Graph y=-1/2|x|-2

$y = - \frac{1}{2} | x | - 2$

Step 1

Find the absolute value vertex. In this case, the vertex for $y = - \frac{| x |}{2} - 2$ is $(0, - 2)$ .

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

$x = 0$

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

$y = - \frac{| 0 |}{2} - 2$

Simplify $- \frac{| 0 |}{2} - 2$ .

Simplify each term.

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

$y = - \frac{0}{2} - 2$

Divide $0$ by $2$ .

$y = - 0 - 2$

Multiply $- 1$ by $0$ .

$y = 0 - 2$

Subtract $2$ from $0$ .

$y = - 2$

The absolute value vertex is $(0, - 2)$ .

$(0, - 2)$

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 $- 2$ into $f (x) = - \frac{| x |}{2} - 2$ . In this case, the point is $(- 2, - 3)$ .

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

$f (- 2) = - \frac{| - 2 |}{2} - 2$

Simplify the result.

Simplify each term.

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

$f (- 2) = - \frac{2}{2} - 2$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (- 2) = - \frac{2}{2} - 2$

Rewrite the expression.

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

Multiply $- 1$ by $1$ .

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

Subtract $2$ from $- 1$ .

$f (- 2) = - 3$

The final answer is $- 3$ .

$y = - 3$

Substitute the $x$ value $- 1$ into $f (x) = - \frac{| x |}{2} - 2$ . In this case, the point is $(- 1, - \frac{5}{2})$ .

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

$f (- 1) = - \frac{| - 1 |}{2} - 2$

Simplify the result.

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

$f (- 1) = - \frac{1}{2} - 2$

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- 1) = - \frac{1}{2} - 2 \cdot \frac{2}{2}$

Combine $- 2$ and $\frac{2}{2}$ .

$f (- 1) = - \frac{1}{2} + \frac{- 2 \cdot 2}{2}$

Combine the numerators over the common denominator.

$f (- 1) = \frac{- 1 - 2 \cdot 2}{2}$

Simplify the numerator.

Multiply $- 2$ by $2$ .

$f (- 1) = \frac{- 1 - 4}{2}$

Subtract $4$ from $- 1$ .

$f (- 1) = \frac{- 5}{2}$

Move the negative in front of the fraction.

$f (- 1) = - \frac{5}{2}$

The final answer is $- \frac{5}{2}$ .

$y = - \frac{5}{2}$

Substitute the $x$ value $2$ into $f (x) = - \frac{| x |}{2} - 2$ . In this case, the point is $(2, - 3)$ .

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

$f (2) = - \frac{| 2 |}{2} - 2$

Simplify the result.

Simplify each term.

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

$f (2) = - \frac{2}{2} - 2$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (2) = - \frac{2}{2} - 2$

Rewrite the expression.

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

Multiply $- 1$ by $1$ .

$f (2) = - 1 - 2$

Subtract $2$ from $- 1$ .

$f (2) = - 3$

The final answer is $- 3$ .

$y = - 3$

The absolute value can be graphed using the points around the vertex $(0, - 2), (- 2, - 3), (- 1, - 2.5), (1, - 2.5), (2, - 3)$

$\begin{array}{c} x & y \\ - 2 & - 3 \\ - 1 & - 2.5 \\ 0 & - 2 \\ 1 & - 2.5 \\ 2 & - 3 \end{array}$

Step 4

Graph y=-1/2|x|-2

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