# 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 one hundred million two hundred eighty-one thousand five hundred fifty-six

### Interesting facts

• 100281556 has 8 divisors, whose sum is 225633510
• The reverse of 100281556 is 655182001
• Previous prime number is 2

### Basic properties

• Is Prime? no
• Number parity even
• Number length 9
• Sum of Digits 28
• Digital Root 1

### Name

Name eight hundred thirty-five million three hundred sixty-eight thousand one hundred forty-seven

### Interesting facts

• 835368147 has 32 divisors, whose sum is 1544696832
• The reverse of 835368147 is 741863538
• Previous prime number is 347

### Basic properties

• Is Prime? no
• Number parity odd
• Number length 9
• Sum of Digits 45
• Digital Root 9

### Name

Name four hundred forty-nine million six hundred eighty thousand eight hundred eighty-seven

### Interesting facts

• 449680887 has 16 divisors, whose sum is 801665280
• The reverse of 449680887 is 788086944
• Previous prime number is 359

### Basic properties

• Is Prime? no
• Number parity odd
• Number length 9
• Sum of Digits 54
• Digital Root 9