Precalculus Examples

y = \tan (x - \frac{π}{4})

$y = \tan (x - \frac{π}{4})$

Step 1

Find the asymptotes.

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Step 1.1

For any

y = \tan (x)

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

x = \frac{π}{2} + n π

$x = \frac{π}{2} + n π$ , where

n

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

y = \tan (x)

$y = \tan (x)$ ,

(- \frac{π}{2}, \frac{π}{2})

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

y = \tan (x - \frac{π}{4})

$y = \tan (x - \frac{π}{4})$ . Set the inside of the tangent function,

b x + c

$b x + c$ , for

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

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

- \frac{π}{2}

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

y = \tan (x - \frac{π}{4})

$y = \tan (x - \frac{π}{4})$ .

x - \frac{π}{4} = - \frac{π}{2}

$x - \frac{π}{4} = - \frac{π}{2}$

Step 1.2

Move all terms not containing

x

$x$ to the right side of the equation.

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Step 1.2.1

Add

\frac{π}{4}

$\frac{π}{4}$ to both sides of the equation.

x = - \frac{π}{2} + \frac{π}{4}

$x = - \frac{π}{2} + \frac{π}{4}$

Step 1.2.2

To write

- \frac{π}{2}

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

\frac{2}{2}

$\frac{2}{2}$ .

x = - \frac{π}{2} \cdot \frac{2}{2} + \frac{π}{4}

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

Step 1.2.3

Write each expression with a common denominator of

4

$4$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 1.2.3.1

Multiply

\frac{π}{2}

$\frac{π}{2}$ by

\frac{2}{2}

$\frac{2}{2}$ .

x = - \frac{π \cdot 2}{2 \cdot 2} + \frac{π}{4}

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

Step 1.2.3.2

Multiply

2

$2$ by

2

$2$ .

x = - \frac{π \cdot 2}{4} + \frac{π}{4}

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

x = - \frac{π \cdot 2}{4} + \frac{π}{4}

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

Step 1.2.4

Combine the numerators over the common denominator.

x = \frac{- π \cdot 2 + π}{4}

$x = \frac{- π \cdot 2 + π}{4}$

Step 1.2.5

Simplify the numerator.

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Step 1.2.5.1

Multiply

2

$2$ by

- 1

$- 1$ .

x = \frac{- 2 π + π}{4}

$x = \frac{- 2 π + π}{4}$

Step 1.2.5.2

Add

- 2 π

$- 2 π$ and

π

$π$ .

x = \frac{- π}{4}

$x = \frac{- π}{4}$

x = \frac{- π}{4}

$x = \frac{- π}{4}$

Step 1.2.6

Move the negative in front of the fraction.

x = - \frac{π}{4}

$x = - \frac{π}{4}$

x = - \frac{π}{4}

$x = - \frac{π}{4}$

Step 1.3

Set the inside of the tangent function

x - \frac{π}{4}

$x - \frac{π}{4}$ equal to

\frac{π}{2}

$\frac{π}{2}$ .

x - \frac{π}{4} = \frac{π}{2}

$x - \frac{π}{4} = \frac{π}{2}$

Step 1.4

Move all terms not containing

x

$x$ to the right side of the equation.

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Step 1.4.1

Add

\frac{π}{4}

$\frac{π}{4}$ to both sides of the equation.

x = \frac{π}{2} + \frac{π}{4}

$x = \frac{π}{2} + \frac{π}{4}$

Step 1.4.2

To write

\frac{π}{2}

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

\frac{2}{2}

$\frac{2}{2}$ .

x = \frac{π}{2} \cdot \frac{2}{2} + \frac{π}{4}

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

Step 1.4.3

Write each expression with a common denominator of

4

$4$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 1.4.3.1

Multiply

\frac{π}{2}

$\frac{π}{2}$ by

\frac{2}{2}

$\frac{2}{2}$ .

x = \frac{π \cdot 2}{2 \cdot 2} + \frac{π}{4}

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

Step 1.4.3.2

Multiply

2

$2$ by

2

$2$ .

x = \frac{π \cdot 2}{4} + \frac{π}{4}

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

x = \frac{π \cdot 2}{4} + \frac{π}{4}

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

Step 1.4.4

Combine the numerators over the common denominator.

x = \frac{π \cdot 2 + π}{4}

$x = \frac{π \cdot 2 + π}{4}$

Step 1.4.5

Simplify the numerator.

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Step 1.4.5.1

Move

2

$2$ to the left of

π

$π$ .

x = \frac{2 \cdot π + π}{4}

$x = \frac{2 \cdot π + π}{4}$

Step 1.4.5.2

Add

2 π

$2 π$ and

π

$π$ .

x = \frac{3 π}{4}

$x = \frac{3 π}{4}$

x = \frac{3 π}{4}

$x = \frac{3 π}{4}$

x = \frac{3 π}{4}

$x = \frac{3 π}{4}$

Step 1.5

The basic period for

y = \tan (x - \frac{π}{4})

$y = \tan (x - \frac{π}{4})$ will occur at

(- \frac{π}{4}, \frac{3 π}{4})

$(- \frac{π}{4}, \frac{3 π}{4})$ , where

- \frac{π}{4}

$- \frac{π}{4}$ and

\frac{3 π}{4}

$\frac{3 π}{4}$ are vertical asymptotes.

(- \frac{π}{4}, \frac{3 π}{4})

$(- \frac{π}{4}, \frac{3 π}{4})$

Step 1.6

Find the period

\frac{π}{| b |}

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

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Step 1.6.1

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

0

$0$ and

1

$1$ is

1

$1$ .

\frac{π}{1}

$\frac{π}{1}$

Step 1.6.2

Divide

π

$π$ by

1

$1$ .

π

$π$

π

$π$

Step 1.7

The vertical asymptotes for

y = \tan (x - \frac{π}{4})

$y = \tan (x - \frac{π}{4})$ occur at

- \frac{π}{4}

$- \frac{π}{4}$ ,

\frac{3 π}{4}

$\frac{3 π}{4}$ , and every

x = - \frac{π}{4} + π n

$x = - \frac{π}{4} + π n$ , where

n

$n$ is an integer.

x = - \frac{π}{4} + π n

$x = - \frac{π}{4} + π n$

Step 1.8

Tangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes:

x = - \frac{π}{4} + π n

$x = - \frac{π}{4} + π n$ where

n

$n$ is an integer

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes:

x = - \frac{π}{4} + π n

$x = - \frac{π}{4} + π n$ where

n

$n$ is an integer

Step 2

Use the form

a \tan (b x - c) + d

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

a = 1

$a = 1$

b = 1

$b = 1$

c = \frac{π}{4}

$c = \frac{π}{4}$

d = 0

$d = 0$

Step 3

Since the graph of the function

t a n

$t a n$ 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 (x - \frac{π}{4})

$\tan (x - \frac{π}{4})$ .

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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

1

$1$ in the formula for period.

\frac{π}{| 1 |}

$\frac{π}{| 1 |}$

Step 4.3

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

0

$0$ and

1

$1$ is

1

$1$ .

\frac{π}{1}

$\frac{π}{1}$

Step 4.4

Divide

π

$π$ by

1

$1$ .

π

$π$

π

$π$

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{\frac{π}{4}}{1}

$\frac{\frac{π}{4}}{1}$

Step 5.3

Divide

\frac{π}{4}

$\frac{π}{4}$ by

1

$1$ .

Phase Shift:

\frac{π}{4}

$\frac{π}{4}$

Phase Shift:

\frac{π}{4}

$\frac{π}{4}$

Step 6

List the properties of the trigonometric function.

Amplitude: None

Period:

π

$π$

Phase Shift:

\frac{π}{4}

$\frac{π}{4}$ (

\frac{π}{4}

$\frac{π}{4}$ to the right)

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}{4} + π n

$x = - \frac{π}{4} + π n$ where

n

$n$ is an integer

Amplitude: None

Period:

π

$π$

Phase Shift:

\frac{π}{4}

$\frac{π}{4}$ (

\frac{π}{4}

$\frac{π}{4}$ to the right)

Vertical Shift: None

Step 8