Graph f(x)=|2cos((pix)/2)|

Graph f(x)=|2cos((pix)/2)|
Find the absolute value vertex. In this case, the vertex for is .
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To find the coordinate of the vertex, set the inside of the absolute value equal to . In this case, .
Solve the equation to find the coordinate for the absolute value vertex.
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Take the inverse cosine of both sides of the equation to extract from inside the cosine.
The exact value of is .
Since the expression on each side of the equation has the same denominator, the numerators must be equal.
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 .
Cancel the common factor of .
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Cancel the common factor.
Rewrite the expression.
The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from to find the solution in the fourth quadrant.
Solve for .
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Multiply both sides of the equation by .
Simplify both sides of the equation.
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Simplify .
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Cancel the common factor of .
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Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Simplify .
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To write as a fraction with a common denominator, multiply by .
Combine fractions.
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Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by .
Subtract from .
Reduce the expression by cancelling the common factors.
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Cancel the common factor of .
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Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Find the period of .
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The period of the function can be calculated using .
Replace with in the formula for period.
is approximately which is positive so remove the absolute value
Multiply the numerator by the reciprocal of the denominator.
Cancel the common factor of .
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply by .
The period of the function is so values will repeat every radians in both directions.
, for any integer
Consolidate the answers.
, for any integer
, for any integer
Replace the variable with in the expression.
The absolute value vertex is .
The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
Interval Notation:
Set-Builder Notation:
The absolute value can be graphed using the points around the vertex
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Name

Name seven hundred ninety-nine million ninety thousand six hundred seventy-eight

Interesting facts

  • 799090678 has 8 divisors, whose sum is 1269144072
  • The reverse of 799090678 is 876090997
  • Previous prime number is 17

Basic properties

  • Is Prime? no
  • Number parity even
  • Number length 9
  • Sum of Digits 55
  • Digital Root 1

Name

Name two billion thirty-nine million eight hundred ninety thousand seven hundred nine

Interesting facts

  • 2039890709 has 4 divisors, whose sum is 2040093804
  • The reverse of 2039890709 is 9070989302
  • Previous prime number is 10597

Basic properties

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

Name

Name eleven million eight hundred fifty-seven thousand seven hundred seventy-four

Interesting facts

  • 11857774 has 16 divisors, whose sum is 18670176
  • The reverse of 11857774 is 47775811
  • Previous prime number is 41

Basic properties

  • Is Prime? no
  • Number parity even
  • Number length 8
  • Sum of Digits 40
  • Digital Root 4