For any , verticalasymptotes occur at , where is an integer. Use the basic period for , , to find the verticalasymptotes for . Set the inside of the cosecantfunction, , for equal to to find where the verticalasymptote occurs for .
Set the inside of the cosecantfunction equal to .
The basic period for will occur at , where and are verticalasymptotes.
Find the period to find where the verticalasymptotes exist. Verticalasymptotes occur every half period.
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The absolute value is the distance between a number and zero. The distance between and is .
Divide by .
The verticalasymptotes for occur at , , and every , where is an integer. This is half of the period.
There are only verticalasymptotes for secant and cosecantfunctions.
VerticalAsymptotes: for any integer
No HorizontalAsymptotes
No Oblique Asymptotes
VerticalAsymptotes: for any integer
No HorizontalAsymptotes
No Oblique Asymptotes
Rewrite the expression as .
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
Find the period of .
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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 .
Divide by .
Find the phase shift using the formula .
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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.
VerticalAsymptotes: for any integer
Amplitude: None
Period:
Phase Shift: ( to the right)
Vertical Shift:
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