Calculus Examples

$f (x) = 4 \csc (\frac{1}{4} π x + \frac{1}{3} π)$

Step 1

Remove parentheses.

$y = 4 \csc (\frac{1}{4} \cdot (π x) + \frac{1}{3} \cdot π)$

Step 2

Remove parentheses.

$y = 4 \csc (\frac{1}{4} \cdot (π x) + \frac{1}{3} \cdot π)$

Step 3

Simplify each term.

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Step 3.1

Multiply $\frac{1}{4} (π x)$ .

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Step 3.1.1

Combine $π$ and $\frac{1}{4}$ .

$y = 4 \csc (\frac{π}{4} x + \frac{1}{3} \cdot π)$

Step 3.1.2

Combine $\frac{π}{4}$ and $x$ .

$y = 4 \csc (\frac{π x}{4} + \frac{1}{3} \cdot π)$

Step 3.2

Combine $\frac{1}{3}$ and $π$ .

$y = 4 \csc (\frac{π x}{4} + \frac{π}{3})$

Step 4

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 $y = 4 \csc (\frac{π x}{4} + \frac{π}{3})$ . 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 $y = 4 \csc (\frac{π x}{4} + \frac{π}{3})$ .

$\frac{π x}{4} + \frac{π}{3} = 0$

Step 5

Solve for $x$ .

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Step 5.1

Subtract $\frac{π}{3}$ from both sides of the equation.

$\frac{π x}{4} = - \frac{π}{3}$

Step 5.2

Multiply both sides of the equation by $\frac{4}{π}$ .

$\frac{4}{π} \cdot \frac{π x}{4} = \frac{4}{π} (- \frac{π}{3})$

Step 5.3

Simplify both sides of the equation.

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Step 5.3.1

Simplify the left side.

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Step 5.3.1.1

Simplify $\frac{4}{π} \cdot \frac{π x}{4}$ .

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Step 5.3.1.1.1

Cancel the common factor of $4$ .

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Step 5.3.1.1.1.1

Cancel the common factor.

$\frac{4}{π} \cdot \frac{π x}{4} = \frac{4}{π} (- \frac{π}{3})$

Step 5.3.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{4}{π} (- \frac{π}{3})$

Step 5.3.1.1.2

Cancel the common factor of $π$ .

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Step 5.3.1.1.2.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{4}{π} (- \frac{π}{3})$

Step 5.3.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{4}{π} (- \frac{π}{3})$

Step 5.3.1.1.2.3

Rewrite the expression.

$x = \frac{4}{π} (- \frac{π}{3})$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Simplify $\frac{4}{π} (- \frac{π}{3})$ .

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Step 5.3.2.1.1

Cancel the common factor of $π$ .

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Step 5.3.2.1.1.1

Move the leading negative in $- \frac{π}{3}$ into the numerator.

$x = \frac{4}{π} \cdot \frac{- π}{3}$

Step 5.3.2.1.1.2

Factor $π$ out of $- π$ .

$x = \frac{4}{π} \cdot \frac{π \cdot - 1}{3}$

Step 5.3.2.1.1.3

Cancel the common factor.

$x = \frac{4}{π} \cdot \frac{π \cdot - 1}{3}$

Step 5.3.2.1.1.4

Rewrite the expression.

$x = 4 (\frac{- 1}{3})$

Step 5.3.2.1.2

Combine $4$ and $\frac{- 1}{3}$ .

$x = \frac{4 \cdot - 1}{3}$

Step 5.3.2.1.3

Simplify the expression.

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Step 5.3.2.1.3.1

Multiply $4$ by $- 1$ .

$x = \frac{- 4}{3}$

Step 5.3.2.1.3.2

Move the negative in front of the fraction.

$x = - \frac{4}{3}$

Step 6

Set the inside of the cosecant function $\frac{π x}{4} + \frac{π}{3}$ equal to $2 π$ .

$\frac{π x}{4} + \frac{π}{3} = 2 π$

Step 7

Solve for $x$ .

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Step 7.1

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

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Step 7.1.1

Subtract $\frac{π}{3}$ from both sides of the equation.

$\frac{π x}{4} = 2 π - \frac{π}{3}$

Step 7.1.2

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{π x}{4} = 2 π \cdot \frac{3}{3} - \frac{π}{3}$

Step 7.1.3

Combine $2 π$ and $\frac{3}{3}$ .

$\frac{π x}{4} = \frac{2 π \cdot 3}{3} - \frac{π}{3}$

Step 7.1.4

Combine the numerators over the common denominator.

$\frac{π x}{4} = \frac{2 π \cdot 3 - π}{3}$

Step 7.1.5

Simplify the numerator.

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Step 7.1.5.1

Multiply $3$ by $2$ .

$\frac{π x}{4} = \frac{6 π - π}{3}$

Step 7.1.5.2

Subtract $π$ from $6 π$ .

$\frac{π x}{4} = \frac{5 π}{3}$

Step 7.2

Multiply both sides of the equation by $\frac{4}{π}$ .

$\frac{4}{π} \cdot \frac{π x}{4} = \frac{4}{π} \cdot \frac{5 π}{3}$

Step 7.3

Simplify both sides of the equation.

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Step 7.3.1

Simplify the left side.

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Step 7.3.1.1

Simplify $\frac{4}{π} \cdot \frac{π x}{4}$ .

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Step 7.3.1.1.1

Cancel the common factor of $4$ .

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Step 7.3.1.1.1.1

Cancel the common factor.

$\frac{4}{π} \cdot \frac{π x}{4} = \frac{4}{π} \cdot \frac{5 π}{3}$

Step 7.3.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{4}{π} \cdot \frac{5 π}{3}$

Step 7.3.1.1.2

Cancel the common factor of $π$ .

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Step 7.3.1.1.2.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{4}{π} \cdot \frac{5 π}{3}$

Step 7.3.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{4}{π} \cdot \frac{5 π}{3}$

Step 7.3.1.1.2.3

Rewrite the expression.

$x = \frac{4}{π} \cdot \frac{5 π}{3}$

Step 7.3.2

Simplify the right side.

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Step 7.3.2.1

Simplify $\frac{4}{π} \cdot \frac{5 π}{3}$ .

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Step 7.3.2.1.1

Cancel the common factor of $π$ .

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Step 7.3.2.1.1.1

Factor $π$ out of $5 π$ .

$x = \frac{4}{π} \cdot \frac{π \cdot 5}{3}$

Step 7.3.2.1.1.2

Cancel the common factor.

$x = \frac{4}{π} \cdot \frac{π \cdot 5}{3}$

Step 7.3.2.1.1.3

Rewrite the expression.

$x = 4 (\frac{5}{3})$

Step 7.3.2.1.2

Combine $4$ and $\frac{5}{3}$ .

$x = \frac{4 \cdot 5}{3}$

Step 7.3.2.1.3

Multiply $4$ by $5$ .

$x = \frac{20}{3}$

Step 8

The basic period for $y = 4 \csc (\frac{π x}{4} + \frac{π}{3})$ will occur at $(- \frac{4}{3}, \frac{20}{3})$ , where $- \frac{4}{3}$ and $\frac{20}{3}$ are vertical asymptotes.

$(- \frac{4}{3}, \frac{20}{3})$

Step 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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Step 9.1

$\frac{π}{4}$ is approximately $0.78539816$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{4}}$

Step 9.2

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{4}{π}$

Step 9.3

Cancel the common factor of $π$ .

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Step 9.3.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{4}{π}$

Step 9.3.2

Cancel the common factor.

$π \cdot 2 \frac{4}{π}$

Step 9.3.3

Rewrite the expression.

$2 \cdot 4$

Step 9.4

Multiply $2$ by $4$ .

$8$

Step 10

The vertical asymptotes for $y = 4 \csc (\frac{π x}{4} + \frac{π}{3})$ occur at $- \frac{4}{3}$ , $\frac{20}{3}$ , and every $x = - \frac{4}{3} + 4 n$ , where $n$ is an integer. This is half of the period.

$x = - \frac{4}{3} + 4 n$

Step 11

Cosecant only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = - \frac{4}{3} + 4 n$ where $n$ is an integer

Step 12