y=|x-2|+2

To find the x coordinate of the vertex, set the inside of the absolute value x-2 equal to 0. In this case, x-2=0.

x-2=0

Add 2 to both sides of the equation.

x=2

Replace the variable x with 2 in the expression.

y=|(2)-2|+2

Simplify |(2)-2|+2.

Simplify each term.

Subtract 2 from 2.

y=|0|+2

The absolute value is the distance between a number and zero. The distance between 0 and 0 is 0.

y=0+2

y=0+2

Add 0 and 2.

y=2

y=2

The absolute value vertex is (2,2).

(2,2)

(2,2)

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:

{x|x∈ℝ}

Substitute the x value 0 into f(x)=|x-2|+2. In this case, the point is (0,4).

Replace the variable x with 0 in the expression.

f(0)=|(0)-2|+2

Simplify the result.

Simplify each term.

Subtract 2 from 0.

f(0)=|-2|+2

The absolute value is the distance between a number and zero. The distance between -2 and 0 is 2.

f(0)=2+2

f(0)=2+2

Add 2 and 2.

f(0)=4

The final answer is 4.

y=4

y=4

y=4

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

Replace the variable x with 1 in the expression.

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

Simplify the result.

Simplify each term.

Subtract 2 from 1.

f(1)=|-1|+2

The absolute value is the distance between a number and zero. The distance between -1 and 0 is 1.

f(1)=1+2

f(1)=1+2

Add 1 and 2.

f(1)=3

The final answer is 3.

y=3

y=3

y=3

Substitute the x value 4 into f(x)=|x-2|+2. In this case, the point is (4,4).

Replace the variable x with 4 in the expression.

f(4)=|(4)-2|+2

Simplify the result.

Simplify each term.

Subtract 2 from 4.

f(4)=|2|+2

The absolute value is the distance between a number and zero. The distance between 0 and 2 is 2.

f(4)=2+2

f(4)=2+2

Add 2 and 2.

f(4)=4

The final answer is 4.

y=4

y=4

y=4

The absolute value can be graphed using the points around the vertex (2,2),(0,4),(1,3),(3,3),(4,4)

xy0413223344

xy0413223344

Graph y=|x-2|+2