Graph y=|3-x|

$y = | 3 - x |$

Step 1

Find the absolute value vertex. In this case, the vertex for $y = | 3 - x |$ is $(3, 0)$ .

To find the $x$ coordinate of the vertex, set the inside of the absolute value $3 - x$ equal to $0$ . In this case, $3 - x = 0$ .

$3 - x = 0$

Solve the equation $3 - x = 0$ to find the $x$ coordinate for the absolute value vertex.

Subtract $3$ from both sides of the equation.

$- x = - 3$

Divide each term in $- x = - 3$ by $- 1$ and simplify.

Divide each term in $- x = - 3$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 3}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 3}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 3}{- 1}$

Simplify the right side.

Divide $- 3$ by $- 1$ .

$x = 3$

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

$y = | 3 - (3) |$

Simplify $| 3 - (3) |$ .

Multiply $- 1$ by $3$ .

$y = | 3 - 3 |$

Subtract $3$ from $3$ .

$y = | 0 |$

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 $(3, 0)$ .

$(3, 0)$

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

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

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

Simplify the result.

Multiply $- 1$ by $1$ .

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

Subtract $1$ from $3$ .

$f (1) = | 2 |$

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

$f (1) = 2$

The final answer is $2$ .

$y = 2$

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

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

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

Simplify the result.

Multiply $- 1$ by $2$ .

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

Subtract $2$ from $3$ .

$f (2) = | 1 |$

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

$f (2) = 1$

The final answer is $1$ .

$y = 1$

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

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

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

Simplify the result.

Multiply $- 1$ by $5$ .

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

Subtract $5$ from $3$ .

$f (5) = | - 2 |$

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 $2$ .

$y = 2$

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

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

Step 4

Graph y=|3-x|

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