Graph y=1/3|x-4|

$y = \frac{1}{3} | x - 4 |$

Step 1

Find the absolute value vertex. In this case, the vertex for $y = \frac{| x - 4 |}{3}$ is $(4, 0)$ .

To find the $x$ coordinate of the vertex, set the inside of the absolute value $x - 4$ equal to $0$ . In this case, $x - 4 = 0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

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

$y = \frac{| (4) - 4 |}{3}$

Simplify $\frac{| (4) - 4 |}{3}$ .

Simplify the numerator.

Subtract $4$ from $4$ .

$y = \frac{| 0 |}{3}$

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

$y = \frac{0}{3}$

Divide $0$ by $3$ .

$y = 0$

The absolute value vertex is $(4, 0)$ .

$(4, 0)$

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

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

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

Simplify the result.

Simplify the numerator.

Subtract $4$ from $2$ .

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

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

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

The final answer is $\frac{2}{3}$ .

$y = \frac{2}{3}$

Substitute the $x$ value $3$ into $f (x) = \frac{| x - 4 |}{3}$ . In this case, the point is $(3, \frac{1}{3})$ .

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

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

Simplify the result.

Simplify the numerator.

Subtract $4$ from $3$ .

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

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

$f (3) = \frac{1}{3}$

The final answer is $\frac{1}{3}$ .

$y = \frac{1}{3}$

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

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

$f (6) = \frac{| (6) - 4 |}{3}$

Simplify the result.

Simplify the numerator.

Subtract $4$ from $6$ .

$f (6) = \frac{| 2 |}{3}$

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

$f (6) = \frac{2}{3}$

The final answer is $\frac{2}{3}$ .

$y = \frac{2}{3}$

The absolute value can be graphed using the points around the vertex $(4, 0), (2, 0.67), (3, 0.33), (5, 0.33), (6, 0.67)$

$\begin{array}{c} x & y \\ 2 & 0.67 \\ 3 & 0.33 \\ 4 & 0 \\ 5 & 0.33 \\ 6 & 0.67 \end{array}$

Step 4

Graph y=1/3|x-4|

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