Graph y=2tan(x+3)

$y = 2 \tan (x + 3)$

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 $y = 2 \tan (x + 3)$ . 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 $y = 2 \tan (x + 3)$ .

$x + 3 = - \frac{π}{2}$

Subtract $3$ from both sides of the equation.

$x = - \frac{π}{2} - 3$

Set the inside of the tangent function $x + 3$ equal to $\frac{π}{2}$ .

$x + 3 = \frac{π}{2}$

Subtract $3$ from both sides of the equation.

$x = \frac{π}{2} - 3$

The basic period for $y = 2 \tan (x + 3)$ will occur at $(- \frac{π}{2} - 3, \frac{π}{2} - 3)$ , where $- \frac{π}{2} - 3$ and $\frac{π}{2} - 3$ are vertical asymptotes.

$(- \frac{π}{2} - 3, \frac{π}{2} - 3)$

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

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The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Divide $π$ by $1$ .

$π$

The vertical asymptotes for $y = 2 \tan (x + 3)$ occur at $- \frac{π}{2} - 3$ , $\frac{π}{2} - 3$ , and every $x = - \frac{π}{2} - 3 + π n$ , where $n$ is an integer.

$x = - \frac{π}{2} - 3 + π n$

Tangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = - \frac{π}{2} - 3 + π n$ where $n$ is an integer

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = - \frac{π}{2} - 3 + π n$ 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.

$a = 2$

$b = 1$

$c = - 3$

$d = 0$

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 $2 \tan (x + 3)$ .

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

$\frac{π}{| b |}$

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Divide $π$ by $1$ .

$π$

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: $\frac{- 3}{1}$

Divide $- 3$ by $1$ .

Phase Shift: $- 3$

Find the vertical shift $d$ .

Vertical Shift: $0$

List the properties of the trigonometric function.

Amplitude: None

Period: $π$

Phase Shift: $- 3$ ( $3$ 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: $x = - \frac{π}{2} - 3 + π n$ where $n$ is an integer

Amplitude: None

Period: $π$

Phase Shift: $- 3$ ( $3$ to the left)

Vertical Shift: $0$

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