Precalculus Examples

$y = \tan (\frac{3}{4} x)$

Step 1

Combine $\frac{3}{4}$ and $x$ .

$y = \tan (\frac{3 x}{4})$

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 $y = \tan (\frac{3 x}{4})$ . 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 $y = \tan (\frac{3 x}{4})$ .

$\frac{3 x}{4} = - \frac{π}{2}$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides of the equation by $\frac{4}{3}$ .

$\frac{4}{3} \cdot \frac{3 x}{4} = \frac{4}{3} (- \frac{π}{2})$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $\frac{4}{3} \cdot \frac{3 x}{4}$ .

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Step 3.2.1.1.1

Cancel the common factor of $4$ .

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Step 3.2.1.1.1.1

Cancel the common factor.

$\frac{4}{3} \cdot \frac{3 x}{4} = \frac{4}{3} (- \frac{π}{2})$

Step 3.2.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 x) = \frac{4}{3} (- \frac{π}{2})$

Step 3.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.2.1.1.2.1

Factor $3$ out of $3 x$ .

$\frac{1}{3} (3 (x)) = \frac{4}{3} (- \frac{π}{2})$

Step 3.2.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 x) = \frac{4}{3} (- \frac{π}{2})$

Step 3.2.1.1.2.3

Rewrite the expression.

$x = \frac{4}{3} (- \frac{π}{2})$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $\frac{4}{3} (- \frac{π}{2})$ .

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Step 3.2.2.1.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1.1

Move the leading negative in $- \frac{π}{2}$ into the numerator.

$x = \frac{4}{3} \cdot \frac{- π}{2}$

Step 3.2.2.1.1.2

Factor $2$ out of $4$ .

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

Step 3.2.2.1.1.3

Cancel the common factor.

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

Step 3.2.2.1.1.4

Rewrite the expression.

$x = \frac{2}{3} (- π)$

Step 3.2.2.1.2

Combine $\frac{2}{3}$ and $π$ .

$x = - \frac{2 π}{3}$

Step 4

Set the inside of the tangent function $\frac{3 x}{4}$ equal to $\frac{π}{2}$ .

$\frac{3 x}{4} = \frac{π}{2}$

Step 5

Solve for $x$ .

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Step 5.1

Multiply both sides of the equation by $\frac{4}{3}$ .

$\frac{4}{3} \cdot \frac{3 x}{4} = \frac{4}{3} \cdot \frac{π}{2}$

Step 5.2

Simplify both sides of the equation.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Simplify $\frac{4}{3} \cdot \frac{3 x}{4}$ .

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Step 5.2.1.1.1

Cancel the common factor of $4$ .

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Step 5.2.1.1.1.1

Cancel the common factor.

$\frac{4}{3} \cdot \frac{3 x}{4} = \frac{4}{3} \cdot \frac{π}{2}$

Step 5.2.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 x) = \frac{4}{3} \cdot \frac{π}{2}$

Step 5.2.1.1.2

Cancel the common factor of $3$ .

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Step 5.2.1.1.2.1

Factor $3$ out of $3 x$ .

$\frac{1}{3} (3 (x)) = \frac{4}{3} \cdot \frac{π}{2}$

Step 5.2.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 x) = \frac{4}{3} \cdot \frac{π}{2}$

Step 5.2.1.1.2.3

Rewrite the expression.

$x = \frac{4}{3} \cdot \frac{π}{2}$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $\frac{4}{3} \cdot \frac{π}{2}$ .

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Step 5.2.2.1.1

Cancel the common factor of $2$ .

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Step 5.2.2.1.1.1

Factor $2$ out of $4$ .

$x = \frac{2 (2)}{3} \cdot \frac{π}{2}$

Step 5.2.2.1.1.2

Cancel the common factor.

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

Step 5.2.2.1.1.3

Rewrite the expression.

$x = \frac{2}{3} π$

Step 5.2.2.1.2

Combine $\frac{2}{3}$ and $π$ .

$x = \frac{2 π}{3}$

Step 6

The basic period for $y = \tan (\frac{3 x}{4})$ will occur at $(- \frac{2 π}{3}, \frac{2 π}{3})$ , where $- \frac{2 π}{3}$ and $\frac{2 π}{3}$ are vertical asymptotes.

$(- \frac{2 π}{3}, \frac{2 π}{3})$

Step 7

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

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Step 7.1

$\frac{3}{4}$ is approximately $0.75$ which is positive so remove the absolute value

$\frac{π}{\frac{3}{4}}$

Step 7.2

Multiply the numerator by the reciprocal of the denominator.

$π \frac{4}{3}$

Step 7.3

Combine $π$ and $\frac{4}{3}$ .

$\frac{π \cdot 4}{3}$

Step 7.4

Move $4$ to the left of $π$ .

$\frac{4 π}{3}$

Step 8

The vertical asymptotes for $y = \tan (\frac{3 x}{4})$ occur at $- \frac{2 π}{3}$ , $\frac{2 π}{3}$ , and every $\frac{4 π n}{3}$ , where $n$ is an integer.

$x = \frac{2 π}{3} + \frac{4 π n}{3}$

Step 9

Tangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = \frac{2 π}{3} + \frac{4 π n}{3}$ where $n$ is an integer

Step 10