Trigonometry Examples

Graph y=2|-3x+4|-3
Step 1
Find the absolute value vertex. In this case, the vertex for is .
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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.
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Step 1.2.1
Subtract from both sides of the equation.
Step 1.2.2
Divide each term in by and simplify.
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Step 1.2.2.1
Divide each term in by .
Step 1.2.2.2
Simplify the left side.
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Step 1.2.2.2.1
Cancel the common factor of .
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Step 1.2.2.2.1.1
Cancel the common factor.
Step 1.2.2.2.1.2
Divide by .
Step 1.2.2.3
Simplify the right side.
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Step 1.2.2.3.1
Dividing two negative values results in a positive value.
Step 1.3
Replace the variable with in the expression.
Step 1.4
Simplify .
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Step 1.4.1
Simplify each term.
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Step 1.4.1.1
Simplify each term.
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Step 1.4.1.1.1
Cancel the common factor of .
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Step 1.4.1.1.1.1
Factor out of .
Step 1.4.1.1.1.2
Cancel the common factor.
Step 1.4.1.1.1.3
Rewrite the expression.
Step 1.4.1.1.2
Multiply 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 1.4.1.4
Multiply by .
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
For each value, there is one value. Select a few values from the domain. It would be more useful to select the values so that they are around the value of the absolute value vertex.
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Step 3.1
Substitute the value into . In this case, the point is .
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Step 3.1.1
Replace the variable with in the expression.
Step 3.1.2
Simplify the result.
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Step 3.1.2.1
Simplify each term.
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Step 3.1.2.1.1
Multiply by .
Step 3.1.2.1.2
Add and .
Step 3.1.2.1.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 3.1.2.1.4
Multiply by .
Step 3.1.2.2
Subtract from .
Step 3.1.2.3
The final answer is .
Step 3.2
Substitute the value into . In this case, the point is .
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Step 3.2.1
Replace the variable with in the expression.
Step 3.2.2
Simplify the result.
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Step 3.2.2.1
Simplify each term.
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Step 3.2.2.1.1
Multiply by .
Step 3.2.2.1.2
Add and .
Step 3.2.2.1.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 3.2.2.1.4
Multiply by .
Step 3.2.2.2
Subtract from .
Step 3.2.2.3
The final answer is .
Step 3.3
Substitute the value into . In this case, the point is .
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Step 3.3.1
Replace the variable with in the expression.
Step 3.3.2
Simplify the result.
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Step 3.3.2.1
Simplify each term.
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Step 3.3.2.1.1
Multiply by .
Step 3.3.2.1.2
Add and .
Step 3.3.2.1.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 3.3.2.1.4
Multiply by .
Step 3.3.2.2
Subtract from .
Step 3.3.2.3
The final answer is .
Step 3.4
Substitute the value into . In this case, the point is .
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Step 3.4.1
Replace the variable with in the expression.
Step 3.4.2
Simplify the result.
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Step 3.4.2.1
Simplify each term.
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Step 3.4.2.1.1
Multiply by .
Step 3.4.2.1.2
Add and .
Step 3.4.2.1.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 3.4.2.1.4
Multiply by .
Step 3.4.2.2
Subtract from .
Step 3.4.2.3
The final answer is .
Step 3.5
The absolute value can be graphed using the points around the vertex
Step 4