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Trigonometry Examples
Step 1
Step 1.1
To find the coordinate of the vertex, set the inside of the absolute value equal to . In this case, .
Step 1.2
Solve the equation to find the coordinate for the absolute value vertex.
Step 1.2.1
Subtract from both sides of the equation.
Step 1.2.2
Multiply both sides of the equation by .
Step 1.2.3
Simplify both sides of the equation.
Step 1.2.3.1
Simplify the left side.
Step 1.2.3.1.1
Cancel the common factor of .
Step 1.2.3.1.1.1
Cancel the common factor.
Step 1.2.3.1.1.2
Rewrite the expression.
Step 1.2.3.2
Simplify the right side.
Step 1.2.3.2.1
Multiply by .
Step 1.3
Replace the variable with in the expression.
Step 1.4
Simplify .
Step 1.4.1
Simplify each term.
Step 1.4.1.1
Divide by .
Step 1.4.1.2
Add and .
Step 1.4.1.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 1.4.2
Subtract from .
Step 1.5
The absolute value vertex is .
Step 2
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:
Step 3
Step 3.1
Substitute the value into . In this case, the point is .
Step 3.1.1
Replace the variable with in the expression.
Step 3.1.2
Simplify the result.
Step 3.1.2.1
Simplify each term.
Step 3.1.2.1.1
Simplify each term.
Step 3.1.2.1.1.1
Cancel the common factor of and .
Step 3.1.2.1.1.1.1
Factor out of .
Step 3.1.2.1.1.1.2
Cancel the common factors.
Step 3.1.2.1.1.1.2.1
Factor out of .
Step 3.1.2.1.1.1.2.2
Cancel the common factor.
Step 3.1.2.1.1.1.2.3
Rewrite the expression.
Step 3.1.2.1.1.2
Move the negative in front of the fraction.
Step 3.1.2.1.2
Write as a fraction with a common denominator.
Step 3.1.2.1.3
Combine the numerators over the common denominator.
Step 3.1.2.1.4
Add and .
Step 3.1.2.1.5
Move the negative in front of the fraction.
Step 3.1.2.1.6
is approximately which is negative so negate and remove the absolute value
Step 3.1.2.2
To write as a fraction with a common denominator, multiply by .
Step 3.1.2.3
Combine and .
Step 3.1.2.4
Combine the numerators over the common denominator.
Step 3.1.2.5
Simplify the numerator.
Step 3.1.2.5.1
Multiply by .
Step 3.1.2.5.2
Subtract from .
Step 3.1.2.6
Move the negative in front of the fraction.
Step 3.1.2.7
The final answer is .
Step 3.2
Substitute the value into . In this case, the point is .
Step 3.2.1
Replace the variable with in the expression.
Step 3.2.2
Simplify the result.
Step 3.2.2.1
Simplify each term.
Step 3.2.2.1.1
Move the negative in front of the fraction.
Step 3.2.2.1.2
Write as a fraction with a common denominator.
Step 3.2.2.1.3
Combine the numerators over the common denominator.
Step 3.2.2.1.4
Add and .
Step 3.2.2.1.5
Move the negative in front of the fraction.
Step 3.2.2.1.6
is approximately which is negative so negate and remove the absolute value
Step 3.2.2.2
To write as a fraction with a common denominator, multiply by .
Step 3.2.2.3
Combine and .
Step 3.2.2.4
Combine the numerators over the common denominator.
Step 3.2.2.5
Simplify the numerator.
Step 3.2.2.5.1
Multiply by .
Step 3.2.2.5.2
Subtract from .
Step 3.2.2.6
Move the negative in front of the fraction.
Step 3.2.2.7
The final answer is .
Step 3.3
Substitute the value into . In this case, the point is .
Step 3.3.1
Replace the variable with in the expression.
Step 3.3.2
Simplify the result.
Step 3.3.2.1
Simplify each term.
Step 3.3.2.1.1
Simplify each term.
Step 3.3.2.1.1.1
Cancel the common factor of and .
Step 3.3.2.1.1.1.1
Factor out of .
Step 3.3.2.1.1.1.2
Cancel the common factors.
Step 3.3.2.1.1.1.2.1
Factor out of .
Step 3.3.2.1.1.1.2.2
Cancel the common factor.
Step 3.3.2.1.1.1.2.3
Rewrite the expression.
Step 3.3.2.1.1.2
Move the negative in front of the fraction.
Step 3.3.2.1.2
Write as a fraction with a common denominator.
Step 3.3.2.1.3
Combine the numerators over the common denominator.
Step 3.3.2.1.4
Add and .
Step 3.3.2.1.5
is approximately which is positive so remove the absolute value
Step 3.3.2.2
To write as a fraction with a common denominator, multiply by .
Step 3.3.2.3
Combine and .
Step 3.3.2.4
Combine the numerators over the common denominator.
Step 3.3.2.5
Simplify the numerator.
Step 3.3.2.5.1
Multiply by .
Step 3.3.2.5.2
Subtract from .
Step 3.3.2.6
Move the negative in front of the fraction.
Step 3.3.2.7
The final answer is .
Step 3.4
The absolute value can be graphed using the points around the vertex
Step 4