Graph -2-2tan(4x-2pi)

Graph -2-2tan(4x-2pi)
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 tangent function, , for equal to to find where the vertical asymptote occurs for .
Solve for .
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Move all terms not containing to the right side of the equation.
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Add to both sides of the equation.
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by .
Add and .
Divide each term by and simplify.
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Divide each term in by .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Multiply .
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Multiply and .
Multiply by .
Set the inside of the tangent function equal to .
Solve for .
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Move all terms not containing to the right side of the equation.
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Add to both sides of the equation.
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by .
Add and .
Divide each term by and simplify.
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Divide each term in by .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Multiply .
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Multiply and .
Multiply by .
The basic period for will occur at , where and are vertical asymptotes.
The absolute value is the distance between a number and zero. The distance between and is .
The vertical asymptotes for occur at , , and every , where is an integer.
Tangent 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
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 .
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:
Cancel the common factor of and .
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Factor out of .
Phase Shift:
Cancel the common factors.
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Factor out of .
Phase Shift:
Cancel the common factor.
Phase Shift:
Rewrite the expression.
Phase Shift:
Phase Shift:
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 billion six hundred seven million eight hundred eighty-four thousand eighty-five

Interesting facts

  • 1607884085 has 16 divisors, whose sum is 1978808832
  • The reverse of 1607884085 is 5804887061
  • Previous prime number is 967

Basic properties

  • Is Prime? no
  • Number parity odd
  • Number length 10
  • Sum of Digits 47
  • Digital Root 2

Name

Name one billion nine hundred thirty-five million eight hundred eighty-three thousand eight hundred thirty-six

Interesting facts

  • 1935883836 has 16 divisors, whose sum is 4839709680
  • The reverse of 1935883836 is 6383885391
  • Previous prime number is 9

Basic properties

  • Is Prime? no
  • Number parity even
  • Number length 10
  • Sum of Digits 54
  • Digital Root 9

Name

Name two hundred thirty-nine million seven hundred six thousand forty-four

Interesting facts

  • 239706044 has 16 divisors, whose sum is 540291600
  • The reverse of 239706044 is 440607932
  • Previous prime number is 569

Basic properties

  • Is Prime? no
  • Number parity even
  • Number length 9
  • Sum of Digits 35
  • Digital Root 8