Algebra Examples

$g (x) = 4 | 1 - x |$

Step 1

Find the absolute value vertex. In this case, the vertex for $y = 4 | 1 - x |$ is $(1, 0)$ .

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Step 1.1

To find the $x$ coordinate of the vertex, set the inside of the absolute value $1 - x$ equal to $0$ . In this case, $1 - x = 0$ .

$1 - x = 0$

Step 1.2

Solve the equation $1 - x = 0$ to find the $x$ coordinate for the absolute value vertex.

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Step 1.2.1

Subtract $1$ from both sides of the equation.

$- x = - 1$

Step 1.2.2

Divide each term in $- x = - 1$ by $- 1$ and simplify.

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Step 1.2.2.1

Divide each term in $- x = - 1$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 1}{- 1}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 1}{- 1}$

Step 1.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 1}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $- 1$ by $- 1$ .

$x = 1$

Step 1.3

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

$y = 4 | 1 - (1) |$

Step 1.4

Simplify $4 | 1 - (1) |$ .

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Step 1.4.1

Multiply $- 1$ by $1$ .

$y = 4 | 1 - 1 |$

Step 1.4.2

Subtract $1$ from $1$ .

$y = 4 | 0 |$

Step 1.4.3

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

$y = 4 \cdot 0$

Step 1.4.4

Multiply $4$ by $0$ .

$y = 0$

Step 1.5

The absolute value vertex is $(1, 0)$ .

$(1, 0)$

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 $- 1$ into $f (x) = 4 | 1 - x |$ . In this case, the point is $(- 1, 8)$ .

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Step 3.1.1

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

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

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 3.1.2.2

Add $1$ and $1$ .

$f (- 1) = 4 | 2 |$

Step 3.1.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$f (- 1) = 4 \cdot 2$

Step 3.1.2.4

Multiply $4$ by $2$ .

$f (- 1) = 8$

Step 3.1.2.5

The final answer is $8$ .

$y = 8$

Step 3.2

Substitute the $x$ value $0$ into $f (x) = 4 | 1 - x |$ . In this case, the point is $(0, 4)$ .

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Step 3.2.1

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

$f (0) = 4 | 1 - (0) |$

Step 3.2.2

Simplify the result.

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Step 3.2.2.1

Subtract $0$ from $1$ .

$f (0) = 4 | 1 |$

Step 3.2.2.2

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$f (0) = 4 \cdot 1$

Step 3.2.2.3

Multiply $4$ by $1$ .

$f (0) = 4$

Step 3.2.2.4

The final answer is $4$ .

$y = 4$

Step 3.3

Substitute the $x$ value $3$ into $f (x) = 4 | 1 - x |$ . In this case, the point is $(3, 8)$ .

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Step 3.3.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = 4 | 1 - (3) |$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Multiply $- 1$ by $3$ .

$f (3) = 4 | 1 - 3 |$

Step 3.3.2.2

Subtract $3$ from $1$ .

$f (3) = 4 | - 2 |$

Step 3.3.2.3

The absolute value is the distance between a number and zero. The distance between $- 2$ and $0$ is $2$ .

$f (3) = 4 \cdot 2$

Step 3.3.2.4

Multiply $4$ by $2$ .

$f (3) = 8$

Step 3.3.2.5

The final answer is $8$ .

$y = 8$

Step 3.4

The absolute value can be graphed using the points around the vertex $(1, 0), (- 1, 8), (0, 4), (2, 4), (3, 8)$

$\begin{array}{c} x & y \\ - 1 & 8 \\ 0 & 4 \\ 1 & 0 \\ 2 & 4 \\ 3 & 8 \end{array}$

Step 4