Graph 9x+|y|=0

$9 x + | y | = 0$

Subtract $| y |$ from both sides of the equation.

$9 x = - | y |$

Divide each term in $9 x = - | y |$ by $9$ and simplify.

Divide each term in $9 x = - | y |$ by $9$ .

$\frac{9 x}{9} = \frac{- | y |}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 x}{9} = \frac{- | y |}{9}$

Divide $x$ by $1$ .

$x = \frac{- | y |}{9}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{| y |}{9}$

Find the absolute value vertex. In this case, the vertex for $x = - \frac{| y |}{9}$ is $(0, 0)$ .

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

$y = 0$

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

$x = - \frac{| 0 |}{9}$

Simplify $- \frac{| 0 |}{9}$ .

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

$x = - \frac{0}{9}$

Divide $0$ by $9$ .

$x = - 0$

Multiply $- 1$ by $0$ .

$x = 0$

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

$(0, 0)$

The domain is the set of all valid $x$ values. Use the graph to find the domain.

$(- \infty, 0]$

${x | x \leq 0}$

For each $x$ value, there is one positive $y$ value and one negative $y$ 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 $f (x) = - 9 x$ . In this case, the point is $(- 2, 18)$ .

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

$f (- 2) = - 9 \cdot - 2$

Simplify the result.

Multiply $- 9$ by $- 2$ .

$f (- 2) = 18$

The final answer is $18$ .

$y = 18$

Substitute the $x$ value $- 2$ into $f (x) = 9 x$ . In this case, the point is $(- 2, - 18)$ .

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

$f (- 2) = 9 (- 2)$

Simplify the result.

Multiply $9$ by $- 2$ .

$f (- 2) = - 18$

The final answer is $- 18$ .

$y = - 18$

Substitute the $x$ value $- 1$ into $f (x) = - 9 x$ . In this case, the point is $(- 1, 9)$ .

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

$f (- 1) = - 9 \cdot - 1$

Simplify the result.

Multiply $- 9$ by $- 1$ .

$f (- 1) = 9$

The final answer is $9$ .

$y = 9$

Substitute the $x$ value $- 1$ into $f (x) = 9 x$ . In this case, the point is $(- 1, - 9)$ .

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

$f (- 1) = 9 (- 1)$

Simplify the result.

Multiply $9$ by $- 1$ .

$f (- 1) = - 9$

The final answer is $- 9$ .

$y = - 9$

The absolute value can be graphed using the points around the vertex $(0, 0), (- 2, 18), (- 2, - 18), (- 1, 9), (- 1, - 9)$

$\begin{array}{c} x & y \\ - 2 & 18 \\ - 2 & - 18 \\ - 1 & 9 \\ - 1 & - 9 \\ 0 & 0 \end{array}$

Graph 9x+|y|=0

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