Precalculus Examples

$y = \cot (x - \frac{π}{2})$

Step 1

Find the asymptotes.

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Step 1.1

For any $y = \cot (x)$ , vertical asymptotes occur at $x = n π$ , where $n$ is an integer. Use the basic period for $y = \cot (x)$ , $(0, π)$ , to find the vertical asymptotes for $y = \cot (x - \frac{π}{2})$ . Set the inside of the cotangent function, $b x + c$ , for $y = a \cot (b x + c) + d$ equal to $0$ to find where the vertical asymptote occurs for $y = \cot (x - \frac{π}{2})$ .

$x - \frac{π}{2} = 0$

Step 1.2

Add $\frac{π}{2}$ to both sides of the equation.

$x = \frac{π}{2}$

Step 1.3

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

$x - \frac{π}{2} = π$

Step 1.4

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

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Step 1.4.1

Add $\frac{π}{2}$ to both sides of the equation.

$x = π + \frac{π}{2}$

Step 1.4.2

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

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

Step 1.4.3

Combine $π$ and $\frac{2}{2}$ .

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

Step 1.4.4

Combine the numerators over the common denominator.

$x = \frac{π \cdot 2 + π}{2}$

Step 1.4.5

Simplify the numerator.

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Step 1.4.5.1

Move $2$ to the left of $π$ .

$x = \frac{2 \cdot π + π}{2}$

Step 1.4.5.2

Add $2 π$ and $π$ .

$x = \frac{3 π}{2}$

Step 1.5

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

$(\frac{π}{2}, \frac{3 π}{2})$

Step 1.6

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

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Step 1.6.1

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

$\frac{π}{1}$

Step 1.6.2

Divide $π$ by $1$ .

$π$

Step 1.7

The vertical asymptotes for $y = \cot (x - \frac{π}{2})$ occur at $\frac{π}{2}$ , $\frac{3 π}{2}$ , and every $x = \frac{π}{2} + π n$ , where $n$ is an integer.

$x = \frac{π}{2} + π n$

Step 1.8

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

$a = 1$

$b = 1$

$c = \frac{π}{2}$

$d = 0$

Step 3

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

Amplitude: None

Step 4

Find the period of $\cot (x - \frac{π}{2})$ .

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Step 4.1

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

$\frac{π}{| b |}$

Step 4.2

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

$\frac{π}{| 1 |}$

Step 4.3

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

$\frac{π}{1}$

Step 4.4

Divide $π$ by $1$ .

$π$

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: $\frac{\frac{π}{2}}{1}$

Step 5.3

Divide $\frac{π}{2}$ by $1$ .

Phase Shift: $\frac{π}{2}$

Step 6

List the properties of the trigonometric function.

Amplitude: None

Period: $π$

Phase Shift: $\frac{π}{2}$ ( $\frac{π}{2}$ 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: $π$

Phase Shift: $\frac{π}{2}$ ( $\frac{π}{2}$ to the right)

Vertical Shift: None

Step 8