Precalculus Examples

$f (x) = \cot (3 x - \frac{π}{2})$

Step 1

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

$3 x - \frac{π}{2} = 0$

Step 2

Solve for $x$ .

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Step 2.1

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

$3 x = \frac{π}{2}$

Step 2.2

Divide each term in $3 x = \frac{π}{2}$ by $3$ and simplify.

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Step 2.2.1

Divide each term in $3 x = \frac{π}{2}$ by $3$ .

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $3$ .

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Step 2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{3}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{3}$

Step 2.2.3.2

Multiply $\frac{π}{2} \cdot \frac{1}{3}$ .

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Step 2.2.3.2.1

Multiply $\frac{π}{2}$ by $\frac{1}{3}$ .

$x = \frac{π}{2 \cdot 3}$

Step 2.2.3.2.2

Multiply $2$ by $3$ .

$x = \frac{π}{6}$

Step 3

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

$3 x - \frac{π}{2} = π$

Step 4

Solve for $x$ .

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Step 4.1

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

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Step 4.1.1

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

$3 x = π + \frac{π}{2}$

Step 4.1.2

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

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

Step 4.1.3

Combine $π$ and $\frac{2}{2}$ .

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

Step 4.1.4

Combine the numerators over the common denominator.

$3 x = \frac{π \cdot 2 + π}{2}$

Step 4.1.5

Simplify the numerator.

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Step 4.1.5.1

Move $2$ to the left of $π$ .

$3 x = \frac{2 \cdot π + π}{2}$

Step 4.1.5.2

Add $2 π$ and $π$ .

$3 x = \frac{3 π}{2}$

Step 4.2

Divide each term in $3 x = \frac{3 π}{2}$ by $3$ and simplify.

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Step 4.2.1

Divide each term in $3 x = \frac{3 π}{2}$ by $3$ .

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

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.1.1

Cancel the common factor.

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

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{2}}{3}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{2} \cdot \frac{1}{3}$

Step 4.2.3.2

Cancel the common factor of $3$ .

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Step 4.2.3.2.1

Factor $3$ out of $3 π$ .

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

Step 4.2.3.2.2

Cancel the common factor.

$x = \frac{3 π}{2} \cdot \frac{1}{3}$

Step 4.2.3.2.3

Rewrite the expression.

$x = \frac{π}{2}$

Step 5

The basic period for $y = \cot (3 x - \frac{π}{2})$ will occur at $(\frac{π}{6}, \frac{π}{2})$ , where $\frac{π}{6}$ and $\frac{π}{2}$ are vertical asymptotes.

$(\frac{π}{6}, \frac{π}{2})$

Step 6

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{π}{3}$

Step 7

The vertical asymptotes for $y = \cot (3 x - \frac{π}{2})$ occur at $\frac{π}{6}$ , $\frac{π}{2}$ , and every $x = \frac{π}{6} + \frac{π n}{3}$ , where $n$ is an integer.

$x = \frac{π}{6} + \frac{π n}{3}$

Step 8

Cotangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = \frac{π}{6} + \frac{π n}{3}$ where $n$ is an integer

Step 9