Graph y+4=|x+2|

$y + 4 = | x + 2 |$

Step 1

Subtract $4$ from both sides of the equation.

$y = | x + 2 | - 4$

Step 2

Find the absolute value vertex. In this case, the vertex for $y = | x + 2 | - 4$ is $(- 2, - 4)$ .

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

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Replace the variable $x$ with $- 2$ in the expression.

$y = | (- 2) + 2 | - 4$

Simplify $| (- 2) + 2 | - 4$ .

Simplify each term.

Add $- 2$ and $2$ .

$y = | 0 | - 4$

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

$y = 0 - 4$

Subtract $4$ from $0$ .

$y = - 4$

The absolute value vertex is $(- 2, - 4)$ .

$(- 2, - 4)$

Step 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:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4

For each $x$ value, there is one $y$ value. Select a 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.

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

Replace the variable $x$ with $- 4$ in the expression.

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

Simplify the result.

Simplify each term.

Add $- 4$ and $2$ .

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

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

$f (- 4) = 2 - 4$

Subtract $4$ from $2$ .

$f (- 4) = - 2$

The final answer is $- 2$ .

$y = - 2$

Substitute the $x$ value $- 3$ into $f (x) = | x + 2 | - 4$ . In this case, the point is $(- 3, - 3)$ .

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = | (- 3) + 2 | - 4$

Simplify the result.

Simplify each term.

Add $- 3$ and $2$ .

$f (- 3) = | - 1 | - 4$

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

$f (- 3) = 1 - 4$

Subtract $4$ from $1$ .

$f (- 3) = - 3$

The final answer is $- 3$ .

$y = - 3$

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

Replace the variable $x$ with $0$ in the expression.

$f (0) = | (0) + 2 | - 4$

Simplify the result.

Simplify each term.

Add $0$ and $2$ .

$f (0) = | 2 | - 4$

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

$f (0) = 2 - 4$

Subtract $4$ from $2$ .

$f (0) = - 2$

The final answer is $- 2$ .

$y = - 2$

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

$\begin{array}{c} x & y \\ - 4 & - 2 \\ - 3 & - 3 \\ - 2 & - 4 \\ - 1 & - 3 \\ 0 & - 2 \end{array}$

Step 5

Graph y+4=|x+2|

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