Precalculus Examples

f (x) = 3 cot (4 x)

$f (x) = 3 \cot (4 x)$

Step 1

Find the asymptotes.

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Step 1.1

For any

y = cot (x)

$y = \cot (x)$ , vertical asymptotes occur at

x = n π

$x = n π$ , where

n

$n$ is an integer. Use the basic period for

y = cot (x)

$y = \cot (x)$ ,

(0, π)

$(0, π)$ , to find the vertical asymptotes for

y = 3 cot (4 x)

$y = 3 \cot (4 x)$ . Set the inside of the cotangent function,

b x + c

$b x + c$ , for

y = a cot (b x + c) + d

$y = a \cot (b x + c) + d$ equal to

0

$0$ to find where the vertical asymptote occurs for

y = 3 cot (4 x)

$y = 3 \cot (4 x)$ .

4 x = 0

$4 x = 0$

Step 1.2

Divide each term in

4 x = 0

$4 x = 0$ by

4

$4$ and simplify.

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Step 1.2.1

Divide each term in

4 x = 0

$4 x = 0$ by

4

$4$ .

\frac{4 x}{4} = \frac{0}{4}

$\frac{4 x}{4} = \frac{0}{4}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of

4

$4$ .

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Step 1.2.2.1.1

Cancel the common factor.

\frac{4 x}{4} = \frac{0}{4}

$\frac{4 x}{4} = \frac{0}{4}$

Step 1.2.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{0}{4}

$x = \frac{0}{4}$

x = \frac{0}{4}

$x = \frac{0}{4}$

x = \frac{0}{4}

$x = \frac{0}{4}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide

0

$0$ by

4

$4$ .

x = 0

$x = 0$

x = 0

$x = 0$

x = 0

$x = 0$

Step 1.3

Set the inside of the cotangent function

4 x

$4 x$ equal to

π

$π$ .

4 x = π

$4 x = π$

Step 1.4

Divide each term in

4 x = π

$4 x = π$ by

4

$4$ and simplify.

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Step 1.4.1

Divide each term in

4 x = π

$4 x = π$ by

4

$4$ .

\frac{4 x}{4} = \frac{π}{4}

$\frac{4 x}{4} = \frac{π}{4}$

Step 1.4.2

Simplify the left side.

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Step 1.4.2.1

Cancel the common factor of

4

$4$ .

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Step 1.4.2.1.1

Cancel the common factor.

\frac{4 x}{4} = \frac{π}{4}

$\frac{4 x}{4} = \frac{π}{4}$

Step 1.4.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{π}{4}

$x = \frac{π}{4}$

x = \frac{π}{4}

$x = \frac{π}{4}$

x = \frac{π}{4}

$x = \frac{π}{4}$

x = \frac{π}{4}

$x = \frac{π}{4}$

Step 1.5

The basic period for

y = 3 cot (4 x)

$y = 3 \cot (4 x)$ will occur at

(0, \frac{π}{4})

$(0, \frac{π}{4})$ , where

0

$0$ and

\frac{π}{4}

$\frac{π}{4}$ are vertical asymptotes.

(0, \frac{π}{4})

$(0, \frac{π}{4})$

Step 1.6

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

0

$0$ and

4

$4$ is

4

$4$ .

\frac{π}{4}

$\frac{π}{4}$

Step 1.7

The vertical asymptotes for

y = 3 cot (4 x)

$y = 3 \cot (4 x)$ occur at

0

$0$ ,

\frac{π}{4}

$\frac{π}{4}$ , and every

\frac{π n}{4}

$\frac{π n}{4}$ , where

n

$n$ is an integer.

x = \frac{π n}{4}

$x = \frac{π n}{4}$

Step 1.8

Cotangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes:

x = \frac{π n}{4}

$x = \frac{π n}{4}$ where

n

$n$ is an integer

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes:

x = \frac{π n}{4}

$x = \frac{π n}{4}$ where

n

$n$ is an integer

Step 2

Use the form

a cot (b x - c) + d

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

a = 3

$a = 3$

b = 4

$b = 4$

c = 0

$c = 0$

d = 0

$d = 0$

Step 3

Since the graph of the function

c o t

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

Amplitude: None

Step 4

Find the period of

3 cot (4 x)

$3 \cot (4 x)$ .

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Step 4.1

The period of the function can be calculated using

\frac{π}{| b |}

$\frac{π}{| b |}$ .

\frac{π}{| b |}

$\frac{π}{| b |}$

Step 4.2

Replace

b

$b$ with

4

$4$ in the formula for period.

\frac{π}{| 4 |}

$\frac{π}{| 4 |}$

Step 4.3

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

0

$0$ and

4

$4$ is

4

$4$ .

\frac{π}{4}

$\frac{π}{4}$

\frac{π}{4}

$\frac{π}{4}$

Step 5

Find the phase shift using the formula

\frac{c}{b}

$\frac{c}{b}$ .

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Step 5.1

The phase shift of the function can be calculated from

\frac{c}{b}

$\frac{c}{b}$ .

Phase Shift:

\frac{c}{b}

$\frac{c}{b}$

Step 5.2

Replace the values of

c

$c$ and

b

$b$ in the equation for phase shift.

Phase Shift:

\frac{0}{4}

$\frac{0}{4}$

Step 5.3

Divide

0

$0$ by

4

$4$ .

Phase Shift:

0

$0$

Phase Shift:

0

$0$

Step 6

List the properties of the trigonometric function.

Amplitude: None

Period:

\frac{π}{4}

$\frac{π}{4}$

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}{4}

$x = \frac{π n}{4}$ where

n

$n$ is an integer

Amplitude: None

Period:

\frac{π}{4}

$\frac{π}{4}$

Phase Shift: None

Vertical Shift: None

Step 8

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