Examples

$f (x) = \cot (3 x)$

Step 1

Find the asymptotes.

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Step 1.1

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 $y = \cot (3 x)$ . 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 $y = \cot (3 x)$ .

$3 x = 0$

Step 1.2

Divide each term in $3 x = 0$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x = 0$ by $3$ .

$\frac{3 x}{3} = \frac{0}{3}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{0}{3}$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{3}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $0$ by $3$ .

$x = 0$

Step 1.3

Set the inside of the cotangent function $3 x$ equal to $π$ .

$3 x = π$

Step 1.4

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

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Step 1.4.1

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

$\frac{3 x}{3} = \frac{π}{3}$

Step 1.4.2

Simplify the left side.

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Step 1.4.2.1

Cancel the common factor of $3$ .

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Step 1.4.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{π}{3}$

Step 1.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{3}$

Step 1.5

The basic period for $y = \cot (3 x)$ will occur at $(0, \frac{π}{3})$ , where $0$ and $\frac{π}{3}$ are vertical asymptotes.

$(0, \frac{π}{3})$

Step 1.6

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

$\frac{π}{3}$

Step 1.7

The vertical asymptotes for $y = \cot (3 x)$ occur at $0$ , $\frac{π}{3}$ , and every $\frac{π n}{3}$ , where $n$ is an integer.

$x = \frac{π n}{3}$

Step 1.8

Cotangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

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

No Horizontal Asymptotes

No Oblique Asymptotes

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

Step 2

Use the form $a \cot (b x - c) + d$ to find the variables used to find the amplitude, period, phase shift, and vertical shift.

$a = 1$

$b = 3$

$c = 0$

$d = 0$

Step 3

Since the graph of the function $c o t$ does not have a maximum or minimum value, there can be no value for the amplitude.

Amplitude: None

Step 4

Find the period of $\cot (3 x)$ .

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Step 4.1

The period of the function can be calculated using $\frac{π}{| b |}$ .

$\frac{π}{| b |}$

Step 4.2

Replace $b$ with $3$ in the formula for period.

$\frac{π}{| 3 |}$

Step 4.3

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

$\frac{π}{3}$

Step 5

Find the phase shift using the formula $\frac{c}{b}$ .

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Step 5.1

The phase shift of the function can be calculated from $\frac{c}{b}$ .

Phase Shift: $\frac{c}{b}$

Step 5.2

Replace the values of $c$ and $b$ in the equation for phase shift.

Phase Shift: $\frac{0}{3}$

Step 5.3

Divide $0$ by $3$ .

Phase Shift: $0$

Step 6

List the properties of the trigonometric function.

Amplitude: None

Period: $\frac{π}{3}$

Phase Shift: None

Vertical Shift: None

Step 7

The trig function can be graphed using the amplitude, period, phase shift, vertical shift, and the points.

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

Amplitude: None

Period: $\frac{π}{3}$

Phase Shift: None

Vertical Shift: None

Step 8

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