Precalculus Examples

$k (x) = 3 \tan (\frac{3 π}{2} x)$

Step 1

Combine $\frac{3 π}{2}$ and $x$ .

$y = 3 \tan (\frac{3 π x}{2})$

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 $y = 3 \tan (\frac{3 π x}{2})$ . 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 $y = 3 \tan (\frac{3 π x}{2})$ .

$\frac{3 π x}{2} = - \frac{π}{2}$

Step 3

Solve for $x$ .

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Step 3.1

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$3 π x = - π$

Step 3.2

Divide each term in $3 π x = - π$ by $3 π$ and simplify.

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Step 3.2.1

Divide each term in $3 π x = - π$ by $3 π$ .

$\frac{3 π x}{3 π} = \frac{- π}{3 π}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{3 π x}{3 π} = \frac{- π}{3 π}$

Step 3.2.2.1.2

Rewrite the expression.

$\frac{π x}{π} = \frac{- π}{3 π}$

Step 3.2.2.2

Cancel the common factor of $π$ .

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Step 3.2.2.2.1

Cancel the common factor.

$\frac{π x}{π} = \frac{- π}{3 π}$

Step 3.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- π}{3 π}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Cancel the common factor of $π$ .

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Step 3.2.3.1.1

Cancel the common factor.

$x = \frac{- π}{3 π}$

Step 3.2.3.1.2

Rewrite the expression.

$x = \frac{- 1}{3}$

Step 3.2.3.2

Move the negative in front of the fraction.

$x = - \frac{1}{3}$

Step 4

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

$\frac{3 π x}{2} = \frac{π}{2}$

Step 5

Solve for $x$ .

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Step 5.1

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$3 π x = π$

Step 5.2

Divide each term in $3 π x = π$ by $3 π$ and simplify.

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Step 5.2.1

Divide each term in $3 π x = π$ by $3 π$ .

$\frac{3 π x}{3 π} = \frac{π}{3 π}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $3$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{3 π x}{3 π} = \frac{π}{3 π}$

Step 5.2.2.1.2

Rewrite the expression.

$\frac{π x}{π} = \frac{π}{3 π}$

Step 5.2.2.2

Cancel the common factor of $π$ .

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Step 5.2.2.2.1

Cancel the common factor.

$\frac{π x}{π} = \frac{π}{3 π}$

Step 5.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{π}{3 π}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Cancel the common factor of $π$ .

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Step 5.2.3.1.1

Cancel the common factor.

$x = \frac{π}{3 π}$

Step 5.2.3.1.2

Rewrite the expression.

$x = \frac{1}{3}$

Step 6

The basic period for $y = 3 \tan (\frac{3 π x}{2})$ will occur at $(- \frac{1}{3}, \frac{1}{3})$ , where $- \frac{1}{3}$ and $\frac{1}{3}$ are vertical asymptotes.

$(- \frac{1}{3}, \frac{1}{3})$

Step 7

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

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Step 7.1

$\frac{3 π}{2}$ is approximately $4.71238898$ which is positive so remove the absolute value

$\frac{π}{\frac{3 π}{2}}$

Step 7.2

Multiply the numerator by the reciprocal of the denominator.

$π \frac{2}{3 π}$

Step 7.3

Cancel the common factor of $π$ .

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Step 7.3.1

Factor $π$ out of $3 π$ .

$π \frac{2}{π \cdot 3}$

Step 7.3.2

Cancel the common factor.

$π \frac{2}{π \cdot 3}$

Step 7.3.3

Rewrite the expression.

$\frac{2}{3}$

Step 8

The vertical asymptotes for $y = 3 \tan (\frac{3 π x}{2})$ occur at $- \frac{1}{3}$ , $\frac{1}{3}$ , and every $\frac{2 n}{3}$ , where $n$ is an integer.

$x = \frac{1}{3} + \frac{2 n}{3}$

Step 9

Tangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = \frac{1}{3} + \frac{2 n}{3}$ where $n$ is an integer

Step 10