Graph y=-|x+4|

$y = - | x + 4 |$

Step 1

Find the absolute value vertex. In this case, the vertex for $y = - | x + 4 |$ is $(- 4, 0)$ .

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.

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

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

Add $- 4$ and $4$ .

$y = - | 0 |$

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

$y = - 0$

Multiply $- 1$ by $0$ .

$y = 0$

The absolute value vertex is $(- 4, 0)$ .

$(- 4, 0)$

Step 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:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 3

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

Replace the variable $x$ with $- 6$ in the expression.

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

Simplify the result.

Add $- 6$ and $4$ .

$f (- 6) = - | - 2 |$

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

$f (- 6) = - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$f (- 6) = - 2$

The final answer is $- 2$ .

$y = - 2$

Substitute the $x$ value $- 5$ into $f (x) = - | x + 4 |$ . In this case, the point is $(- 5, - 1)$ .

Replace the variable $x$ with $- 5$ in the expression.

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

Simplify the result.

Add $- 5$ and $4$ .

$f (- 5) = - | - 1 |$

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

$f (- 5) = - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$f (- 5) = - 1$

The final answer is $- 1$ .

$y = - 1$

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

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

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

Simplify the result.

Add $- 2$ and $4$ .

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

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

$f (- 2) = - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$f (- 2) = - 2$

The final answer is $- 2$ .

$y = - 2$

The absolute value can be graphed using the points around the vertex $(- 4, 0), (- 6, - 2), (- 5, - 1), (- 3, - 1), (- 2, - 2)$

$\begin{array}{c} x & y \\ - 6 & - 2 \\ - 5 & - 1 \\ - 4 & 0 \\ - 3 & - 1 \\ - 2 & - 2 \end{array}$

Step 4

Graph y=-|x+4|

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