Trigonometry Examples

$y = - 6 \sec (\frac{π}{3} x)$

Step 1

Find the asymptotes.

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Step 1.1

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 $y = - 6 \sec (\frac{π x}{3})$ . 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 $y = - 6 \sec (\frac{π x}{3})$ .

$\frac{π x}{3} = - \frac{π}{2}$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Multiply both sides of the equation by $\frac{3}{π}$ .

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

Step 1.2.2

Simplify both sides of the equation.

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Step 1.2.2.1

Simplify the left side.

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Step 1.2.2.1.1

Simplify $\frac{3}{π} \cdot \frac{π x}{3}$ .

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Step 1.2.2.1.1.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1.1.1

Cancel the common factor.

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

Step 1.2.2.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{3}{π} (- \frac{π}{2})$

Step 1.2.2.1.1.2

Cancel the common factor of $π$ .

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Step 1.2.2.1.1.2.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{3}{π} (- \frac{π}{2})$

Step 1.2.2.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{3}{π} (- \frac{π}{2})$

Step 1.2.2.1.1.2.3

Rewrite the expression.

$x = \frac{3}{π} (- \frac{π}{2})$

Step 1.2.2.2

Simplify the right side.

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Step 1.2.2.2.1

Simplify $\frac{3}{π} (- \frac{π}{2})$ .

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Step 1.2.2.2.1.1

Cancel the common factor of $π$ .

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Step 1.2.2.2.1.1.1

Move the leading negative in $- \frac{π}{2}$ into the numerator.

$x = \frac{3}{π} \cdot \frac{- π}{2}$

Step 1.2.2.2.1.1.2

Factor $π$ out of $- π$ .

$x = \frac{3}{π} \cdot \frac{π \cdot - 1}{2}$

Step 1.2.2.2.1.1.3

Cancel the common factor.

$x = \frac{3}{π} \cdot \frac{π \cdot - 1}{2}$

Step 1.2.2.2.1.1.4

Rewrite the expression.

$x = 3 (\frac{- 1}{2})$

Step 1.2.2.2.1.2

Combine $3$ and $\frac{- 1}{2}$ .

$x = \frac{3 \cdot - 1}{2}$

Step 1.2.2.2.1.3

Simplify the expression.

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Step 1.2.2.2.1.3.1

Multiply $3$ by $- 1$ .

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

Step 1.2.2.2.1.3.2

Move the negative in front of the fraction.

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

Step 1.3

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

$\frac{π x}{3} = \frac{3 π}{2}$

Step 1.4

Solve for $x$ .

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Step 1.4.1

Multiply both sides of the equation by $\frac{3}{π}$ .

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

Step 1.4.2

Simplify both sides of the equation.

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Step 1.4.2.1

Simplify the left side.

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Step 1.4.2.1.1

Simplify $\frac{3}{π} \cdot \frac{π x}{3}$ .

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Step 1.4.2.1.1.1

Cancel the common factor of $3$ .

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Step 1.4.2.1.1.1.1

Cancel the common factor.

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

Step 1.4.2.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{3}{π} \cdot \frac{3 π}{2}$

Step 1.4.2.1.1.2

Cancel the common factor of $π$ .

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Step 1.4.2.1.1.2.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{3}{π} \cdot \frac{3 π}{2}$

Step 1.4.2.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{3}{π} \cdot \frac{3 π}{2}$

Step 1.4.2.1.1.2.3

Rewrite the expression.

$x = \frac{3}{π} \cdot \frac{3 π}{2}$

Step 1.4.2.2

Simplify the right side.

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Step 1.4.2.2.1

Simplify $\frac{3}{π} \cdot \frac{3 π}{2}$ .

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Step 1.4.2.2.1.1

Cancel the common factor of $π$ .

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Step 1.4.2.2.1.1.1

Factor $π$ out of $3 π$ .

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

Step 1.4.2.2.1.1.2

Cancel the common factor.

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

Step 1.4.2.2.1.1.3

Rewrite the expression.

$x = 3 (\frac{3}{2})$

Step 1.4.2.2.1.2

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

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

Step 1.4.2.2.1.3

Multiply $3$ by $3$ .

$x = \frac{9}{2}$

Step 1.5

The basic period for $y = - 6 \sec (\frac{π x}{3})$ will occur at $(- \frac{3}{2}, \frac{9}{2})$ , where $- \frac{3}{2}$ and $\frac{9}{2}$ are vertical asymptotes.

$(- \frac{3}{2}, \frac{9}{2})$

Step 1.6

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 1.6.1

$\frac{π}{3}$ is approximately $1.04719755$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{3}}$

Step 1.6.2

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{π}$

Step 1.6.3

Cancel the common factor of $π$ .

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Step 1.6.3.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{3}{π}$

Step 1.6.3.2

Cancel the common factor.

$π \cdot 2 \frac{3}{π}$

Step 1.6.3.3

Rewrite the expression.

$2 \cdot 3$

Step 1.6.4

Multiply $2$ by $3$ .

$6$

Step 1.7

The vertical asymptotes for $y = - 6 \sec (\frac{π x}{3})$ occur at $- \frac{3}{2}$ , $\frac{9}{2}$ , and every $3 n$ , where $n$ is an integer. This is half of the period.

$x = \frac{9}{2} + 3 n$

Step 1.8

Secant only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = \frac{9}{2} + 3 n$ where $n$ is an integer

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = \frac{9}{2} + 3 n$ 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.

$a = - 6$

$b = \frac{π}{3}$

$c = 0$

$d = 0$

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 $- 6 \sec (\frac{π x}{3})$ .

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Step 4.1

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

$\frac{2 π}{| b |}$

Step 4.2

Replace $b$ with $\frac{π}{3}$ in the formula for period.

$\frac{2 π}{| \frac{π}{3} |}$

Step 4.3

$\frac{π}{3}$ is approximately $1.04719755$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{3}}$

Step 4.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{π}$

Step 4.5

Cancel the common factor of $π$ .

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Step 4.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{3}{π}$

Step 4.5.2

Cancel the common factor.

$π \cdot 2 \frac{3}{π}$

Step 4.5.3

Rewrite the expression.

$2 \cdot 3$

Step 4.6

Multiply $2$ by $3$ .

$6$

Step 5

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

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Step 5.1

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

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

Step 5.2

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

Phase Shift: $\frac{0}{\frac{π}{3}}$

Step 5.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $0 (\frac{3}{π})$

Step 5.4

Multiply $0$ by $\frac{3}{π}$ .

Phase Shift: $0$

Step 6

List the properties of the trigonometric function.

Amplitude: None

Period: $6$

Phase Shift: None

Vertical Shift: None

Step 7

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

Vertical Asymptotes: $x = \frac{9}{2} + 3 n$ where $n$ is an integer

Amplitude: None

Period: $6$

Phase Shift: None

Vertical Shift: None

Step 8