Graph x|2x-1|-3

Math
x|2x-1|-3
Find the absolute value vertex. In this case, the vertex for y=x|2x-1|-3 is (12,-3).
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To find the x coordinate of the vertex, set the inside of the absolute value 2x-1 equal to 0. In this case, 2x-1=0.
2x-1=0
Solve the equation 2x-1=0 to find the x coordinate for the absolute value vertex.
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Add 1 to both sides of the equation.
2x=1
Divide each term by 2 and simplify.
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Divide each term in 2x=1 by 2.
2×2=12
Cancel the common factor of 2.
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Cancel the common factor.
2×2=12
Divide x by 1.
x=12
x=12
x=12
x=12
Replace the variable x with 12 in the expression.
y=(12)|2(12)-1|-3
Simplify (12)|2(12)-1|-3.
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Simplify each term.
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Cancel the common factor of 2.
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Cancel the common factor.
y=12⋅|2(12)-1|-3
Rewrite the expression.
y=12⋅|1-1|-3
y=12⋅|1-1|-3
Subtract 1 from 1.
y=12⋅|0|-3
Remove non-negative terms from the absolute value.
y=12⋅0-3
Multiply 12 by 0.
y=0-3
y=0-3
Subtract 3 from 0.
y=-3
y=-3
The absolute value vertex is (12,-3).
(12,-3)
(12,-3)
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:
{x|x∈ℝ}
For each x value, there is one y value. Select few x values from the domain. It would be more useful to select the values so that they are around the x value of the absolute value vertex.
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Substitute the x value -2 into f(x)=x|2x-1|-3. In this case, the point is (-2,-13).
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Replace the variable x with -2 in the expression.
f(-2)=(-2)|2(-2)-1|-3
Simplify the result.
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Simplify each term.
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Multiply 2 by -2.
f(-2)=-2|-4-1|-3
Subtract 1 from -4.
f(-2)=-2|-5|-3
The absolute value is the distance between a number and zero. The distance between -5 and 0 is 5.
f(-2)=-2⋅5-3
Multiply -2 by 5.
f(-2)=-10-3
f(-2)=-10-3
Subtract 3 from -10.
f(-2)=-13
The final answer is -13.
y=-13
y=-13
y=-13
Substitute the x value -1 into f(x)=x|2x-1|-3. In this case, the point is (-1,-6).
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Replace the variable x with -1 in the expression.
f(-1)=(-1)|2(-1)-1|-3
Simplify the result.
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Simplify each term.
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Multiply 2 by -1.
f(-1)=-1|-2-1|-3
Subtract 1 from -2.
f(-1)=-1|-3|-3
The absolute value is the distance between a number and zero. The distance between -3 and 0 is 3.
f(-1)=-1⋅3-3
Multiply -1 by 3.
f(-1)=-3-3
f(-1)=-3-3
Subtract 3 from -3.
f(-1)=-6
The final answer is -6.
y=-6
y=-6
y=-6
Substitute the x value 0 into f(x)=x|2x-1|-3. In this case, the point is (0,-3).
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Replace the variable x with 0 in the expression.
f(0)=(0)|2(0)-1|-3
Simplify the result.
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Simplify each term.
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Multiply 2 by 0.
f(0)=0|0-1|-3
Subtract 1 from 0.
f(0)=0|-1|-3
The absolute value is the distance between a number and zero. The distance between -1 and 0 is 1.
f(0)=0⋅1-3
Multiply 0 by 1.
f(0)=0-3
f(0)=0-3
Subtract 3 from 0.
f(0)=-3
The final answer is -3.
y=-3
y=-3
y=-3
Substitute the x value 1 into f(x)=x|2x-1|-3. In this case, the point is (1,-2).
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Replace the variable x with 1 in the expression.
f(1)=(1)|2(1)-1|-3
Simplify the result.
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Simplify each term.
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Multiply |2(1)-1| by 1.
f(1)=|2(1)-1|-3
Multiply 2 by 1.
f(1)=|2-1|-3
Subtract 1 from 2.
f(1)=|1|-3
The absolute value is the distance between a number and zero. The distance between 0 and 1 is 1.
f(1)=1-3
f(1)=1-3
Subtract 3 from 1.
f(1)=-2
The final answer is -2.
y=-2
y=-2
y=-2
The absolute value can be graphed using the points around the vertex (12,-3),(-2,-13),(-1,-6),(0,-3),(1,-2)
xy-2-13-1-60-312-31-2
xy-2-13-1-60-312-31-2
Graph x|2x-1|-3

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