Graph y=4csc(4x)

$y = 4 \csc (4 x)$

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 $y = 4 \csc (4 x)$ . 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 $y = 4 \csc (4 x)$ .

$4 x = 0$

Divide each term by $4$ and simplify.

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Divide each term in $4 x = 0$ by $4$ .

$\frac{4 x}{4} = \frac{0}{4}$

Cancel the common factor of $4$ .

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Cancel the common factor.

$\frac{4 x}{4} = \frac{0}{4}$

Divide $x$ by $1$ .

$x = \frac{0}{4}$

Divide $0$ by $4$ .

$x = 0$

Set the inside of the cosecant function $4 x$ equal to $2 π$ .

$4 x = 2 π$

Divide each term by $4$ and simplify.

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Divide each term in $4 x = 2 π$ by $4$ .

$\frac{4 x}{4} = \frac{2 π}{4}$

Cancel the common factor of $4$ .

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Cancel the common factor.

$\frac{4 x}{4} = \frac{2 π}{4}$

Divide $x$ by $1$ .

$x = \frac{2 π}{4}$

Cancel the common factor of $2$ and $4$ .

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Factor $2$ out of $2 π$ .

$x = \frac{2 (π)}{4}$

Cancel the common factors.

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Factor $2$ out of $4$ .

$x = \frac{2 π}{2 \cdot 2}$

Cancel the common factor.

$x = \frac{2 π}{2 \cdot 2}$

Rewrite the expression.

$x = \frac{π}{2}$

The basic period for $y = 4 \csc (4 x)$ will occur at $(0, \frac{π}{2})$ , where $0$ and $\frac{π}{2}$ are vertical asymptotes.

$(0, \frac{π}{2})$

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

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The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{2 π}{4}$

Cancel the common factor of $2$ and $4$ .

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Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Cancel the common factors.

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Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Rewrite the expression.

$\frac{π}{2}$

The vertical asymptotes for $y = 4 \csc (4 x)$ occur at $0$ , $\frac{π}{2}$ , and every $x = \frac{π n}{4}$ , where $n$ is an integer. This is half of the period.

$x = \frac{π n}{4}$

Cosecant only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

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

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = \frac{π n}{4}$ 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.

$a = 4$

$b = 4$

$c = 0$

$d = 0$

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 $4 \csc (4 x)$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $4$ in the formula for period.

$\frac{2 π}{| 4 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{2 π}{4}$

Cancel the common factor of $2$ and $4$ .

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Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Cancel the common factors.

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Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Rewrite the expression.

$\frac{π}{2}$

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: $\frac{0}{4}$

Divide $0$ by $4$ .

Phase Shift: $0$

Find the vertical shift $d$ .

Vertical Shift: $0$

List the properties of the trigonometric function.

Amplitude: None

Period: $\frac{π}{2}$

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: $x = \frac{π n}{4}$ where $n$ is an integer

Amplitude: None

Period: $\frac{π}{2}$

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

Vertical Shift: $0$

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