Graph 4/5*csc(3/2x-pi/6)

$\frac{4}{5} \cdot \csc (\frac{3}{2} x - \frac{π}{6})$

Find the asymptotes.

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For any $y = \csc (x)$ , vertical asymptotes occur at $x = n π$ , where $n$ is an integer. Use the basic period for $y = c s c (x)$ , $(0, 2 π)$ , to find the vertical asymptotes for $y = \frac{4 \csc (\frac{3 x}{2} - \frac{π}{6})}{5}$ . Set the inside of the cosecant function, $b x + c$ , for $y = a \csc (b x + c) + d$ equal to $0$ to find where the vertical asymptote occurs for $y = \frac{4 \csc (\frac{3 x}{2} - \frac{π}{6})}{5}$ .

$\frac{3 x}{2} - \frac{π}{6} = 0$

Solve for $x$ .

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Add $\frac{π}{6}$ to both sides of the equation.

$\frac{3 x}{2} = \frac{π}{6}$

Multiply both sides of the equation by $\frac{2}{3}$ .

$\frac{2}{3} \cdot \frac{3 x}{2} = \frac{2}{3} \cdot \frac{π}{6}$

Simplify both sides of the equation.

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Simplify $\frac{2}{3} \cdot \frac{3 x}{2}$ .

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Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2}{3} \cdot \frac{3 x}{2} = \frac{2}{3} \cdot \frac{π}{6}$

Rewrite the expression.

$\frac{1}{3} \cdot (3 x) = \frac{2}{3} \cdot \frac{π}{6}$

Cancel the common factor of $3$ .

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Factor $3$ out of $3 x$ .

$\frac{1}{3} \cdot (3 (x)) = \frac{2}{3} \cdot \frac{π}{6}$

Cancel the common factor.

$\frac{1}{3} \cdot (3 x) = \frac{2}{3} \cdot \frac{π}{6}$

Rewrite the expression.

$x = \frac{2}{3} \cdot \frac{π}{6}$

Simplify $\frac{2}{3} \cdot \frac{π}{6}$ .

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Cancel the common factor of $2$ .

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Factor $2$ out of $6$ .

$x = \frac{2}{3} \cdot \frac{π}{2 (3)}$

Cancel the common factor.

$x = \frac{2}{3} \cdot \frac{π}{2 \cdot 3}$

Rewrite the expression.

$x = \frac{1}{3} \cdot \frac{π}{3}$

Multiply $\frac{1}{3}$ and $\frac{π}{3}$ .

$x = \frac{π}{3 \cdot 3}$

Multiply $3$ by $3$ .

$x = \frac{π}{9}$

Set the inside of the cosecant function $\frac{3 x}{2} - \frac{π}{6}$ equal to $2 π$ .

$\frac{3 x}{2} - \frac{π}{6} = 2 π$

Solve for $x$ .

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Move all terms not containing $x$ to the right side of the equation.

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Add $\frac{π}{6}$ to both sides of the equation.

$\frac{3 x}{2} = 2 π + \frac{π}{6}$

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{3 x}{2} = 2 π \cdot \frac{6}{6} + \frac{π}{6}$

Combine $2 π$ and $\frac{6}{6}$ .

$\frac{3 x}{2} = \frac{2 π \cdot 6}{6} + \frac{π}{6}$

Combine the numerators over the common denominator.

$\frac{3 x}{2} = \frac{2 π \cdot 6 + π}{6}$

Simplify the numerator.

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Multiply $6$ by $2$ .

$\frac{3 x}{2} = \frac{12 π + π}{6}$

Add $12 π$ and $π$ .

$\frac{3 x}{2} = \frac{13 π}{6}$

Multiply both sides of the equation by $\frac{2}{3}$ .

$\frac{2}{3} \cdot \frac{3 x}{2} = \frac{2}{3} \cdot \frac{13 π}{6}$

Simplify both sides of the equation.

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Simplify $\frac{2}{3} \cdot \frac{3 x}{2}$ .

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Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2}{3} \cdot \frac{3 x}{2} = \frac{2}{3} \cdot \frac{13 π}{6}$

Rewrite the expression.

$\frac{1}{3} \cdot (3 x) = \frac{2}{3} \cdot \frac{13 π}{6}$

Cancel the common factor of $3$ .

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Factor $3$ out of $3 x$ .

$\frac{1}{3} \cdot (3 (x)) = \frac{2}{3} \cdot \frac{13 π}{6}$

Cancel the common factor.

$\frac{1}{3} \cdot (3 x) = \frac{2}{3} \cdot \frac{13 π}{6}$

Rewrite the expression.

$x = \frac{2}{3} \cdot \frac{13 π}{6}$

Simplify $\frac{2}{3} \cdot \frac{13 π}{6}$ .

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Cancel the common factor of $2$ .

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Factor $2$ out of $6$ .

$x = \frac{2}{3} \cdot \frac{13 π}{2 (3)}$

Cancel the common factor.

$x = \frac{2}{3} \cdot \frac{13 π}{2 \cdot 3}$

Rewrite the expression.

$x = \frac{1}{3} \cdot \frac{13 π}{3}$

Multiply $\frac{1}{3}$ and $\frac{13 π}{3}$ .

$x = \frac{13 π}{3 \cdot 3}$

Multiply $3$ by $3$ .

$x = \frac{13 π}{9}$

The basic period for $y = \frac{4 \csc (\frac{3 x}{2} - \frac{π}{6})}{5}$ will occur at $(\frac{π}{9}, \frac{13 π}{9})$ , where $\frac{π}{9}$ and $\frac{13 π}{9}$ are vertical asymptotes.

$(\frac{π}{9}, \frac{13 π}{9})$

Find the period $\frac{2 π}{| b |}$ to find where the vertical asymptotes exist. Vertical asymptotes occur every half period.

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$\frac{3}{2}$ is approximately $1.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{3}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{2}{3}$

Multiply $2 π \frac{2}{3}$ .

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Combine $\frac{2}{3}$ and $2$ .

$\frac{2 \cdot 2}{3} π$

Multiply $2$ by $2$ .

$\frac{4}{3} π$

Combine $\frac{4}{3}$ and $π$ .

$\frac{4 π}{3}$

The vertical asymptotes for $y = \frac{4 \csc (\frac{3 x}{2} - \frac{π}{6})}{5}$ occur at $\frac{π}{9}$ , $\frac{13 π}{9}$ , and every $x = \frac{π}{9} + \frac{2 π n}{3}$ , where $n$ is an integer. This is half of the period.

$x = \frac{π}{9} + \frac{2 π n}{3}$

Cosecant only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = \frac{π}{9} + \frac{2 π n}{3}$ where $n$ is an integer

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = \frac{π}{9} + \frac{2 π n}{3}$ where $n$ is an integer

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

$a = \frac{4}{5}$

$b = \frac{3}{2}$

$c = \frac{π}{6}$

$d = 0$

Since the graph of the function $c s c$ does not have a maximum or minimum value, there can be no value for the amplitude.

Amplitude: None

Find the period of $\frac{4 \csc (\frac{3 x}{2} - \frac{π}{6})}{5}$ .

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The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{3}{2}$ in the formula for period.

$\frac{2 π}{| \frac{3}{2} |}$

$\frac{3}{2}$ is approximately $1.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{3}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{2}{3}$

Multiply $2 π \frac{2}{3}$ .

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Combine $\frac{2}{3}$ and $2$ .

$\frac{2 \cdot 2}{3} π$

Multiply $2$ by $2$ .

$\frac{4}{3} π$

Combine $\frac{4}{3}$ and $π$ .

$\frac{4 π}{3}$

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: $\frac{\frac{π}{6}}{\frac{3}{2}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{π}{6} \cdot \frac{2}{3}$

Cancel the common factor of $2$ .

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Factor $2$ out of $6$ .

Phase Shift: $\frac{π}{2 (3)} \cdot \frac{2}{3}$

Cancel the common factor.

Phase Shift: $\frac{π}{2 \cdot 3} \cdot \frac{2}{3}$

Rewrite the expression.

Phase Shift: $\frac{π}{3} \cdot \frac{1}{3}$

Multiply $\frac{π}{3}$ and $\frac{1}{3}$ .

Phase Shift: $\frac{π}{3 \cdot 3}$

Multiply $3$ by $3$ .

Phase Shift: $\frac{π}{9}$

Find the vertical shift $d$ .

Vertical Shift: $0$

List the properties of the trigonometric function.

Amplitude: None

Period: $\frac{4 π}{3}$

Phase Shift: $\frac{π}{9}$ ( $\frac{π}{9}$ 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: $x = \frac{π}{9} + \frac{2 π n}{3}$ where $n$ is an integer

Amplitude: None

Period: $\frac{4 π}{3}$

Phase Shift: $\frac{π}{9}$ ( $\frac{π}{9}$ to the right)

Vertical Shift: $0$

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