Graph f(x)=|x| g(x)=|x|-4

$f (x) = | x |$ $g (x) = | x | - 4$

Step 1

Graph $f (x) = | x |$ .

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 $(0, 0), (- 2, 2), (- 1, 1), (1, 1), (2, 2)$

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

Step 2

Graph $g (x) = | x | - 4$ .

Find the absolute value vertex. In this case, the vertex for $y = | x | - 4$ is $(0, - 4)$ .

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.

$y = | 0 | - 4$

Simplify $| 0 | - 4$ .

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

$y = 0 - 4$

Subtract $4$ from $0$ .

$y = - 4$

The absolute value vertex is $(0, - 4)$ .

$(0, - 4)$

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

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

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

Simplify the result.

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

$f (- 2) = 2 - 4$

Subtract $4$ from $2$ .

$f (- 2) = - 2$

The final answer is $- 2$ .

$y = - 2$

Substitute the $x$ value $- 1$ into $f (x) = | x | - 4$ . In this case, the point is $(- 1, - 3)$ .

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

$f (- 1) = | - 1 | - 4$

Simplify the result.

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

$f (- 1) = 1 - 4$

Subtract $4$ from $1$ .

$f (- 1) = - 3$

The final answer is $- 3$ .

$y = - 3$

Substitute the $x$ value $2$ into $f (x) = | x | - 4$ . In this case, the point is $(2, - 2)$ .

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

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

Simplify the result.

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

$f (2) = 2 - 4$

Subtract $4$ from $2$ .

$f (2) = - 2$

The final answer is $- 2$ .

$y = - 2$

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

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

Step 3

Plot each graph on the same coordinate system.

$f (x) = | x |$

$g (x) = | x | - 4$

Step 4

Graph f(x)=|x| g(x)=|x|-4

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