# Graph f(x) equals | two cos((pix) divide two)|

To find the coordinate of the vertex, set the inside of the absolute value equal to . In this case, .

Find solution to 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 .

Because the term on each side of the equation has similar denominators, the numerators must be equal.

First, divide each term by and simplify.

Divide each term in by .

The common factor should be canceled of .

Cancel the common factor.

Divide by .

The common factor should be canceled.

Cancel the common factor.

Reformulate the expression.

The 1st and 4th quadrants of the cosine function are positive. To calculate the second answer, subtract the reference angle from to find the solution in the fourth quadrant.

Calculate .

First, multiply both sides of the equation by .

Clarify both sides of the equation.

Clarify .

The common factor should be canceled.

Cancel the common factor.

Should be rewritten 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 .

Add fractions.

Add and .

Add numerators over the common denominator.

Clarify the numerator.

First, multiply by .

Subtract from .

Remove the common factors to reduce the expression.

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.

Calculate the period of .

Evaluate the period of function using .

Replace with in the formula for period.

is approximately which is positive so remove the absolute value

Multiply the numerator by the denominator's reciprocal.

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 two directions at once.

, 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 presented by all real numbers except where we can't define the expression. 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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