Graph g(x)=-7/6|x-2|

$g (x) = - \frac{7}{6} | x - 2 |$

Find the absolute value vertex. In this case, the vertex for $y = - \frac{7 | x - 2 |}{6}$ is $(2, 0)$ .

To find the $x$ coordinate of the vertex, set the inside of the absolute value $x - 2$ equal to $0$ . In this case, $x - 2 = 0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

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

$y = - \frac{7 | (2) - 2 |}{6}$

Simplify $- \frac{7 | (2) - 2 |}{6}$ .

Simplify the numerator.

Subtract $2$ from $2$ .

$y = - \frac{7 | 0 |}{6}$

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

$y = - \frac{7 \cdot 0}{6}$

Simplify the expression.

Multiply $7$ by $0$ .

$y = - \frac{0}{6}$

Divide $0$ by $6$ .

$y = - 0$

Multiply $- 1$ by $0$ .

$y = 0$

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

$(2, 0)$

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 $0$ into $f (x) = - \frac{7 | x - 2 |}{6}$ . In this case, the point is $(0, - \frac{7}{3})$ .

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

$f (0) = - \frac{7 | (0) - 2 |}{6}$

Simplify the result.

Simplify the numerator.

Subtract $2$ from $0$ .

$f (0) = - \frac{7 | - 2 |}{6}$

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

$f (0) = - \frac{7 \cdot 2}{6}$

Reduce the expression by cancelling the common factors.

Multiply $7$ by $2$ .

$f (0) = - \frac{14}{6}$

Cancel the common factor of $14$ and $6$ .

Factor $2$ out of $14$ .

$f (0) = - \frac{2 (7)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$f (0) = - \frac{2 \cdot 7}{2 \cdot 3}$

Cancel the common factor.

$f (0) = - \frac{2 \cdot 7}{2 \cdot 3}$

Rewrite the expression.

$f (0) = - \frac{7}{3}$

The final answer is $- \frac{7}{3}$ .

$y = - \frac{7}{3}$

Substitute the $x$ value $1$ into $f (x) = - \frac{7 | x - 2 |}{6}$ . In this case, the point is $(1, - \frac{7}{6})$ .

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

$f (1) = - \frac{7 | (1) - 2 |}{6}$

Simplify the result.

Simplify the numerator.

Subtract $2$ from $1$ .

$f (1) = - \frac{7 | - 1 |}{6}$

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

$f (1) = - \frac{7 \cdot 1}{6}$

Multiply $7$ by $1$ .

$f (1) = - \frac{7}{6}$

The final answer is $- \frac{7}{6}$ .

$y = - \frac{7}{6}$

Substitute the $x$ value $4$ into $f (x) = - \frac{7 | x - 2 |}{6}$ . In this case, the point is $(4, - \frac{7}{3})$ .

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

$f (4) = - \frac{7 | (4) - 2 |}{6}$

Simplify the result.

Simplify the numerator.

Subtract $2$ from $4$ .

$f (4) = - \frac{7 | 2 |}{6}$

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

$f (4) = - \frac{7 \cdot 2}{6}$

Reduce the expression by cancelling the common factors.

Multiply $7$ by $2$ .

$f (4) = - \frac{14}{6}$

Cancel the common factor of $14$ and $6$ .

Factor $2$ out of $14$ .

$f (4) = - \frac{2 (7)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$f (4) = - \frac{2 \cdot 7}{2 \cdot 3}$

Cancel the common factor.

$f (4) = - \frac{2 \cdot 7}{2 \cdot 3}$

Rewrite the expression.

$f (4) = - \frac{7}{3}$

The final answer is $- \frac{7}{3}$ .

$y = - \frac{7}{3}$

The absolute value can be graphed using the points around the vertex $(2, 0), (0, - 2.33), (1, - 1.17), (3, - 1.17), (4, - 2.33)$

$\begin{array}{c} x & y \\ 0 & - 2.33 \\ 1 & - 1.17 \\ 2 & 0 \\ 3 & - 1.17 \\ 4 & - 2.33 \end{array}$

Graph g(x)=-7/6|x-2|

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