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

Graph f(x)=|2cos((pix)/2)|
Find the absolute value vertex. In this case, the vertex for is .
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.
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.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Cancel the common factor of .
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 .
Multiply both sides of the equation by .
Simplify both sides of the equation.
Simplify .
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Simplify .
To write as a fraction with a common denominator, multiply by .
Combine fractions.
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Subtract from .
Reduce the expression by cancelling the common factors.
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Find the period of .
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 .
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
, 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 one billion one hundred ten million nine hundred twenty-four thousand nine hundred seventy-five

### Interesting facts

• 1110924975 has 32 divisors, whose sum is 2134605312
• The reverse of 1110924975 is 5794290111
• Previous prime number is 1511

### Basic properties

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

### Name

Name one billion one hundred six million one hundred fifty-four thousand thirteen

### Interesting facts

• 1106154013 has 4 divisors, whose sum is 1106340508
• The reverse of 1106154013 is 3104516011
• Previous prime number is 6133

### Basic properties

• Is Prime? no
• Number parity odd
• Number length 10
• Sum of Digits 22
• Digital Root 4

### Name

Name nine hundred forty-two million seven hundred four thousand one hundred seventy

### Interesting facts

• 942704170 has 16 divisors, whose sum is 1700588808
• The reverse of 942704170 is 071407249
• Previous prime number is 457

### Basic properties

• Is Prime? no
• Number parity even
• Number length 9
• Sum of Digits 34
• Digital Root 7