Algebra Examples

$| y |$

Step 1

Find the absolute value vertex. In this case, the vertex for $x = | y |$ is $(0, 0)$ .

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Step 1.1

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

$y = 0$

Step 1.2

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

$x = | 0 |$

Step 1.3

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

$x = 0$

Step 1.4

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

$(0, 0)$

Step 2

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

$[0, \infty)$

${x | x \geq 0}$

Step 3

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.

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Step 3.1

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

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Step 3.1.1

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

$f (1) = 1$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Remove parentheses.

$f (1) = 1$

Step 3.1.2.2

The final answer is $1$ .

$y = 1$

Step 3.2

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

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Step 3.2.1

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

$f (1) = - (1)$

Step 3.2.2

Simplify the result.

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Step 3.2.2.1

Multiply $- 1$ by $1$ .

$f (1) = - 1$

Step 3.2.2.2

The final answer is $- 1$ .

$y = - 1$

Step 3.3

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

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Step 3.3.1

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

$f (2) = 2$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Remove parentheses.

$f (2) = 2$

Step 3.3.2.2

The final answer is $2$ .

$y = 2$

Step 3.4

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

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Step 3.4.1

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

$f (2) = - (2)$

Step 3.4.2

Simplify the result.

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Step 3.4.2.1

Multiply $- 1$ by $2$ .

$f (2) = - 2$

Step 3.4.2.2

The final answer is $- 2$ .

$y = - 2$

Step 3.5

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

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

Step 4