x|2x-1|-3

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.

Add 1 to both sides of the equation.

2x=1

Divide each term by 2 and simplify.

Divide each term in 2x=1 by 2.

2×2=12

Cancel the common factor of 2.

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.

Simplify each term.

Cancel the common factor of 2.

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}

Substitute the x value -2 into f(x)=x|2x-1|-3. In this case, the point is (-2,-13).

Replace the variable x with -2 in the expression.

f(-2)=(-2)|2(-2)-1|-3

Simplify the result.

Simplify each term.

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).

Replace the variable x with -1 in the expression.

f(-1)=(-1)|2(-1)-1|-3

Simplify the result.

Simplify each term.

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).

Replace the variable x with 0 in the expression.

f(0)=(0)|2(0)-1|-3

Simplify the result.

Simplify each term.

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).

Replace the variable x with 1 in the expression.

f(1)=(1)|2(1)-1|-3

Simplify the result.

Simplify each term.

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