y=-|x+4|+4

To find the x coordinate of the vertex, set the inside of the absolute value x+4 equal to 0. In this case, x+4=0.

x+4=0

Subtract 4 from both sides of the equation.

x=-4

Replace the variable x with -4 in the expression.

y=-|(-4)+4|+4

Simplify -|(-4)+4|+4.

Simplify each term.

Add -4 and 4.

y=-|0|+4

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

y=-0+4

Multiply -1 by 0.

y=0+4

y=0+4

Add 0 and 4.

y=4

y=4

The absolute value vertex is (-4,4).

(-4,4)

(-4,4)

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 -6 into f(x)=-|x+4|+4. In this case, the point is (-6,2).

Replace the variable x with -6 in the expression.

f(-6)=-|(-6)+4|+4

Simplify the result.

Simplify each term.

Add -6 and 4.

f(-6)=-|-2|+4

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

f(-6)=-1⋅2+4

Multiply -1 by 2.

f(-6)=-2+4

f(-6)=-2+4

Add -2 and 4.

f(-6)=2

The final answer is 2.

y=2

y=2

y=2

Substitute the x value -5 into f(x)=-|x+4|+4. In this case, the point is (-5,3).

Replace the variable x with -5 in the expression.

f(-5)=-|(-5)+4|+4

Simplify the result.

Simplify each term.

Add -5 and 4.

f(-5)=-|-1|+4

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

f(-5)=-1⋅1+4

Multiply -1 by 1.

f(-5)=-1+4

f(-5)=-1+4

Add -1 and 4.

f(-5)=3

The final answer is 3.

y=3

y=3

y=3

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

Replace the variable x with -2 in the expression.

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

Simplify the result.

Simplify each term.

Add -2 and 4.

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

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

f(-2)=-1⋅2+4

Multiply -1 by 2.

f(-2)=-2+4

f(-2)=-2+4

Add -2 and 4.

f(-2)=2

The final answer is 2.

y=2

y=2

y=2

The absolute value can be graphed using the points around the vertex (-4,4),(-6,2),(-5,3),(-3,3),(-2,2)

xy-62-53-44-33-22

xy-62-53-44-33-22

Graph y=-|x+4|+4