Graph f(x)=|x-1|-3

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

Step 1

Find the absolute value vertex. In this case, the vertex for $y = | x - 1 | - 3$ is $(1, - 3)$ .

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

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

$y = | (1) - 1 | - 3$

Simplify $| (1) - 1 | - 3$ .

Simplify each term.

Subtract $1$ from $1$ .

$y = | 0 | - 3$

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

$y = 0 - 3$

Subtract $3$ from $0$ .

$y = - 3$

The absolute value vertex is $(1, - 3)$ .

$(1, - 3)$

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.

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

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

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

Simplify the result.

Simplify each term.

Subtract $1$ from $- 1$ .

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

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

$f (- 1) = 2 - 3$

Subtract $3$ from $2$ .

$f (- 1) = - 1$

The final answer is $- 1$ .

$y = - 1$

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

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

$f (0) = | (0) - 1 | - 3$

Simplify the result.

Simplify each term.

Subtract $1$ from $0$ .

$f (0) = | - 1 | - 3$

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

$f (0) = 1 - 3$

Subtract $3$ from $1$ .

$f (0) = - 2$

The final answer is $- 2$ .

$y = - 2$

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

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

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

Simplify the result.

Simplify each term.

Subtract $1$ from $3$ .

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

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

$f (3) = 2 - 3$

Subtract $3$ from $2$ .

$f (3) = - 1$

The final answer is $- 1$ .

$y = - 1$

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

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

Step 4

Graph f(x)=|x-1|-3

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