2 \frac{- 7 π}{12}$

Step 3.3.2.1.1.2

Factor $2$ out of $12$ .

$x = 2 \frac{- 7 π}{2 (6)}$

Step 3.3.2.1.1.3

Cancel the common factor.

$x = 2 \frac{- 7 π}{2 \cdot 6}$

Step 3.3.2.1.1.4

Rewrite the expression.

$x = \frac{- 7 π}{6}$

Step 3.3.2.1.2

Move the negative in front of the fraction.

$x = - \frac{7 π}{6}$

Step 4

Set the inside of the tangent function $\frac{x}{2} + \frac{π}{12}$ equal to $\frac{π}{2}$ .

$\frac{x}{2} + \frac{π}{12} = \frac{π}{2}$

Step 5

Solve for $x$ .

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Step 5.1

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

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Step 5.1.1

Subtract $\frac{π}{12}$ from both sides of the equation.

$\frac{x}{2} = \frac{π}{2} - \frac{π}{12}$

Step 5.1.2

To write $\frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{x}{2} = \frac{π}{2} \cdot \frac{6}{6} - \frac{π}{12}$

Step 5.1.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.1.3.1

Multiply $\frac{π}{2}$ by $\frac{6}{6}$ .

$\frac{x}{2} = \frac{π \cdot 6}{2 \cdot 6} - \frac{π}{12}$

Step 5.1.3.2

Multiply $2$ by $6$ .

$\frac{x}{2} = \frac{π \cdot 6}{12} - \frac{π}{12}$

Step 5.1.4

Combine the numerators over the common denominator.

$\frac{x}{2} = \frac{π \cdot 6 - π}{12}$

Step 5.1.5

Simplify the numerator.

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Step 5.1.5.1

Move $6$ to the left of $π$ .

$\frac{x}{2} = \frac{6 \cdot π - π}{12}$

Step 5.1.5.2

Subtract $π$ from $6 π$ .

$\frac{x}{2} = \frac{5 π}{12}$

Step 5.2

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \frac{5 π}{12}$

Step 5.3

Simplify both sides of the equation.

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Step 5.3.1

Simplify the left side.

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Step 5.3.1.1

Cancel the common factor of $2$ .

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Step 5.3.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \frac{5 π}{12}$

Step 5.3.1.1.2

Rewrite the expression.

$x = 2 \frac{5 π}{12}$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Cancel the common factor of $2$ .

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Step 5.3.2.1.1

Factor $2$ out of $12$ .

$x = 2 \frac{5 π}{2 (6)}$

Step 5.3.2.1.2

Cancel the common factor.

$x = 2 \frac{5 π}{2 \cdot 6}$

Step 5.3.2.1.3

Rewrite the expression.

$x = \frac{5 π}{6}$

Step 6

The basic period for $y = \tan (\frac{x}{2} + \frac{π}{12})$ will occur at $(- \frac{7 π}{6}, \frac{5 π}{6})$ , where $- \frac{7 π}{6}$ and $\frac{5 π}{6}$ are vertical asymptotes.

$(- \frac{7 π}{6}, \frac{5 π}{6})$

Step 7

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

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Step 7.1

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

$\frac{π}{\frac{1}{2}}$

Step 7.2

Multiply the numerator by the reciprocal of the denominator.

$π \cdot 2$

Step 7.3

Move $2$ to the left of $π$ .

$2 π$

Step 8

The vertical asymptotes for $y = \tan (\frac{x}{2} + \frac{π}{12})$ occur at $- \frac{7 π}{6}$ , $\frac{5 π}{6}$ , and every $x = - \frac{7 π}{6} + 2 π n$ , where $n$ is an integer.

$x = - \frac{7 π}{6} + 2 π n$

Step 9

Tangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = - \frac{7 π}{6} + 2 π n$ where $n$ is an integer

Step 10