Graph n(x)=1/2|x-6|

$n (x) = \frac{1}{2} | x - 6 |$

Find the absolute value vertex. In this case, the vertex for $y = \frac{| x - 6 |}{2}$ is $(6, 0)$ .

To find the $x$ coordinate of the vertex, set the inside of the absolute value $x - 6$ equal to $0$ . In this case, $x - 6 = 0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

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

$y = \frac{| (6) - 6 |}{2}$

Simplify $\frac{| (6) - 6 |}{2}$ .

Simplify the numerator.

Subtract $6$ from $6$ .

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

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 $(6, 0)$ .

$(6, 0)$

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 $4$ into $f (x) = \frac{| x - 6 |}{2}$ . In this case, the point is $(4, 1)$ .

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

$f (4) = \frac{| (4) - 6 |}{2}$

Simplify the result.

Simplify the numerator.

Subtract $6$ from $4$ .

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

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

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

Divide $2$ by $2$ .

$f (4) = 1$

The final answer is $1$ .

$y = 1$

Substitute the $x$ value $5$ into $f (x) = \frac{| x - 6 |}{2}$ . In this case, the point is $(5, \frac{1}{2})$ .

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

$f (5) = \frac{| (5) - 6 |}{2}$

Simplify the result.

Simplify the numerator.

Subtract $6$ from $5$ .

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

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

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

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

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

Substitute the $x$ value $8$ into $f (x) = \frac{| x - 6 |}{2}$ . In this case, the point is $(8, 1)$ .

Replace the variable $x$ with $8$ in the expression.

$f (8) = \frac{| (8) - 6 |}{2}$

Simplify the result.

Simplify the numerator.

Subtract $6$ from $8$ .

$f (8) = \frac{| 2 |}{2}$

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

$f (8) = \frac{2}{2}$

Divide $2$ by $2$ .

$f (8) = 1$

The final answer is $1$ .

$y = 1$

The absolute value can be graphed using the points around the vertex $(6, 0), (4, 1), (5, 0.5), (7, 0.5), (8, 1)$

$\begin{array}{c} x & y \\ 4 & 1 \\ 5 & 0.5 \\ 6 & 0 \\ 7 & 0.5 \\ 8 & 1 \end{array}$

Graph n(x)=1/2|x-6|

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