Pre-Algebra Examples

$| y | + 12$

Step 1

Find the absolute value vertex. In this case, the vertex for $x = | y | + 12$ is $(12, 0)$ .

Tap for more steps...

Step 1.1

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

$y = 0$

Step 1.2

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

$x = | 0 | + 12$

Step 1.3

Simplify $| 0 | + 12$ .

Tap for more steps...

Step 1.3.1

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

$x = 0 + 12$

Step 1.3.2

Add $0$ and $12$ .

$x = 12$

Step 1.4

The absolute value vertex is $(12, 0)$ .

$(12, 0)$

Step 2

The domain is the set of all valid $x$ values. Use the graph to find the domain.

$[12, \infty)$

${x | x \geq 12}$

Step 3

For each $x$ value, there is one positive $y$ value and one negative $y$ 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.

Tap for more steps...

Step 3.1

Substitute the $x$ value $13$ into $f (x) = x - 12$ . In this case, the point is $(13, 1)$ .

Tap for more steps...

Step 3.1.1

Replace the variable $x$ with $13$ in the expression.

$f (13) = (13) - 12$

Step 3.1.2

Simplify the result.

Tap for more steps...

Step 3.1.2.1

Subtract $12$ from $13$ .

$f (13) = 1$

Step 3.1.2.2

The final answer is $1$ .

$y = 1$

Step 3.2

Substitute the $x$ value $13$ into $f (x) = - (x - 12)$ . In this case, the point is $(13, - 1)$ .

Tap for more steps...

Step 3.2.1

Replace the variable $x$ with $13$ in the expression.

$f (13) = - ((13) - 12)$

Step 3.2.2

Simplify the result.

Tap for more steps...

Step 3.2.2.1

Subtract $12$ from $13$ .

$f (13) = - 1 \cdot 1$

Step 3.2.2.2

Multiply $- 1$ by $1$ .

$f (13) = - 1$

Step 3.2.2.3

The final answer is $- 1$ .

$y = - 1$

Step 3.3

Substitute the $x$ value $14$ into $f (x) = x - 12$ . In this case, the point is $(14, 2)$ .

Tap for more steps...

Step 3.3.1

Replace the variable $x$ with $14$ in the expression.

$f (14) = (14) - 12$

Step 3.3.2

Simplify the result.

Tap for more steps...

Step 3.3.2.1

Subtract $12$ from $14$ .

$f (14) = 2$

Step 3.3.2.2

The final answer is $2$ .

$y = 2$

Step 3.4

Substitute the $x$ value $14$ into $f (x) = - (x - 12)$ . In this case, the point is $(14, - 2)$ .

Tap for more steps...

Step 3.4.1

Replace the variable $x$ with $14$ in the expression.

$f (14) = - ((14) - 12)$

Step 3.4.2

Simplify the result.

Tap for more steps...

Step 3.4.2.1

Subtract $12$ from $14$ .

$f (14) = - 1 \cdot 2$

Step 3.4.2.2

Multiply $- 1$ by $2$ .

$f (14) = - 2$

Step 3.4.2.3

The final answer is $- 2$ .

$y = - 2$

Step 3.5

The absolute value can be graphed using the points around the vertex $(12, 0), (13, 1), (13, - 1), (14, 2), (14, - 2)$

$\begin{array}{c} x & y \\ 12 & 0 \\ 13 & 1 \\ 13 & - 1 \\ 14 & 2 \\ 14 & - 2 \end{array}$

Step 4