Graph f(x)=-0.5|x+3|-2

$f (x) = - 0.5 | x + 3 | - 2$

Step 1

Find the absolute value vertex. In this case, the vertex for $y = - 0.5 | x + 3 | - 2$ is $(- 3, - 2)$ .

To find the $x$ coordinate of the vertex, set the inside of the absolute value $x + 3$ equal to $0$ . In this case, $x + 3 = 0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

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

$y = - 0.5 | (- 3) + 3 | - 2$

Simplify $- 0.5 | (- 3) + 3 | - 2$ .

Simplify each term.

Add $- 3$ and $3$ .

$y = - 0.5 | 0 | - 2$

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

$y = - 0.5 \cdot 0 - 2$

Multiply $- 0.5$ by $0$ .

$y = 0 - 2$

Subtract $2$ from $0$ .

$y = - 2$

The absolute value vertex is $(- 3, - 2)$ .

$(- 3, - 2)$

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 $- 5$ into $f (x) = - 0.5 | x + 3 | - 2$ . In this case, the point is $(- 5, - 3)$ .

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

$f (- 5) = - 0.5 | (- 5) + 3 | - 2$

Simplify the result.

Simplify each term.

Add $- 5$ and $3$ .

$f (- 5) = - 0.5 | - 2 | - 2$

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

$f (- 5) = - 0.5 \cdot 2 - 2$

Multiply $- 0.5$ by $2$ .

$f (- 5) = - 1 - 2$

Subtract $2$ from $- 1$ .

$f (- 5) = - 3$

The final answer is $- 3$ .

$y = - 3$

Substitute the $x$ value $- 4$ into $f (x) = - 0.5 | x + 3 | - 2$ . In this case, the point is $(- 4, - 2.5)$ .

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

$f (- 4) = - 0.5 | (- 4) + 3 | - 2$

Simplify the result.

Simplify each term.

Add $- 4$ and $3$ .

$f (- 4) = - 0.5 | - 1 | - 2$

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

$f (- 4) = - 0.5 \cdot 1 - 2$

Multiply $- 0.5$ by $1$ .

$f (- 4) = - 0.5 - 2$

Subtract $2$ from $- 0.5$ .

$f (- 4) = - 2.5$

The final answer is $- 2.5$ .

$y = - 2.5$

Substitute the $x$ value $- 1$ into $f (x) = - 0.5 | x + 3 | - 2$ . In this case, the point is $(- 1, - 3)$ .

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = - 0.5 | (- 1) + 3 | - 2$

Simplify the result.

Simplify each term.

Add $- 1$ and $3$ .

$f (- 1) = - 0.5 | 2 | - 2$

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

$f (- 1) = - 0.5 \cdot 2 - 2$

Multiply $- 0.5$ by $2$ .

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

Subtract $2$ from $- 1$ .

$f (- 1) = - 3$

The final answer is $- 3$ .

$y = - 3$

The absolute value can be graphed using the points around the vertex $(- 3, - 2), (- 5, - 3), (- 4, - 2.5), (- 2, - 2.5), (- 1, - 3)$

$\begin{array}{c} x & y \\ - 5 & - 3 \\ - 4 & - 2.5 \\ - 3 & - 2 \\ - 2 & - 2.5 \\ - 1 & - 3 \end{array}$

Step 4

Graph f(x)=-0.5|x+3|-2

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