Graph y=4csc(4x)

Graph y=4csc(4x)
Find the asymptotes.
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 .
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 cosecant 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 .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
The basic period for will occur at , where and are vertical asymptotes.
Find 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 .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
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
Find the period of .
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 .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Find the phase shift using the formula .
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:
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Name

Name four hundred forty million five hundred fifteen thousand one hundred forty-two

Interesting facts

• 440515142 has 16 divisors, whose sum is 673373952
• The reverse of 440515142 is 241515044
• Previous prime number is 191

Basic properties

• Is Prime? no
• Number parity even
• Number length 9
• Sum of Digits 26
• Digital Root 8

Name

Name one billion eight hundred ninety-eight million one hundred ninety-seven thousand one hundred thirteen

Interesting facts

• 1898197113 has 8 divisors, whose sum is 2532886048
• The reverse of 1898197113 is 3117918981
• Previous prime number is 1297

Basic properties

• Is Prime? no
• Number parity odd
• Number length 10
• Sum of Digits 48
• Digital Root 3

Name

Name eight hundred sixty-three million five hundred fifty-one thousand three hundred ninety-three

Interesting facts

• 863551393 has 4 divisors, whose sum is 863968048
• The reverse of 863551393 is 393155368
• Previous prime number is 2083

Basic properties

• Is Prime? no
• Number parity odd
• Number length 9
• Sum of Digits 43
• Digital Root 7