Find the Asymptotes y=-sec(7x)

$y = - \sec (7 x)$

Step 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 $y = - \sec (7 x)$ . 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 $y = - \sec (7 x)$ .

$7 x = - \frac{π}{2}$

Step 2

Divide each term in $7 x = - \frac{π}{2}$ by $7$ and simplify.

Divide each term in $7 x = - \frac{π}{2}$ by $7$ .

$\frac{7 x}{7} = \frac{- \frac{π}{2}}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x}{7} = \frac{- \frac{π}{2}}{7}$

Divide $x$ by $1$ .

$x = \frac{- \frac{π}{2}}{7}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{π}{2} \cdot \frac{1}{7}$

Multiply $- \frac{π}{2} \cdot \frac{1}{7}$ .

Multiply $\frac{1}{7}$ by $\frac{π}{2}$ .

$x = - \frac{π}{7 \cdot 2}$

Multiply $7$ by $2$ .

$x = - \frac{π}{14}$

Step 3

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

$7 x = \frac{3 π}{2}$

Step 4

Divide each term in $7 x = \frac{3 π}{2}$ by $7$ and simplify.

Divide each term in $7 x = \frac{3 π}{2}$ by $7$ .

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

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{2}}{7}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{2} \cdot \frac{1}{7}$

Multiply $\frac{3 π}{2} \cdot \frac{1}{7}$ .

Multiply $\frac{3 π}{2}$ by $\frac{1}{7}$ .

$x = \frac{3 π}{2 \cdot 7}$

Multiply $2$ by $7$ .

$x = \frac{3 π}{14}$

Step 5

The basic period for $y = - \sec (7 x)$ will occur at $(- \frac{π}{14}, \frac{3 π}{14})$ , where $- \frac{π}{14}$ and $\frac{3 π}{14}$ are vertical asymptotes.

$(- \frac{π}{14}, \frac{3 π}{14})$

Step 6

The absolute value is the distance between a number and zero. The distance between $0$ and $7$ is $7$ .

$\frac{2 π}{7}$

Step 7

The vertical asymptotes for $y = - \sec (7 x)$ occur at $- \frac{π}{14}$ , $\frac{3 π}{14}$ , and every $x = \frac{π n}{7}$ , where $n$ is an integer. This is half of the period.

$x = \frac{π n}{7}$

Step 8

Secant only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = \frac{π n}{7}$ where $n$ is an integer

Step 9

Find the Asymptotes y=-sec(7x)

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