Precalculus Examples

$y = - | x + 8 |$

Step 1

Find the absolute value vertex. In this case, the vertex for $y = - | x + 8 |$ is $(- 8, 0)$ .

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Step 1.1

To find the $x$ coordinate of the vertex, set the inside of the absolute value $x + 8$ equal to $0$ . In this case, $x + 8 = 0$ .

$x + 8 = 0$

Step 1.2

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 1.3

Replace the variable $x$ with $- 8$ in the expression.

$y = - | (- 8) + 8 |$

Step 1.4

Simplify $- | (- 8) + 8 |$ .

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Step 1.4.1

Add $- 8$ and $8$ .

$y = - | 0 |$

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 1.4.3

Multiply $- 1$ by $0$ .

$y = 0$

Step 1.5

The absolute value vertex is $(- 8, 0)$ .

$(- 8, 0)$

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 $- 10$ into $f (x) = - | x + 8 |$ . In this case, the point is $(- 10, - 2)$ .

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Step 3.1.1

Replace the variable $x$ with $- 10$ in the expression.

$f (- 10) = - | (- 10) + 8 |$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Add $- 10$ and $8$ .

$f (- 10) = - | - 2 |$

Step 3.1.2.2

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

$f (- 10) = - 1 \cdot 2$

Step 3.1.2.3

Multiply $- 1$ by $2$ .

$f (- 10) = - 2$

Step 3.1.2.4

The final answer is $- 2$ .

$y = - 2$

Step 3.2

Substitute the $x$ value $- 9$ into $f (x) = - | x + 8 |$ . In this case, the point is $(- 9, - 1)$ .

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Step 3.2.1

Replace the variable $x$ with $- 9$ in the expression.

$f (- 9) = - | (- 9) + 8 |$

Step 3.2.2

Simplify the result.

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Step 3.2.2.1

Add $- 9$ and $8$ .

$f (- 9) = - | - 1 |$

Step 3.2.2.2

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

$f (- 9) = - 1 \cdot 1$

Step 3.2.2.3

Multiply $- 1$ by $1$ .

$f (- 9) = - 1$

Step 3.2.2.4

The final answer is $- 1$ .

$y = - 1$

Step 3.3

Substitute the $x$ value $- 6$ into $f (x) = - | x + 8 |$ . In this case, the point is $(- 6, - 2)$ .

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Step 3.3.1

Replace the variable $x$ with $- 6$ in the expression.

$f (- 6) = - | (- 6) + 8 |$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Add $- 6$ and $8$ .

$f (- 6) = - | 2 |$

Step 3.3.2.2

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

$f (- 6) = - 1 \cdot 2$

Step 3.3.2.3

Multiply $- 1$ by $2$ .

$f (- 6) = - 2$

Step 3.3.2.4

The final answer is $- 2$ .

$y = - 2$

Step 3.4

The absolute value can be graphed using the points around the vertex $(- 8, 0), (- 10, - 2), (- 9, - 1), (- 7, - 1), (- 6, - 2)$

$\begin{array}{c} x & y \\ - 10 & - 2 \\ - 9 & - 1 \\ - 8 & 0 \\ - 7 & - 1 \\ - 6 & - 2 \end{array}$

Step 4