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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
To solve for , rewrite the equation using properties of logarithms.
Step 1.2.2
Rewrite in exponential form using the definition of a logarithm. If and are positive real numbers and , then is equivalent to .
Step 1.2.3
Solve for .
Step 1.2.3.1
Rewrite the equation as .
Step 1.2.3.2
Anything raised to is .
Step 1.2.3.3
Move all terms not containing to the right side of the equation.
Step 1.2.3.3.1
Add to both sides of the equation.
Step 1.2.3.3.2
Add and .
Step 1.3
Replace the variable with in the expression.
Step 1.4
Simplify .
Step 1.4.1
Subtract from .
Step 1.4.2
The natural logarithm of is .
Step 1.4.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 1.4.4
Multiply by .
Step 1.5
The absolute value vertex is .
Step 2
Step 2.1
Set the argument in greater than to find where the expression is defined.
Step 2.2
Add to both sides of the inequality.
Step 2.3
The domain is all values of that make the expression defined.
Interval Notation:
Set-Builder Notation:
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
Subtract from .
Step 3.1.2.2
is approximately which is positive so remove the absolute value
Step 3.1.2.3
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
Subtract from .
Step 3.2.2.2
is approximately which is positive so remove the absolute value
Step 3.2.2.3
The final answer is .
Step 3.3
The absolute value can be graphed using the points around the vertex
Step 4