Precalculus Examples

$y = - \tan (x - \frac{π}{3})$

Step 1

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

$x - \frac{π}{3} = - \frac{π}{2}$

Step 2

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

Tap for more steps...

Step 2.1

Add $\frac{π}{3}$ to both sides of the equation.

$x = - \frac{π}{2} + \frac{π}{3}$

Step 2.2

To write $- \frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 2.3

To write $\frac{π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 2.4.1

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

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

Step 2.4.2

Multiply $2$ by $3$ .

$x = - \frac{π \cdot 3}{6} + \frac{π}{3} \cdot \frac{2}{2}$

Step 2.4.3

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

$x = - \frac{π \cdot 3}{6} + \frac{π \cdot 2}{3 \cdot 2}$

Step 2.4.4

Multiply $3$ by $2$ .

$x = - \frac{π \cdot 3}{6} + \frac{π \cdot 2}{6}$

Step 2.5

Combine the numerators over the common denominator.

$x = \frac{- π \cdot 3 + π \cdot 2}{6}$

Step 2.6

Simplify the numerator.

Tap for more steps...

Step 2.6.1

Multiply $3$ by $- 1$ .

$x = \frac{- 3 π + π \cdot 2}{6}$

Step 2.6.2

Move $2$ to the left of $π$ .

$x = \frac{- 3 π + 2 \cdot π}{6}$

Step 2.6.3

Add $- 3 π$ and $2 π$ .

$x = \frac{- π}{6}$

Step 2.7

Move the negative in front of the fraction.

$x = - \frac{π}{6}$

Step 3

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

$x - \frac{π}{3} = \frac{π}{2}$

Step 4

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

Tap for more steps...

Step 4.1

Add $\frac{π}{3}$ to both sides of the equation.

$x = \frac{π}{2} + \frac{π}{3}$

Step 4.2

To write $\frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 4.3

To write $\frac{π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 4.4.1

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

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

Step 4.4.2

Multiply $2$ by $3$ .

$x = \frac{π \cdot 3}{6} + \frac{π}{3} \cdot \frac{2}{2}$

Step 4.4.3

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

$x = \frac{π \cdot 3}{6} + \frac{π \cdot 2}{3 \cdot 2}$

Step 4.4.4

Multiply $3$ by $2$ .

$x = \frac{π \cdot 3}{6} + \frac{π \cdot 2}{6}$

Step 4.5

Combine the numerators over the common denominator.

$x = \frac{π \cdot 3 + π \cdot 2}{6}$

Step 4.6

Simplify the numerator.

Tap for more steps...

Step 4.6.1

Move $3$ to the left of $π$ .

$x = \frac{3 \cdot π + π \cdot 2}{6}$

Step 4.6.2

Move $2$ to the left of $π$ .

$x = \frac{3 π + 2 \cdot π}{6}$

Step 4.6.3

Add $3 π$ and $2 π$ .

$x = \frac{5 π}{6}$

Step 5

The basic period for $y = - \tan (x - \frac{π}{3})$ will occur at $(- \frac{π}{6}, \frac{5 π}{6})$ , where $- \frac{π}{6}$ and $\frac{5 π}{6}$ are vertical asymptotes.

$(- \frac{π}{6}, \frac{5 π}{6})$

Step 6

Find the period $\frac{π}{| b |}$ to find where the vertical asymptotes exist.

Tap for more steps...

Step 6.1

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

$\frac{π}{1}$

Step 6.2

Divide $π$ by $1$ .

$π$

Step 7

The vertical asymptotes for $y = - \tan (x - \frac{π}{3})$ occur at $- \frac{π}{6}$ , $\frac{5 π}{6}$ , and every $x = - \frac{π}{6} + π n$ , where $n$ is an integer.

$x = - \frac{π}{6} + π n$

Step 8

Tangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = - \frac{π}{6} + π n$ where $n$ is an integer

Step 9