Algebra Examples

$y = | f (x) |$

Step 1

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

$f (x) = 0$

Step 1.2

Solve the equation $f (x) = 0$ to find the $x$ coordinate for the absolute value vertex.

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Step 1.2.1

Subtract $f (x)$ from both sides of the equation.

$0 = - f (x)$

Step 1.2.2

Rewrite the equation as $- f (x) = 0$ .

$- f (x) = 0$

Step 1.3

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

$y = | (0) (x) |$

Step 1.4

Simplify $| (0) (x) |$ .

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Step 1.4.1

Multiply $0$ by $x$ .

$y = | 0 |$

Step 1.4.2

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

$y = 0$

Step 1.5

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:

${f | f \in ℝ}$

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

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Step 3.1.1

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

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

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Move $- 2$ to the left of $f$ .

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

Step 3.1.2.2

Remove non-negative terms from the absolute value.

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

Step 3.1.2.3

The final answer is $2 | f |$ .

$y = 2 | f |$

Step 3.2

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

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Step 3.2.1

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

$f (- 1) = | f (- 1) |$

Step 3.2.2

Simplify the result.

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Step 3.2.2.1

Move $- 1$ to the left of $f$ .

$f (- 1) = | - 1 f |$

Step 3.2.2.2

Rewrite $- 1 f$ as $- f$ .

$f (- 1) = | - f |$

Step 3.2.2.3

Make $- f$ positive.

$f (- 1) = | f |$

Step 3.2.2.4

The final answer is $| f |$ .

$y = | f |$

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Move $2$ to the left of $f$ .

$f (2) = | 2 f |$

Step 3.3.2.2

Remove non-negative terms from the absolute value.

$f (2) = 2 | f |$

Step 3.3.2.3

The final answer is $2 | f |$ .

$y = 2 | f |$

Step 3.4

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

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

Step 4