Trigonometry Examples

$y = - \sec (x - π)$

Step 1

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 $y = - \sec (x - π)$ . 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 $y = - \sec (x - π)$ .

$x - π = - \frac{π}{2}$

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

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Add $π$ to both sides of the equation.

$x = - \frac{π}{2} + π$

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

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

Combine $π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

$x = \frac{- π + π \cdot 2}{2}$

Simplify the numerator.

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Move $2$ to the left of $π$ .

$x = \frac{- π + 2 \cdot π}{2}$

Add $- π$ and $2 π$ .

$x = \frac{π}{2}$

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

$x - π = \frac{3 π}{2}$

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

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Add $π$ to both sides of the equation.

$x = \frac{3 π}{2} + π$

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

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

Combine $π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

$x = \frac{3 π + π \cdot 2}{2}$

Simplify the numerator.

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Move $2$ to the left of $π$ .

$x = \frac{3 π + 2 \cdot π}{2}$

Add $3 π$ and $2 π$ .

$x = \frac{5 π}{2}$

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

$(\frac{π}{2}, \frac{5 π}{2})$

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

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The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

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

$x = \frac{π}{2} + π n$

Secant only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

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

No Horizontal Asymptotes

No Oblique Asymptotes

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

Step 2

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

$a = - 1$

$b = 1$

$c = π$

$d = 0$

Step 3

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

Step 4

Find the period of $- \sec (x - π)$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

Step 5

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: $\frac{π}{1}$

Divide $π$ by $1$ .

Phase Shift: $π$

Step 6

List the properties of the trigonometric function.

Amplitude: None

Period: $2 π$

Phase Shift: $π$ ( $π$ to the right)

Vertical Shift: None

Step 7

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

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

Amplitude: None

Period: $2 π$

Phase Shift: $π$ ( $π$ to the right)

Vertical Shift: None

Step 8