Graph y=4tan(2x+135)

$y = 4 \tan (2 x + 135)$

Find the asymptotes.

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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 $y = 4 \tan (2 x + 135)$ . 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 $y = 4 \tan (2 x + 135)$ .

$2 x + 135 = - \frac{π}{2}$

Solve for $x$ .

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Subtract $135$ from both sides of the equation.

$2 x = - \frac{π}{2} - 135$

Divide each term by $2$ and simplify.

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Divide each term in $2 x = - \frac{π}{2} - 135$ by $2$ .

$\frac{2 x}{2} = - \frac{π}{2} \cdot \frac{1}{2} + \frac{- 135}{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 x}{2} = - \frac{π}{2} \cdot \frac{1}{2} + \frac{- 135}{2}$

Divide $x$ by $1$ .

$x = - \frac{π}{2} \cdot \frac{1}{2} + \frac{- 135}{2}$

Simplify each term.

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Multiply $- \frac{π}{2} \cdot \frac{1}{2}$ .

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Multiply $\frac{1}{2}$ and $\frac{π}{2}$ .

$x = - \frac{π}{2 \cdot 2} + \frac{- 135}{2}$

Multiply $2$ by $2$ .

$x = - \frac{π}{4} + \frac{- 135}{2}$

Move the negative in front of the fraction.

$x = - \frac{π}{4} - \frac{135}{2}$

Set the inside of the tangent function $2 x + 135$ equal to $\frac{π}{2}$ .

$2 x + 135 = \frac{π}{2}$

Solve for $x$ .

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Subtract $135$ from both sides of the equation.

$2 x = \frac{π}{2} - 135$

Divide each term by $2$ and simplify.

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Divide each term in $2 x = \frac{π}{2} - 135$ by $2$ .

$\frac{2 x}{2} = \frac{π}{2} \cdot \frac{1}{2} + \frac{- 135}{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 x}{2} = \frac{π}{2} \cdot \frac{1}{2} + \frac{- 135}{2}$

Divide $x$ by $1$ .

$x = \frac{π}{2} \cdot \frac{1}{2} + \frac{- 135}{2}$

Simplify each term.

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Multiply $\frac{π}{2} \cdot \frac{1}{2}$ .

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Multiply $\frac{π}{2}$ and $\frac{1}{2}$ .

$x = \frac{π}{2 \cdot 2} + \frac{- 135}{2}$

Multiply $2$ by $2$ .

$x = \frac{π}{4} + \frac{- 135}{2}$

Move the negative in front of the fraction.

$x = \frac{π}{4} - \frac{135}{2}$

The basic period for $y = 4 \tan (2 x + 135)$ will occur at $(- \frac{π}{4} - \frac{135}{2}, \frac{π}{4} - \frac{135}{2})$ , where $- \frac{π}{4} - \frac{135}{2}$ and $\frac{π}{4} - \frac{135}{2}$ are vertical asymptotes.

$(- \frac{π}{4} - \frac{135}{2}, \frac{π}{4} - \frac{135}{2})$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{π}{2}$

The vertical asymptotes for $y = 4 \tan (2 x + 135)$ occur at $- \frac{π}{4} - \frac{135}{2}$ , $\frac{π}{4} - \frac{135}{2}$ , and every $x = - \frac{π}{4} - \frac{135}{2} + \frac{π n}{2}$ , where $n$ is an integer.

$x = - \frac{π}{4} - \frac{135}{2} + \frac{π n}{2}$

Tangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = - \frac{π}{4} - \frac{135}{2} + \frac{π n}{2}$ where $n$ is an integer

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = - \frac{π}{4} - \frac{135}{2} + \frac{π n}{2}$ where $n$ is an integer

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

$a = 4$

$b = 2$

$c = - 135$

$d = 0$

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

Find the period of $4 \tan (2 x + 135)$ .

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

$\frac{π}{| b |}$

Replace $b$ with $2$ in the formula for period.

$\frac{π}{| 2 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{π}{2}$

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

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

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

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

Phase Shift: $\frac{- 135}{2}$

Move the negative in front of the fraction.

Phase Shift: $- \frac{135}{2}$

Find the vertical shift $d$ .

Vertical Shift: $0$

List the properties of the trigonometric function.

Amplitude: None

Period: $\frac{π}{2}$

Phase Shift: $- \frac{135}{2}$ ( $\frac{135}{2}$ to the left)

Vertical Shift: $0$

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

Vertical Asymptotes: $x = - \frac{π}{4} - \frac{135}{2} + \frac{π n}{2}$ where $n$ is an integer

Amplitude: None

Period: $\frac{π}{2}$

Phase Shift: $- \frac{135}{2}$ ( $\frac{135}{2}$ to the left)

Vertical Shift: $0$

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