Graph 4x+|y|=0

$4 x + | y | = 0$

Step 1

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

$4 x = - | y |$

Step 2

Divide each term in $4 x = - | y |$ by $4$ and simplify.

Divide each term in $4 x = - | y |$ by $4$ .

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 3

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

$x = - \frac{| 0 |}{4}$

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

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

$x = - \frac{0}{4}$

Divide $0$ by $4$ .

$x = - 0$

Multiply $- 1$ by $0$ .

$x = 0$

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

$(0, 0)$

Step 4

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

$(- \infty, 0]$

${x | x \leq 0}$

Step 5

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 $f (x) = - 4 x$ . In this case, the point is $(- 2, 8)$ .

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

$f (- 2) = - 4 \cdot - 2$

Simplify the result.

Multiply $- 4$ by $- 2$ .

$f (- 2) = 8$

The final answer is $8$ .

$y = 8$

Substitute the $x$ value $- 2$ into $f (x) = 4 x$ . In this case, the point is $(- 2, - 8)$ .

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

$f (- 2) = 4 (- 2)$

Simplify the result.

Multiply $4$ by $- 2$ .

$f (- 2) = - 8$

The final answer is $- 8$ .

$y = - 8$

Substitute the $x$ value $- 1$ into $f (x) = - 4 x$ . In this case, the point is $(- 1, 4)$ .

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

$f (- 1) = - 4 \cdot - 1$

Simplify the result.

Multiply $- 4$ by $- 1$ .

$f (- 1) = 4$

The final answer is $4$ .

$y = 4$

Substitute the $x$ value $- 1$ into $f (x) = 4 x$ . In this case, the point is $(- 1, - 4)$ .

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

$f (- 1) = 4 (- 1)$

Simplify the result.

Multiply $4$ by $- 1$ .

$f (- 1) = - 4$

The final answer is $- 4$ .

$y = - 4$

The absolute value can be graphed using the points around the vertex $(0, 0), (- 2, 8), (- 2, - 8), (- 1, 4), (- 1, - 4)$

$\begin{array}{c} x & y \\ - 2 & 8 \\ - 2 & - 8 \\ - 1 & 4 \\ - 1 & - 4 \\ 0 & 0 \end{array}$

Step 6

Graph 4x+|y|=0

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