y=-|x+7|+3

To find the x coordinate of the vertex, set the inside of the absolute value x+7 equal to 0. In this case, x+7=0.

x+7=0

Subtract 7 from both sides of the equation.

x=-7

Replace the variable x with -7 in the expression.

y=-|(-7)+7|+3

Simplify -|(-7)+7|+3.

Simplify each term.

Add -7 and 7.

y=-|0|+3

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

y=-0+3

Multiply -1 by 0.

y=0+3

y=0+3

Add 0 and 3.

y=3

y=3

The absolute value vertex is (-7,3).

(-7,3)

(-7,3)

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 -9 into f(x)=-|x+7|+3. In this case, the point is (-9,1).

Replace the variable x with -9 in the expression.

f(-9)=-|(-9)+7|+3

Simplify the result.

Simplify each term.

Add -9 and 7.

f(-9)=-|-2|+3

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

f(-9)=-1⋅2+3

Multiply -1 by 2.

f(-9)=-2+3

f(-9)=-2+3

Add -2 and 3.

f(-9)=1

The final answer is 1.

y=1

y=1

y=1

Substitute the x value -8 into f(x)=-|x+7|+3. In this case, the point is (-8,2).

Replace the variable x with -8 in the expression.

f(-8)=-|(-8)+7|+3

Simplify the result.

Simplify each term.

Add -8 and 7.

f(-8)=-|-1|+3

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

f(-8)=-1⋅1+3

Multiply -1 by 1.

f(-8)=-1+3

f(-8)=-1+3

Add -1 and 3.

f(-8)=2

The final answer is 2.

y=2

y=2

y=2

Substitute the x value -5 into f(x)=-|x+7|+3. In this case, the point is (-5,1).

Replace the variable x with -5 in the expression.

f(-5)=-|(-5)+7|+3

Simplify the result.

Simplify each term.

Add -5 and 7.

f(-5)=-|2|+3

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

f(-5)=-1⋅2+3

Multiply -1 by 2.

f(-5)=-2+3

f(-5)=-2+3

Add -2 and 3.

f(-5)=1

The final answer is 1.

y=1

y=1

y=1

The absolute value can be graphed using the points around the vertex (-7,3),(-9,1),(-8,2),(-6,2),(-5,1)

xy-91-82-73-62-51

xy-91-82-73-62-51

Graph y=-|x+7|+3