Algebra Examples

$f (x) = | x |$

Step 1

Find the absolute value vertex. In this case, the vertex for

$y = | x |$ is

$(0, 0)$ .

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Step 1.1

To find the

$x$ coordinate of the vertex, set the inside of the absolute value

$x$ equal to

$0$ . In this case,

$x = 0$ .

$x = 0$

Step 1.2

Replace the variable

$x$ with

$0$ in the expression.

$y = | 0 |$

Step 1.3

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

$0$ and

$0$ is

$0$ .

$y = 0$

Step 1.4

The absolute value vertex is

$(0, 0)$ .

$(0, 0)$

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.

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Step 3.1

Substitute the

$x$ value

$- 2$ into

$f (x) = | x |$ . In this case, the point is

$(- 2, 2)$ .

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Step 3.1.1

Replace the variable

$x$ with

$- 2$ in the expression.

$f (- 2) = | - 2 |$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

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

$- 2$ and

$0$ is

$2$ .

$f (- 2) = 2$

Step 3.1.2.2

The final answer is

$2$ .

$y = 2$

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

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

$- 1$ and

$0$ is

$1$ .

$f (- 1) = 1$

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

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

$0$ and

$2$ is

$2$ .

$f (2) = 2$

Step 3.3.2.2

The final answer is

$2$ .

$y = 2$

Step 3.4

The absolute value can be graphed using the points around the vertex

$(0, 0), (- 2, 2), (- 1, 1), (1, 1), (2, 2)$

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

Step 4