# Precalculus Examples

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 .

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Divide by .

Set the inside of the cotangent function equal to .

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

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

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

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

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

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

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

Amplitude: None

The period of the function can be calculated using .

Replace with in the formula for period.

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

The phase shift of the function can be calculated from .

Phase Shift:

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

Phase Shift:

Divide by .

Phase Shift:

Phase Shift:

Find the vertical shift .

Vertical Shift:

List the properties of the trigonometric function.

Amplitude: None

Period:

Phase Shift: ( to the right)

Vertical Shift:

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: ( to the right)

Vertical Shift: