Find the absolute valuevertex. 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 valuevertex.
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Take the inversecosine 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 cosinefunction is positive in the first and fourth quadrants. To find the secondsolution, 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 .
Combinefractions.
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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 valuevertex 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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