Graph y=|x-7|

$y = | x - 7 |$

Step 1

Find the absolute value vertex. In this case, the vertex for $y = | x - 7 |$ is $(7, 0)$ .

To find the $x$ coordinate of the vertex, set the inside of the absolute value $x - 7$ equal to $0$ . In this case, $x - 7 = 0$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

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

$y = | (7) - 7 |$

Simplify $| (7) - 7 |$ .

Subtract $7$ from $7$ .

$y = | 0 |$

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 $(7, 0)$ .

$(7, 0)$

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

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

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

Simplify the result.

Subtract $7$ from $5$ .

$f (5) = | - 2 |$

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

$f (5) = 2$

The final answer is $2$ .

$y = 2$

Substitute the $x$ value $6$ into $f (x) = | x - 7 |$ . In this case, the point is $(6, 1)$ .

Replace the variable $x$ with $6$ in the expression.

$f (6) = | (6) - 7 |$

Simplify the result.

Subtract $7$ from $6$ .

$f (6) = | - 1 |$

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

$f (6) = 1$

The final answer is $1$ .

$y = 1$

Substitute the $x$ value $9$ into $f (x) = | x - 7 |$ . In this case, the point is $(9, 2)$ .

Replace the variable $x$ with $9$ in the expression.

$f (9) = | (9) - 7 |$

Simplify the result.

Subtract $7$ from $9$ .

$f (9) = | 2 |$

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

$f (9) = 2$

The final answer is $2$ .

$y = 2$

The absolute value can be graphed using the points around the vertex $(7, 0), (5, 2), (6, 1), (8, 1), (9, 2)$

$\begin{array}{c} x & y \\ 5 & 2 \\ 6 & 1 \\ 7 & 0 \\ 8 & 1 \\ 9 & 2 \end{array}$

Step 4

Graph y=|x-7|

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