Precalculus Examples

$y = \tan (4 x)$

Step 1

Find the asymptotes.

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Step 1.1

For any

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

$x = \frac{π}{2} + n π$ , where

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

$y = \tan (x)$ ,

$(- \frac{π}{2}, \frac{π}{2})$ , to find the vertical asymptotes for

$y = \tan (4 x)$ . Set the inside of the tangent function,

$b x + c$ , for

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

$- \frac{π}{2}$ to find where the vertical asymptote occurs for

$y = \tan (4 x)$ .

$4 x = - \frac{π}{2}$

Step 1.2

Divide each term in

$4 x = - \frac{π}{2}$ by

$4$ and simplify.

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Step 1.2.1

Divide each term in

$4 x = - \frac{π}{2}$ by

$4$ .

$\frac{4 x}{4} = \frac{- \frac{π}{2}}{4}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of

$4$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- \frac{π}{2}}{4}$

Step 1.2.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{- \frac{π}{2}}{4}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{π}{2} \cdot \frac{1}{4}$

Step 1.2.3.2

Multiply

$- \frac{π}{2} \cdot \frac{1}{4}$ .

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Step 1.2.3.2.1

Multiply

$\frac{1}{4}$ by

$\frac{π}{2}$ .

$x = - \frac{π}{4 \cdot 2}$

Step 1.2.3.2.2

Multiply

$4$ by

$2$ .

$x = - \frac{π}{8}$

Step 1.3

Set the inside of the tangent function

$4 x$ equal to

$\frac{π}{2}$ .

$4 x = \frac{π}{2}$

Step 1.4

Divide each term in

$4 x = \frac{π}{2}$ by

$4$ and simplify.

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Step 1.4.1

Divide each term in

$4 x = \frac{π}{2}$ by

$4$ .

$\frac{4 x}{4} = \frac{\frac{π}{2}}{4}$

Step 1.4.2

Simplify the left side.

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Step 1.4.2.1

Cancel the common factor of

$4$ .

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Step 1.4.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{\frac{π}{2}}{4}$

Step 1.4.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{\frac{π}{2}}{4}$

Step 1.4.3

Simplify the right side.

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Step 1.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{4}$

Step 1.4.3.2

Multiply

$\frac{π}{2} \cdot \frac{1}{4}$ .

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Step 1.4.3.2.1

Multiply

$\frac{π}{2}$ by

$\frac{1}{4}$ .

$x = \frac{π}{2 \cdot 4}$

Step 1.4.3.2.2

Multiply

$2$ by

$4$ .

$x = \frac{π}{8}$

Step 1.5

The basic period for

$y = \tan (4 x)$ will occur at

$(- \frac{π}{8}, \frac{π}{8})$ , where

$- \frac{π}{8}$ and

$\frac{π}{8}$ are vertical asymptotes.

$(- \frac{π}{8}, \frac{π}{8})$

Step 1.6

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

$0$ and

$4$ is

$4$ .

$\frac{π}{4}$

Step 1.7

The vertical asymptotes for

$y = \tan (4 x)$ occur at

$- \frac{π}{8}$ ,

$\frac{π}{8}$ , and every

$\frac{π n}{4}$ , where

$n$ is an integer.

$x = \frac{π}{8} + \frac{π n}{4}$

Step 1.8

Tangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes:

$x = \frac{π}{8} + \frac{π n}{4}$ where

$n$ is an integer

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes:

$x = \frac{π}{8} + \frac{π n}{4}$ where

$n$ is an integer

Step 2

Use the form

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

$a = 1$

$b = 4$

$c = 0$

$d = 0$

Step 3

Since the graph of the function

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

Amplitude: None

Step 4

Find the period of

$\tan (4 x)$ .

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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

$4$ in the formula for period.

$\frac{π}{| 4 |}$

Step 4.3

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

$0$ and

$4$ is

$4$ .

$\frac{π}{4}$

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:

$\frac{0}{4}$

Step 5.3

Divide

$0$ by

$4$ .

Phase Shift:

$0$

Phase Shift:

$0$

Step 6

List the properties of the trigonometric function.

Amplitude: None

Period:

$\frac{π}{4}$

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:

$x = \frac{π}{8} + \frac{π n}{4}$ where

$n$ is an integer

Amplitude: None

Period:

$\frac{π}{4}$

Phase Shift: None

Vertical Shift: None

Step 8