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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
Subtract from both sides of the equation.
Step 1.2.3.3.2
Subtract from .
Step 1.2.3.4
Divide each term in by and simplify.
Step 1.2.3.4.1
Divide each term in by .
Step 1.2.3.4.2
Simplify the left side.
Step 1.2.3.4.2.1
Dividing two negative values results in a positive value.
Step 1.2.3.4.2.2
Divide by .
Step 1.2.3.4.3
Simplify the right side.
Step 1.2.3.4.3.1
Divide by .
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 1.5
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
Solve for .
Step 2.2.1
Subtract from both sides of the inequality.
Step 2.2.2
Divide each term in by and simplify.
Step 2.2.2.1
Divide each term in by . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.
Step 2.2.2.2
Simplify the left side.
Step 2.2.2.2.1
Dividing two negative values results in a positive value.
Step 2.2.2.2.2
Divide by .
Step 2.2.2.3
Simplify the right side.
Step 2.2.2.3.1
Divide by .
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
Multiply by .
Step 3.1.2.2
Add and .
Step 3.1.2.3
is approximately which is positive so remove the absolute value
Step 3.1.2.4
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
Multiply by .
Step 3.2.2.2
Add and .
Step 3.2.2.3
is approximately which is positive so remove the absolute value
Step 3.2.2.4
The final answer is .
Step 3.3
The absolute value can be graphed using the points around the vertex
Step 4