Pre-Algebra Examples

$y = | x - 4 | - 7$

Step 1

Find the absolute value vertex. In this case, the vertex for $y = | x - 4 | - 7$ is $(4, - 7)$ .

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Step 1.1

To find the $x$ coordinate of the vertex, set the inside of the absolute value $x - 4$ equal to $0$ . In this case, $x - 4 = 0$ .

$x - 4 = 0$

Step 1.2

Add $4$ to both sides of the equation.

$x = 4$

Step 1.3

Replace the variable $x$ with $4$ in the expression.

$y = | (4) - 4 | - 7$

Step 1.4

Simplify $| (4) - 4 | - 7$ .

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Subtract $4$ from $4$ .

$y = | 0 | - 7$

Step 1.4.1.2

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

$y = 0 - 7$

Step 1.4.2

Subtract $7$ from $0$ .

$y = - 7$

Step 1.5

The absolute value vertex is $(4, - 7)$ .

$(4, - 7)$

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 $f (x) = | x - 4 | - 7$ . In this case, the point is $(2, - 5)$ .

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Step 3.1.1

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

$f (2) = | (2) - 4 | - 7$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Subtract $4$ from $2$ .

$f (2) = | - 2 | - 7$

Step 3.1.2.1.2

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

$f (2) = 2 - 7$

Step 3.1.2.2

Subtract $7$ from $2$ .

$f (2) = - 5$

Step 3.1.2.3

The final answer is $- 5$ .

$y = - 5$

Step 3.2

Substitute the $x$ value $3$ into $f (x) = | x - 4 | - 7$ . In this case, the point is $(3, - 6)$ .

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Step 3.2.1

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

$f (3) = | (3) - 4 | - 7$

Step 3.2.2

Simplify the result.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Subtract $4$ from $3$ .

$f (3) = | - 1 | - 7$

Step 3.2.2.1.2

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

$f (3) = 1 - 7$

Step 3.2.2.2

Subtract $7$ from $1$ .

$f (3) = - 6$

Step 3.2.2.3

The final answer is $- 6$ .

$y = - 6$

Step 3.3

Substitute the $x$ value $6$ into $f (x) = | x - 4 | - 7$ . In this case, the point is $(6, - 5)$ .

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Step 3.3.1

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

$f (6) = | (6) - 4 | - 7$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

Subtract $4$ from $6$ .

$f (6) = | 2 | - 7$

Step 3.3.2.1.2

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

$f (6) = 2 - 7$

Step 3.3.2.2

Subtract $7$ from $2$ .

$f (6) = - 5$

Step 3.3.2.3

The final answer is $- 5$ .

$y = - 5$

Step 3.4

The absolute value can be graphed using the points around the vertex $(4, - 7), (2, - 5), (3, - 6), (5, - 6), (6, - 5)$

$\begin{array}{c} x & y \\ 2 & - 5 \\ 3 & - 6 \\ 4 & - 7 \\ 5 & - 6 \\ 6 & - 5 \end{array}$

Step 4