Precalculus Examples

$f (x) = 2 \csc (4 x)$

Step 1

Find the asymptotes.

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Step 1.1

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 $y = 2 \csc (4 x)$ . 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 $y = 2 \csc (4 x)$ .

$4 x = 0$

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

Step 1.2.2.1.2

Divide $x$ by $1$ .

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

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $0$ by $4$ .

$x = 0$

Step 1.3

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

$4 x = 2 π$

Step 1.4

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

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Step 1.4.1

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

$\frac{4 x}{4} = \frac{2 π}{4}$

Step 1.4.2

Simplify the left side.

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Step 1.4.2.1

Cancel the common factor of $4$ .

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Step 1.4.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{2 π}{4}$

Step 1.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{2 π}{4}$

Step 1.4.3

Simplify the right side.

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Step 1.4.3.1

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

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Step 1.4.3.1.1

Factor $2$ out of $2 π$ .

$x = \frac{2 (π)}{4}$

Step 1.4.3.1.2

Cancel the common factors.

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Step 1.4.3.1.2.1

Factor $2$ out of $4$ .

$x = \frac{2 π}{2 \cdot 2}$

Step 1.4.3.1.2.2

Cancel the common factor.

$x = \frac{2 π}{2 \cdot 2}$

Step 1.4.3.1.2.3

Rewrite the expression.

$x = \frac{π}{2}$

Step 1.5

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

$(0, \frac{π}{2})$

Step 1.6

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

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Step 1.6.1

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

$\frac{2 π}{4}$

Step 1.6.2

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

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Step 1.6.2.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Step 1.6.2.2

Cancel the common factors.

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Step 1.6.2.2.1

Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Step 1.6.2.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Step 1.6.2.2.3

Rewrite the expression.

$\frac{π}{2}$

Step 1.7

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

$x = \frac{π n}{4}$

Step 1.8

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

Step 2

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

$a = 2$

$b = 4$

$c = 0$

$d = 0$

Step 3

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

Step 4

Find the period of $2 \csc (4 x)$ .

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Step 4.1

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

$\frac{2 π}{| b |}$

Step 4.2

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

$\frac{2 π}{| 4 |}$

Step 4.3

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

$\frac{2 π}{4}$

Step 4.4

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

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Step 4.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Step 4.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Step 4.4.2.3

Rewrite the expression.

$\frac{π}{2}$

Step 5

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

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Step 5.1

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

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

Step 5.2

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

Phase Shift: $\frac{0}{4}$

Step 5.3

Divide $0$ by $4$ .

Phase Shift: $0$

Step 6

List the properties of the trigonometric function.

Amplitude: None

Period: $\frac{π}{2}$

Phase Shift: None

Vertical Shift: None

Step 7

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: None

Vertical Shift: None

Step 8

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