Graph y=-|1/3x|+4

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

Step 1

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

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.

$y = - \frac{| 0 |}{3} + 4$

Simplify $- \frac{| 0 |}{3} + 4$ .

Simplify each term.

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

$y = - \frac{0}{3} + 4$

Divide $0$ by $3$ .

$y = - 0 + 4$

Multiply $- 1$ by $0$ .

$y = 0 + 4$

Add $0$ and $4$ .

$y = 4$

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

$(0, 4)$

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 $f (x) = - \frac{| x |}{3} + 4$ . In this case, the point is $(- 2, \frac{10}{3})$ .

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

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

Simplify the result.

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

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

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Combine $4$ and $\frac{3}{3}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $4$ by $3$ .

$f (- 2) = \frac{- 2 + 12}{3}$

Add $- 2$ and $12$ .

$f (- 2) = \frac{10}{3}$

The final answer is $\frac{10}{3}$ .

$y = \frac{10}{3}$

Substitute the $x$ value $- 1$ into $f (x) = - \frac{| x |}{3} + 4$ . In this case, the point is $(- 1, \frac{11}{3})$ .

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

$f (- 1) = - \frac{| - 1 |}{3} + 4$

Simplify the result.

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

$f (- 1) = - \frac{1}{3} + 4$

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (- 1) = - \frac{1}{3} + 4 \cdot \frac{3}{3}$

Combine $4$ and $\frac{3}{3}$ .

$f (- 1) = - \frac{1}{3} + \frac{4 \cdot 3}{3}$

Combine the numerators over the common denominator.

$f (- 1) = \frac{- 1 + 4 \cdot 3}{3}$

Simplify the numerator.

Multiply $4$ by $3$ .

$f (- 1) = \frac{- 1 + 12}{3}$

Add $- 1$ and $12$ .

$f (- 1) = \frac{11}{3}$

The final answer is $\frac{11}{3}$ .

$y = \frac{11}{3}$

Substitute the $x$ value $2$ into $f (x) = - \frac{| x |}{3} + 4$ . In this case, the point is $(2, \frac{10}{3})$ .

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

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

Simplify the result.

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

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

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Combine $4$ and $\frac{3}{3}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $4$ by $3$ .

$f (2) = \frac{- 2 + 12}{3}$

Add $- 2$ and $12$ .

$f (2) = \frac{10}{3}$

The final answer is $\frac{10}{3}$ .

$y = \frac{10}{3}$

The absolute value can be graphed using the points around the vertex $(0, 4), (- 2, 3.33), (- 1, 3.67), (1, 3.67), (2, 3.33)$

$\begin{array}{c} x & y \\ - 2 & 3.33 \\ - 1 & 3.67 \\ 0 & 4 \\ 1 & 3.67 \\ 2 & 3.33 \end{array}$

Step 4

Graph y=-|1/3x|+4

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