Graph y=-1+csc(x)

Graph y=-1+csc(x)
Find the asymptotes.
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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 .
Set the inside of the cosecant function equal to .
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.
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The absolute value is the distance between a number and zero. The distance between and is .
Divide by .
The vertical asymptotes for occur at , , and every , where is an integer. This is half of the period.
There are only vertical asymptotes for secant and cosecant functions.
Vertical Asymptotes: for any integer
No Horizontal Asymptotes
No Oblique Asymptotes
Vertical Asymptotes: for any integer
No Horizontal Asymptotes
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.
Vertical Asymptotes: for any integer
Amplitude: None
Period:
Phase Shift: ( to the right)
Vertical Shift:
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Name

Name one billion two hundred nine million two hundred forty-one thousand two hundred forty-three

Interesting facts

  • 1209241243 has 4 divisors, whose sum is 1381990000
  • The reverse of 1209241243 is 3421429021
  • Previous prime number is 7

Basic properties

  • Is Prime? no
  • Number parity odd
  • Number length 10
  • Sum of Digits 28
  • Digital Root 1

Name

Name two billion seventy-two million nine hundred fifty-eight thousand four hundred eleven

Interesting facts

  • 2072958411 has 16 divisors, whose sum is 2842156800
  • The reverse of 2072958411 is 1148592702
  • Previous prime number is 839

Basic properties

  • Is Prime? no
  • Number parity odd
  • Number length 10
  • Sum of Digits 39
  • Digital Root 3

Name

Name one billion two hundred ninety-five million five hundred three thousand nine hundred sixty-two

Interesting facts

  • 1295503962 has 8 divisors, whose sum is 2591007936
  • The reverse of 1295503962 is 2693055921
  • Previous prime number is 3

Basic properties

  • Is Prime? no
  • Number parity even
  • Number length 10
  • Sum of Digits 42
  • Digital Root 6