Calculus Examples

$f (x) = \csc (\frac{π x}{2})$

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 $y = \csc (\frac{π x}{2})$ . 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 $y = \csc (\frac{π x}{2})$ .

$\frac{π x}{2} = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Set the numerator equal to zero.

$π x = 0$

Step 1.2.2

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

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Step 1.2.2.1

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

$\frac{π x}{π} = \frac{0}{π}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $π$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{π x}{π} = \frac{0}{π}$

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{π}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $0$ by $π$ .

$x = 0$

Step 1.3

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

$\frac{π x}{2} = 2 π$

Step 1.4

Solve for $x$ .

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Step 1.4.1

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

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

Step 1.4.2

Simplify both sides of the equation.

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Step 1.4.2.1

Simplify the left side.

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Step 1.4.2.1.1

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

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Step 1.4.2.1.1.1

Cancel the common factor of $2$ .

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Step 1.4.2.1.1.1.1

Cancel the common factor.

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

Step 1.4.2.1.1.1.2

Rewrite the expression.

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

Step 1.4.2.1.1.2

Cancel the common factor of $π$ .

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Step 1.4.2.1.1.2.1

Factor $π$ out of $π x$ .

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

Step 1.4.2.1.1.2.2

Cancel the common factor.

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

Step 1.4.2.1.1.2.3

Rewrite the expression.

$x = \frac{2}{π} (2 π)$

Step 1.4.2.2

Simplify the right side.

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Step 1.4.2.2.1

Simplify $\frac{2}{π} (2 π)$ .

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Step 1.4.2.2.1.1

Cancel the common factor of $π$ .

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Step 1.4.2.2.1.1.1

Factor $π$ out of $2 π$ .

$x = \frac{2}{π} (π \cdot 2)$

Step 1.4.2.2.1.1.2

Cancel the common factor.

$x = \frac{2}{π} (π \cdot 2)$

Step 1.4.2.2.1.1.3

Rewrite the expression.

$x = 2 \cdot 2$

Step 1.4.2.2.1.2

Multiply $2$ by $2$ .

$x = 4$

Step 1.5

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

$(0, 4)$

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

$\frac{π}{2}$ is approximately $1.57079632$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{2}}$

Step 1.6.2

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{2}{π}$

Step 1.6.3

Cancel the common factor of $π$ .

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Step 1.6.3.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{2}{π}$

Step 1.6.3.2

Cancel the common factor.

$π \cdot 2 \frac{2}{π}$

Step 1.6.3.3

Rewrite the expression.

$2 \cdot 2$

Step 1.6.4

Multiply $2$ by $2$ .

$4$

Step 1.7

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

$x = 2 n$

Step 1.8

Cosecant only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = 2 n$ where $n$ is an integer

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = 2 n$ 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.

$a = 1$

$b = \frac{π}{2}$

$c = 0$

$d = 0$

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 $\csc (\frac{π x}{2})$ .

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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 $\frac{π}{2}$ in the formula for period.

$\frac{2 π}{| \frac{π}{2} |}$

Step 4.3

$\frac{π}{2}$ is approximately $1.57079632$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{2}}$

Step 4.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{2}{π}$

Step 4.5

Cancel the common factor of $π$ .

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Step 4.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{2}{π}$

Step 4.5.2

Cancel the common factor.

$π \cdot 2 \frac{2}{π}$

Step 4.5.3

Rewrite the expression.

$2 \cdot 2$

Step 4.6

Multiply $2$ by $2$ .

$4$

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: $\frac{0}{\frac{π}{2}}$

Step 5.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $0 (\frac{2}{π})$

Step 5.4

Multiply $0$ by $\frac{2}{π}$ .

Phase Shift: $0$

Step 6

List the properties of the trigonometric function.

Amplitude: None

Period: $4$

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: $x = 2 n$ where $n$ is an integer

Amplitude: None

Period: $4$

Phase Shift: None

Vertical Shift: None

Step 8