Calculus Examples

$f (x) = 7 \sec (\frac{1}{4} π x - \frac{1}{2} π)$

Step 1

Remove parentheses.

$y = 7 \sec (\frac{1}{4} \cdot (π x) - \frac{1}{2} \cdot π)$

Step 2

Remove parentheses.

$y = 7 \sec (\frac{1}{4} \cdot (π x) - \frac{1}{2} \cdot π)$

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 $\frac{1}{4}$ .

$y = 7 \sec (\frac{π}{4} x - \frac{1}{2} \cdot π)$

Step 3.1.2

Combine $\frac{π}{4}$ and $x$ .

$y = 7 \sec (\frac{π x}{4} - \frac{1}{2} \cdot π)$

Step 3.2

Combine $π$ and $\frac{1}{2}$ .

$y = 7 \sec (\frac{π x}{4} - \frac{π}{2})$

Step 4

For any $y = \sec (x)$ , vertical asymptotes occur at $x = \frac{π}{2} + n π$ , where $n$ is an integer. Use the basic period for $y = s e c (x)$ , $(- \frac{π}{2}, \frac{3 π}{2})$ , to find the vertical asymptotes for $y = 7 \sec (\frac{π x}{4} - \frac{π}{2})$ . Set the inside of the secant function, $b x + c$ , for $y = a \sec (b x + c) + d$ equal to $- \frac{π}{2}$ to find where the vertical asymptote occurs for $y = 7 \sec (\frac{π x}{4} - \frac{π}{2})$ .

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

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

Add $\frac{π}{2}$ to both sides of the equation.

$\frac{π x}{4} = - \frac{π}{2} + \frac{π}{2}$

Step 5.1.2

Combine the numerators over the common denominator.

$\frac{π x}{4} = \frac{- π + π}{2}$

Step 5.1.3

Add $- π$ and $π$ .

$\frac{π x}{4} = \frac{0}{2}$

Step 5.1.4

Divide $0$ by $2$ .

$\frac{π x}{4} = 0$

Step 5.2

Set the numerator equal to zero.

$π x = 0$

Step 5.3

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

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Step 5.3.1

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

$\frac{π x}{π} = \frac{0}{π}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $π$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{π x}{π} = \frac{0}{π}$

Step 5.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{π}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Divide $0$ by $π$ .

$x = 0$

Step 6

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

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

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

Add $\frac{π}{2}$ to both sides of the equation.

$\frac{π x}{4} = \frac{3 π}{2} + \frac{π}{2}$

Step 7.1.2

Combine the numerators over the common denominator.

$\frac{π x}{4} = \frac{3 π + π}{2}$

Step 7.1.3

Add $3 π$ and $π$ .

$\frac{π x}{4} = \frac{4 π}{2}$

Step 7.1.4

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

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Step 7.1.4.1

Factor $2$ out of $4 π$ .

$\frac{π x}{4} = \frac{2 (2 π)}{2}$

Step 7.1.4.2

Cancel the common factors.

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Step 7.1.4.2.1

Factor $2$ out of $2$ .

$\frac{π x}{4} = \frac{2 (2 π)}{2 (1)}$

Step 7.1.4.2.2

Cancel the common factor.

$\frac{π x}{4} = \frac{2 (2 π)}{2 \cdot 1}$

Step 7.1.4.2.3

Rewrite the expression.

$\frac{π x}{4} = \frac{2 π}{1}$

Step 7.1.4.2.4

Divide $2 π$ by $1$ .

$\frac{π x}{4} = 2 π$

Step 7.2

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

$\frac{4}{π} \cdot \frac{π x}{4} = \frac{4}{π} (2 π)$

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.

$\frac{4}{π} \cdot \frac{π x}{4} = \frac{4}{π} (2 π)$

Step 7.3.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{4}{π} (2 π)$

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 $π x$ .

$\frac{1}{π} (π (x)) = \frac{4}{π} (2 π)$

Step 7.3.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{4}{π} (2 π)$

Step 7.3.1.1.2.3

Rewrite the expression.

$x = \frac{4}{π} (2 π)$

Step 7.3.2

Simplify the right side.

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Step 7.3.2.1

Simplify $\frac{4}{π} (2 π)$ .

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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 $2 π$ .

$x = \frac{4}{π} (π \cdot 2)$

Step 7.3.2.1.1.2

Cancel the common factor.

$x = \frac{4}{π} (π \cdot 2)$

Step 7.3.2.1.1.3

Rewrite the expression.

$x = 4 \cdot 2$

Step 7.3.2.1.2

Multiply $4$ by $2$ .

$x = 8$

Step 8

The basic period for $y = 7 \sec (\frac{π x}{4} - \frac{π}{2})$ will occur at $(0, 8)$ , where $0$ and $8$ are vertical asymptotes.

$(0, 8)$

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 $y = 7 \sec (\frac{π x}{4} - \frac{π}{2})$ occur at $0$ , $8$ , and every $x = 0 + 4 n$ , where $n$ is an integer. This is half of the period.

$x = 0 + 4 n$

Step 11

Secant only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = 0 + 4 n$ where $n$ is an integer

Step 12