Algebra Examples

$2 - | x - 1 |$

Step 1

Find the absolute value vertex. In this case, the vertex for $y = 2 - | x - 1 |$ is $(1, 2)$ .

Tap for more steps...

Step 1.1

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

$x - 1 = 0$

Step 1.2

Add $1$ to both sides of the equation.

$x = 1$

Step 1.3

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

$y = 2 - | (1) - 1 |$

Step 1.4

Simplify $2 - | (1) - 1 |$ .

Tap for more steps...

Step 1.4.1

Simplify each term.

Tap for more steps...

Step 1.4.1.1

Subtract $1$ from $1$ .

$y = 2 - | 0 |$

Step 1.4.1.2

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

$y = 2 - 0$

Step 1.4.1.3

Multiply $- 1$ by $0$ .

$y = 2 + 0$

Step 1.4.2

Add $2$ and $0$ .

$y = 2$

Step 1.5

The absolute value vertex is $(1, 2)$ .

$(1, 2)$

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.

Tap for more steps...

Step 3.1

Substitute the $x$ value $- 1$ into $f (x) = 2 - | x - 1 |$ . In this case, the point is $(- 1, 0)$ .

Tap for more steps...

Step 3.1.1

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

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

Step 3.1.2

Simplify the result.

Tap for more steps...

Step 3.1.2.1

Simplify each term.

Tap for more steps...

Step 3.1.2.1.1

Subtract $1$ from $- 1$ .

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

Step 3.1.2.1.2

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

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

Step 3.1.2.1.3

Multiply $- 1$ by $2$ .

$f (- 1) = 2 - 2$

Step 3.1.2.2

Subtract $2$ from $2$ .

$f (- 1) = 0$

Step 3.1.2.3

The final answer is $0$ .

$y = 0$

Step 3.2

Substitute the $x$ value $0$ into $f (x) = 2 - | x - 1 |$ . In this case, the point is $(0, 1)$ .

Tap for more steps...

Step 3.2.1

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

$f (0) = 2 - | (0) - 1 |$

Step 3.2.2

Simplify the result.

Tap for more steps...

Step 3.2.2.1

Simplify each term.

Tap for more steps...

Step 3.2.2.1.1

Subtract $1$ from $0$ .

$f (0) = 2 - | - 1 |$

Step 3.2.2.1.2

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

$f (0) = 2 - 1 \cdot 1$

Step 3.2.2.1.3

Multiply $- 1$ by $1$ .

$f (0) = 2 - 1$

Step 3.2.2.2

Subtract $1$ from $2$ .

$f (0) = 1$

Step 3.2.2.3

The final answer is $1$ .

$y = 1$

Step 3.3

Substitute the $x$ value $3$ into $f (x) = 2 - | x - 1 |$ . In this case, the point is $(3, 0)$ .

Tap for more steps...

Step 3.3.1

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

$f (3) = 2 - | (3) - 1 |$

Step 3.3.2

Simplify the result.

Tap for more steps...

Step 3.3.2.1

Simplify each term.

Tap for more steps...

Step 3.3.2.1.1

Subtract $1$ from $3$ .

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

Step 3.3.2.1.2

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

$f (3) = 2 - 1 \cdot 2$

Step 3.3.2.1.3

Multiply $- 1$ by $2$ .

$f (3) = 2 - 2$

Step 3.3.2.2

Subtract $2$ from $2$ .

$f (3) = 0$

Step 3.3.2.3

The final answer is $0$ .

$y = 0$

Step 3.4

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

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

Step 4