# Precalculus Examples

Step 1

Step 1.1

For any , vertical asymptotes occur at , where is an integer. Use the basic period for , , to find the vertical asymptotes for . Set the inside of the cotangent function, , for equal to to find where the vertical asymptote occurs for .

Step 1.2

Divide each term in by and simplify.

Step 1.2.1

Divide each term in by .

Step 1.2.2

Simplify the left side.

Step 1.2.2.1

Cancel the common factor of .

Step 1.2.2.1.1

Cancel the common factor.

Step 1.2.2.1.2

Divide by .

Step 1.2.3

Simplify the right side.

Step 1.2.3.1

Divide by .

Step 1.3

Set the inside of the cotangent function equal to .

Step 1.4

Divide each term in by and simplify.

Step 1.4.1

Divide each term in by .

Step 1.4.2

Simplify the left side.

Step 1.4.2.1

Cancel the common factor of .

Step 1.4.2.1.1

Cancel the common factor.

Step 1.4.2.1.2

Divide by .

Step 1.5

The basic period for will occur at , where and are vertical asymptotes.

Step 1.6

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

Step 1.7

The vertical asymptotes for occur at , , and every , where is an integer.

Step 1.8

Cotangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: where is an integer

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: where is an integer

Step 2

Use the form to find the variables used to find the amplitude, period, phase shift, and vertical shift.

Step 3

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

Amplitude: None

Step 4

Step 4.1

The period of the function can be calculated using .

Step 4.2

Replace with in the formula for period.

Step 4.3

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

Step 5

Step 5.1

The phase shift of the function can be calculated from .

Phase Shift:

Step 5.2

Replace the values of and in the equation for phase shift.

Phase Shift:

Step 5.3

Divide by .

Phase Shift:

Phase Shift:

Step 6

List the properties of the trigonometric function.

Amplitude: None

Period:

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: where is an integer

Amplitude: None

Period:

Phase Shift: None

Vertical Shift: None

Step 8