Graph f(x)=1/3|x|-7

$f (x) = \frac{1}{3} | x | - 7$

Step 1

Find the absolute value vertex. In this case, the vertex for $y = \frac{| x |}{3} - 7$ is $(0, - 7)$ .

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.

$y = \frac{| 0 |}{3} - 7$

Simplify $\frac{| 0 |}{3} - 7$ .

Simplify each term.

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

$y = \frac{0}{3} - 7$

Divide $0$ by $3$ .

$y = 0 - 7$

Subtract $7$ from $0$ .

$y = - 7$

The absolute value vertex is $(0, - 7)$ .

$(0, - 7)$

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 $f (x) = \frac{| x |}{3} - 7$ . In this case, the point is $(- 2, - \frac{19}{3})$ .

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

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

Simplify the result.

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

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

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Combine $- 7$ and $\frac{3}{3}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 7$ by $3$ .

$f (- 2) = \frac{2 - 21}{3}$

Subtract $21$ from $2$ .

$f (- 2) = \frac{- 19}{3}$

Move the negative in front of the fraction.

$f (- 2) = - \frac{19}{3}$

The final answer is $- \frac{19}{3}$ .

$y = - \frac{19}{3}$

Substitute the $x$ value $- 1$ into $f (x) = \frac{| x |}{3} - 7$ . In this case, the point is $(- 1, - \frac{20}{3})$ .

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

$f (- 1) = \frac{| - 1 |}{3} - 7$

Simplify the result.

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

$f (- 1) = \frac{1}{3} - 7$

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (- 1) = \frac{1}{3} - 7 \cdot \frac{3}{3}$

Combine $- 7$ and $\frac{3}{3}$ .

$f (- 1) = \frac{1}{3} + \frac{- 7 \cdot 3}{3}$

Combine the numerators over the common denominator.

$f (- 1) = \frac{1 - 7 \cdot 3}{3}$

Simplify the numerator.

Multiply $- 7$ by $3$ .

$f (- 1) = \frac{1 - 21}{3}$

Subtract $21$ from $1$ .

$f (- 1) = \frac{- 20}{3}$

Move the negative in front of the fraction.

$f (- 1) = - \frac{20}{3}$

The final answer is $- \frac{20}{3}$ .

$y = - \frac{20}{3}$

Substitute the $x$ value $2$ into $f (x) = \frac{| x |}{3} - 7$ . In this case, the point is $(2, - \frac{19}{3})$ .

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

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

Simplify the result.

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

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

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Combine $- 7$ and $\frac{3}{3}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 7$ by $3$ .

$f (2) = \frac{2 - 21}{3}$

Subtract $21$ from $2$ .

$f (2) = \frac{- 19}{3}$

Move the negative in front of the fraction.

$f (2) = - \frac{19}{3}$

The final answer is $- \frac{19}{3}$ .

$y = - \frac{19}{3}$

The absolute value can be graphed using the points around the vertex $(0, - 7), (- 2, - 6.33), (- 1, - 6.67), (1, - 6.67), (2, - 6.33)$

$\begin{array}{c} x & y \\ - 2 & - 6.33 \\ - 1 & - 6.67 \\ 0 & - 7 \\ 1 & - 6.67 \\ 2 & - 6.33 \end{array}$

Step 4

Graph f(x)=1/3|x|-7

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