Precalculus Examples

$f (x) = \cot (\frac{1}{2} x - π)$

Step 1

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

$y = \cot (\frac{x}{2} - π)$

Step 2

For any $y = \cot (x)$ , vertical asymptotes occur at $x = n π$ , where $n$ is an integer. Use the basic period for $y = \cot (x)$ , $(0, π)$ , to find the vertical asymptotes for $y = \cot (\frac{x}{2} - π)$ . Set the inside of the cotangent function, $b x + c$ , for $y = a \cot (b x + c) + d$ equal to $0$ to find where the vertical asymptote occurs for $y = \cot (\frac{x}{2} - π)$ .

$\frac{x}{2} - π = 0$

Step 3

Solve for $x$ .

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Step 3.1

Add $π$ to both sides of the equation.

$\frac{x}{2} = π$

Step 3.2

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 π$

Step 3.3

Simplify the left side.

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Step 3.3.1

Cancel the common factor of $2$ .

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Step 3.3.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 π$

Step 3.3.1.2

Rewrite the expression.

$x = 2 π$

Step 4

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

$\frac{x}{2} - π = π$

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

Add $π$ to both sides of the equation.

$\frac{x}{2} = π + π$

Step 5.1.2

Add $π$ and $π$ .

$\frac{x}{2} = 2 π$

Step 5.2

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (2 π)$

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.

$2 \frac{x}{2} = 2 (2 π)$

Step 5.3.1.1.2

Rewrite the expression.

$x = 2 (2 π)$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Multiply $2$ by $2$ .

$x = 4 π$

Step 6

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

$(2 π, 4 π)$

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 $y = \cot (\frac{x}{2} - π)$ occur at $2 π$ , $4 π$ , and every $x = 2 π + 2 π n$ , where $n$ is an integer.

$x = 2 π + 2 π n$

Step 9

Cotangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = 2 π + 2 π n$ where $n$ is an integer

Step 10