# Graph y equals csc(x plus(four pi) divide five)

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 cosecant function, , for equal to to find where the vertical asymptote occurs for .

Subtract from both sides of the equation.

Set the inside of the cosecant function equal to .

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

Subtract from both sides of the equation.

To write as a fraction with a common denominator, First, multiply by .

Add and .

Add numerators over the common denominator.

Clarify the numerator.

Multiply by .

Subtract from .

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

Calculate the period to find where the vertical asymptotes exist. Vertical asymptotes occur every half period.

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

First, divide by .

The vertical asymptotes for occur at , , and every , where is an integer. This is half of the period.

Cosecant 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

Evaluate the period of function using .

Replace with in the formula for period.

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

Divide by .

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 left)

Vertical Shift:

Use the amplitude to plot the trig function, period, phase shift, vertical shift, and the points.

Vertical Asymptotes: where is an integer

Amplitude: None

Period:

Phase Shift: ( to the left)

Vertical Shift:

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