Graph f(x)=1/2|x+1|

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

Find the absolute value vertex. In this case, the vertex for $y = \frac{| x + 1 |}{2}$ is $(- 1, 0)$ .

To find the $x$ coordinate of the vertex, set the inside of the absolute value $x + 1$ equal to $0$ . In this case, $x + 1 = 0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

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

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

Simplify $\frac{| (- 1) + 1 |}{2}$ .

Simplify the numerator.

Add $- 1$ and $1$ .

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

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

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

Divide $0$ by $2$ .

$y = 0$

The absolute value vertex is $(- 1, 0)$ .

$(- 1, 0)$

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:

${x | x \in ℝ}$

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 $- 3$ into $f (x) = \frac{| x + 1 |}{2}$ . In this case, the point is $(- 3, 1)$ .

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

$f (- 3) = \frac{| (- 3) + 1 |}{2}$

Simplify the result.

Simplify the numerator.

Add $- 3$ and $1$ .

$f (- 3) = \frac{| - 2 |}{2}$

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

$f (- 3) = \frac{2}{2}$

Divide $2$ by $2$ .

$f (- 3) = 1$

The final answer is $1$ .

$y = 1$

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

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

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

Simplify the result.

Simplify the numerator.

Add $- 2$ and $1$ .

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

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

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

The final answer is $\frac{1}{2}$ .

$y = \frac{1}{2}$

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

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

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

Simplify the result.

Simplify the numerator.

Add $1$ and $1$ .

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

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

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

Divide $2$ by $2$ .

$f (1) = 1$

The final answer is $1$ .

$y = 1$

The absolute value can be graphed using the points around the vertex $(- 1, 0), (- 3, 1), (- 2, 0.5), (0, 0.5), (1, 1)$

$\begin{array}{c} x & y \\ - 3 & 1 \\ - 2 & 0.5 \\ - 1 & 0 \\ 0 & 0.5 \\ 1 & 1 \end{array}$

Graph f(x)=1/2|x+1|

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