Precalculus Examples

$f (x) = 2 \tan (x - \frac{π}{2})$

Step 1

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

$x - \frac{π}{2} = - \frac{π}{2}$

Step 2

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

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Step 2.1

Add $\frac{π}{2}$ to both sides of the equation.

$x = - \frac{π}{2} + \frac{π}{2}$

Step 2.2

Combine the numerators over the common denominator.

$x = \frac{- π + π}{2}$

Step 2.3

Add $- π$ and $π$ .

$x = \frac{0}{2}$

Step 2.4

Divide $0$ by $2$ .

$x = 0$

Step 3

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

$x - \frac{π}{2} = \frac{π}{2}$

Step 4

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

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Step 4.1

Add $\frac{π}{2}$ to both sides of the equation.

$x = \frac{π}{2} + \frac{π}{2}$

Step 4.2

Combine the numerators over the common denominator.

$x = \frac{π + π}{2}$

Step 4.3

Add $π$ and $π$ .

$x = \frac{2 π}{2}$

Step 4.4

Cancel the common factor of $2$ .

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Step 4.4.1

Cancel the common factor.

$x = \frac{2 π}{2}$

Step 4.4.2

Divide $π$ by $1$ .

$x = π$

Step 5

The basic period for $y = 2 \tan (x - \frac{π}{2})$ will occur at $(0, π)$ , where $0$ and $π$ are vertical asymptotes.

$(0, π)$

Step 6

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

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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 $y = 2 \tan (x - \frac{π}{2})$ occur at $0$ , $π$ , and every $x = 0 + π n$ , where $n$ is an integer.

$x = 0 + π n$

Step 8

Tangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = 0 + π n$ where $n$ is an integer

Step 9