Graph y=|x| and y=|x-5|

$y = | x |$ and $y = | x - 5 |$

Graph $y = | x |$ .

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

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

$x = 0$

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

$y = | 0 |$

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

$y = 0$

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

$(0, 0)$

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:

${x | x \in ℝ}$

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.

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

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

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

Simplify the result.

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

$f (- 2) = 2$

The final answer is $2$ .

$y = 2$

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

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

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

Simplify the result.

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

$f (- 1) = 1$

The final answer is $1$ .

$y = 1$

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

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

$f (2) = | 2 |$

Simplify the result.

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

$f (2) = 2$

The final answer is $2$ .

$y = 2$

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

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

Graph $y = | x - 5 |$ .

Find the absolute value vertex. In this case, the vertex for $y = | x - 5 |$ is $(5, 0)$ .

To find the $x$ coordinate of the vertex, set the inside of the absolute value $x - 5$ equal to $0$ . In this case, $x - 5 = 0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Replace the variable $x$ with $5$ in the expression.

$y = | (5) - 5 |$

Simplify $| (5) - 5 |$ .

Subtract $5$ from $5$ .

$y = | 0 |$

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

$y = 0$

The absolute value vertex is $(5, 0)$ .

$(5, 0)$

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:

${x | x \in ℝ}$

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.

Substitute the $x$ value $3$ into $f (x) = | x - 5 |$ . In this case, the point is $(3, 2)$ .

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

$f (3) = | (3) - 5 |$

Simplify the result.

Subtract $5$ from $3$ .

$f (3) = | - 2 |$

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

$f (3) = 2$

The final answer is $2$ .

$y = 2$

Substitute the $x$ value $4$ into $f (x) = | x - 5 |$ . In this case, the point is $(4, 1)$ .

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

$f (4) = | (4) - 5 |$

Simplify the result.

Subtract $5$ from $4$ .

$f (4) = | - 1 |$

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

$f (4) = 1$

The final answer is $1$ .

$y = 1$

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

Replace the variable $x$ with $7$ in the expression.

$f (7) = | (7) - 5 |$

Simplify the result.

Subtract $5$ from $7$ .

$f (7) = | 2 |$

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

$f (7) = 2$

The final answer is $2$ .

$y = 2$

The absolute value can be graphed using the points around the vertex $(5, 0), (3, 2), (4, 1), (6, 1), (7, 2)$

$\begin{array}{c} x & y \\ 3 & 2 \\ 4 & 1 \\ 5 & 0 \\ 6 & 1 \\ 7 & 2 \end{array}$

Plot each graph on the same coordinate system.

$y = | x |$

$y = | x - 5 |$

Graph y=|x| and y=|x-5|

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