Graph 3sec(1/7x)

$3 \sec (\frac{1}{7} x)$

Find the asymptotes.

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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 $y = 3 \sec (\frac{x}{7})$ . 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 $y = 3 \sec (\frac{x}{7})$ .

$\frac{x}{7} = - \frac{π}{2}$

Solve for $x$ .

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Multiply both sides of the equation by $7$ .

$7 \cdot \frac{x}{7} = 7 \cdot (- \frac{π}{2})$

Simplify both sides of the equation.

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Cancel the common factor of $7$ .

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Cancel the common factor.

$7 \cdot \frac{x}{7} = 7 \cdot (- \frac{π}{2})$

Rewrite the expression.

$x = 7 \cdot (- \frac{π}{2})$

Simplify $7 \cdot (- \frac{π}{2})$ .

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Multiply $7 (- \frac{π}{2})$ .

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Multiply $- 1$ by $7$ .

$x = - 7 \frac{π}{2}$

Combine $- 7$ and $\frac{π}{2}$ .

$x = \frac{- 7 π}{2}$

Move the negative in front of the fraction.

$x = - \frac{7 π}{2}$

Set the inside of the secant function $\frac{x}{7}$ equal to $\frac{3 π}{2}$ .

$\frac{x}{7} = \frac{3 π}{2}$

Solve for $x$ .

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Multiply both sides of the equation by $7$ .

$7 \cdot \frac{x}{7} = 7 \cdot \frac{3 π}{2}$

Simplify both sides of the equation.

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Cancel the common factor of $7$ .

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Cancel the common factor.

$7 \cdot \frac{x}{7} = 7 \cdot \frac{3 π}{2}$

Rewrite the expression.

$x = 7 \cdot \frac{3 π}{2}$

Multiply $7 \frac{3 π}{2}$ .

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Combine $7$ and $\frac{3 π}{2}$ .

$x = \frac{7 (3 π)}{2}$

Multiply $3$ by $7$ .

$x = \frac{21 π}{2}$

The basic period for $y = 3 \sec (\frac{x}{7})$ will occur at $(- \frac{7 π}{2}, \frac{21 π}{2})$ , where $- \frac{7 π}{2}$ and $\frac{21 π}{2}$ are vertical asymptotes.

$(- \frac{7 π}{2}, \frac{21 π}{2})$

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

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$\frac{1}{7}$ is approximately $0.\overline{142857}$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{7}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 7$

Multiply $7$ by $2$ .

$14 π$

The vertical asymptotes for $y = 3 \sec (\frac{x}{7})$ occur at $- \frac{7 π}{2}$ , $\frac{21 π}{2}$ , and every $x = 7 π n$ , where $n$ is an integer. This is half of the period.

$x = 7 π n$

Secant only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = 7 π n$ where $n$ is an integer

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = 7 π n$ where $n$ is an integer

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

$a = 3$

$b = \frac{1}{7}$

$c = 0$

$d = 0$

Since the graph of the function $s e c$ does not have a maximum or minimum value, there can be no value for the amplitude.

Amplitude: None

Find the period of $3 \sec (\frac{x}{7})$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{1}{7}$ in the formula for period.

$\frac{2 π}{| \frac{1}{7} |}$

$\frac{1}{7}$ is approximately $0.\overline{142857}$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{7}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 7$

Multiply $7$ by $2$ .

$14 π$

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

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

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

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

Phase Shift: $\frac{0}{\frac{1}{7}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $0 \cdot 7$

Multiply $0$ by $7$ .

Phase Shift: $0$

Find the vertical shift $d$ .

Vertical Shift: $0$

List the properties of the trigonometric function.

Amplitude: None

Period: $14 π$

Phase Shift: $0$ ( $0$ to the right)

Vertical Shift: $0$

The trig function can be graphed using the amplitude, period, phase shift, vertical shift, and the points.

Vertical Asymptotes: $x = 7 π n$ where $n$ is an integer

Amplitude: None

Period: $14 π$

Phase Shift: $0$ ( $0$ to the right)

Vertical Shift: $0$

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