Trigonometry Examples

$f (x) = 2 \tan (x)$

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

$x = - \frac{π}{2}$

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

$x = \frac{π}{2}$

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

$(- \frac{π}{2}, \frac{π}{2})$

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

$π n$

There are only vertical asymptotes for tangent and cotangent functions.

Vertical Asymptotes: $x = \frac{π}{2} + π n$ for any integer $n$

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = \frac{π}{2} + π n$ for any integer $n$

No Horizontal Asymptotes

No Oblique Asymptotes

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

$a = 2$

$b = 1$

$c = 0$

$d = 0$

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 using the formula $\frac{π}{| b |}$ .

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

Period: $\frac{π}{| b |}$

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

Period: $\frac{π}{| 1 |}$

Solve the equation.

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

Period: $\frac{π}{1}$

Divide $π$ by $1$ .

Period: $π$

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: $\frac{0}{1}$

Divide $0$ by $1$ .

Phase Shift: $0$

Find the vertical shift $d$ .

Vertical Shift: $0$

List the properties of the trigonometric function.

Amplitude: None

Period: $π$

Phase Shift: $0$ ( $0$ to the right)

Vertical Shift: $0$

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

Vertical Asymptotes: $x = \frac{π}{2} + π n$ for any integer $n$

Amplitude: None

Period: $π$

Phase Shift: $0$ ( $0$ to the right)

Vertical Shift: $0$

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