Precalculus Examples

$f (x) = 7 \cot (2 π x) - 2$

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 $y = 7 \cot (2 π x) - 2$ . 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 $y = 7 \cot (2 π x) - 2$ .

$2 π x = 0$

Step 2

Divide each term in $2 π x = 0$ by $2 π$ and simplify.

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Step 2.1

Divide each term in $2 π x = 0$ by $2 π$ .

$\frac{2 π x}{2 π} = \frac{0}{2 π}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2 π x}{2 π} = \frac{0}{2 π}$

Step 2.2.1.2

Rewrite the expression.

$\frac{π x}{π} = \frac{0}{2 π}$

Step 2.2.2

Cancel the common factor of $π$ .

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Step 2.2.2.1

Cancel the common factor.

$\frac{π x}{π} = \frac{0}{2 π}$

Step 2.2.2.2

Divide $x$ by $1$ .

$x = \frac{0}{2 π}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $0$ and $2$ .

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Step 2.3.1.1

Factor $2$ out of $0$ .

$x = \frac{2 (0)}{2 π}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $2$ out of $2 π$ .

$x = \frac{2 (0)}{2 (π)}$

Step 2.3.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot 0}{2 π}$

Step 2.3.1.2.3

Rewrite the expression.

$x = \frac{0}{π}$

Step 2.3.2

Divide $0$ by $π$ .

$x = 0$

Step 3

Set the inside of the cotangent function $2 π x$ equal to $π$ .

$2 π x = π$

Step 4

Divide each term in $2 π x = π$ by $2 π$ and simplify.

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Step 4.1

Divide each term in $2 π x = π$ by $2 π$ .

$\frac{2 π x}{2 π} = \frac{π}{2 π}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 π x}{2 π} = \frac{π}{2 π}$

Step 4.2.1.2

Rewrite the expression.

$\frac{π x}{π} = \frac{π}{2 π}$

Step 4.2.2

Cancel the common factor of $π$ .

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Step 4.2.2.1

Cancel the common factor.

$\frac{π x}{π} = \frac{π}{2 π}$

Step 4.2.2.2

Divide $x$ by $1$ .

$x = \frac{π}{2 π}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Cancel the common factor of $π$ .

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Step 4.3.1.1

Cancel the common factor.

$x = \frac{π}{2 π}$

Step 4.3.1.2

Rewrite the expression.

$x = \frac{1}{2}$

Step 5

The basic period for $y = 7 \cot (2 π x) - 2$ will occur at $(0, \frac{1}{2})$ , where $0$ and $\frac{1}{2}$ are vertical asymptotes.

$(0, \frac{1}{2})$

Step 6

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

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Step 6.1

$2 π$ is approximately $6.2831853$ which is positive so remove the absolute value

$\frac{π}{2 π}$

Step 6.2

Cancel the common factor of $π$ .

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Step 6.2.1

Cancel the common factor.

$\frac{π}{2 π}$

Step 6.2.2

Rewrite the expression.

$\frac{1}{2}$

Step 7

The vertical asymptotes for $y = 7 \cot (2 π x) - 2$ occur at $0$ , $\frac{1}{2}$ , and every $\frac{n}{2}$ , where $n$ is an integer.

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

Step 8

Cotangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

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

Step 9