Calculus Examples

$y = 5 \csc (\frac{3}{2} x + \frac{π}{2}) - 2$

Step 1

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

$y = 5 \csc (\frac{3 x}{2} + \frac{π}{2}) - 2$

Step 2

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 $y = 5 \csc (\frac{3 x}{2} + \frac{π}{2}) - 2$ . 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 $y = 5 \csc (\frac{3 x}{2} + \frac{π}{2}) - 2$ .

$\frac{3 x}{2} + \frac{π}{2} = 0$

Step 3

Solve for $x$ .

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Step 3.1

Subtract $\frac{π}{2}$ from both sides of the equation.

$\frac{3 x}{2} = - \frac{π}{2}$

Step 3.2

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$3 x = - π$

Step 3.3

Divide each term in $3 x = - π$ by $3$ and simplify.

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Step 3.3.1

Divide each term in $3 x = - π$ by $3$ .

$\frac{3 x}{3} = \frac{- π}{3}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $3$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- π}{3}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- π}{3}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Move the negative in front of the fraction.

$x = - \frac{π}{3}$

Step 4

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

$\frac{3 x}{2} + \frac{π}{2} = 2 π$

Step 5

Solve for $x$ .

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Step 5.1

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

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Step 5.1.1

Subtract $\frac{π}{2}$ from both sides of the equation.

$\frac{3 x}{2} = 2 π - \frac{π}{2}$

Step 5.1.2

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

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

Step 5.1.3

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 5.1.4

Combine the numerators over the common denominator.

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

Step 5.1.5

Simplify the numerator.

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Step 5.1.5.1

Multiply $2$ by $2$ .

$\frac{3 x}{2} = \frac{4 π - π}{2}$

Step 5.1.5.2

Subtract $π$ from $4 π$ .

$\frac{3 x}{2} = \frac{3 π}{2}$

Step 5.2

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$3 x = 3 π$

Step 5.3

Divide each term in $3 x = 3 π$ by $3$ and simplify.

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Step 5.3.1

Divide each term in $3 x = 3 π$ by $3$ .

$\frac{3 x}{3} = \frac{3 π}{3}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $3$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{3 π}{3}$

Step 5.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{3 π}{3}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Cancel the common factor of $3$ .

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Step 5.3.3.1.1

Cancel the common factor.

$x = \frac{3 π}{3}$

Step 5.3.3.1.2

Divide $π$ by $1$ .

$x = π$

Step 6

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

$(- \frac{π}{3}, π)$

Step 7

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 7.1

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

$\frac{2 π}{\frac{3}{2}}$

Step 7.2

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{2}{3}$

Step 7.3

Multiply $2 π \frac{2}{3}$ .

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Step 7.3.1

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

$\frac{2 \cdot 2}{3} π$

Step 7.3.2

Multiply $2$ by $2$ .

$\frac{4}{3} π$

Step 7.3.3

Combine $\frac{4}{3}$ and $π$ .

$\frac{4 π}{3}$

Step 8

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

$x = - \frac{π}{3} + \frac{2 π n}{3}$

Step 9

Cosecant only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

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

Step 10