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 eight hundred sixty-nine million seventy thousand twenty-five

Interesting facts

  • 869070025 has 16 divisors, whose sum is 1257080832
  • The reverse of 869070025 is 520070968
  • Previous prime number is 223

Basic properties

  • Is Prime? no
  • Number parity odd
  • Number length 9
  • Sum of Digits 37
  • Digital Root 1

Name

Name one billion seven hundred twenty-seven million seven hundred eighty-six thousand six hundred eighty-one

Interesting facts

  • 1727786681 has 8 divisors, whose sum is 1749535200
  • The reverse of 1727786681 is 1866877271
  • Previous prime number is 2393

Basic properties

  • Is Prime? no
  • Number parity odd
  • Number length 10
  • Sum of Digits 53
  • Digital Root 8

Name

Name one hundred fifty-nine million one hundred five thousand two hundred seventy-eight

Interesting facts

  • 159105278 has 32 divisors, whose sum is 267805440
  • The reverse of 159105278 is 872501951
  • Previous prime number is 163

Basic properties

  • Is Prime? no
  • Number parity even
  • Number length 9
  • Sum of Digits 38
  • Digital Root 2