Graph y=|x+2|+3

$y = | x + 2 | + 3$

Step 1

Find the absolute value vertex. In this case, the vertex for $y = | x + 2 | + 3$ is $(- 2, 3)$ .

To find the $x$ coordinate of the vertex, set the inside of the absolute value $x + 2$ equal to $0$ . In this case, $x + 2 = 0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

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

$y = | (- 2) + 2 | + 3$

Simplify $| (- 2) + 2 | + 3$ .

Simplify each term.

Add $- 2$ and $2$ .

$y = | 0 | + 3$

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

$y = 0 + 3$

Add $0$ and $3$ .

$y = 3$

The absolute value vertex is $(- 2, 3)$ .

$(- 2, 3)$

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 $- 4$ into $f (x) = | x + 2 | + 3$ . In this case, the point is $(- 4, 5)$ .

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

$f (- 4) = | (- 4) + 2 | + 3$

Simplify the result.

Simplify each term.

Add $- 4$ and $2$ .

$f (- 4) = | - 2 | + 3$

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

$f (- 4) = 2 + 3$

Add $2$ and $3$ .

$f (- 4) = 5$

The final answer is $5$ .

$y = 5$

Substitute the $x$ value $- 3$ into $f (x) = | x + 2 | + 3$ . In this case, the point is $(- 3, 4)$ .

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

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

Simplify the result.

Simplify each term.

Add $- 3$ and $2$ .

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

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

$f (- 3) = 1 + 3$

Add $1$ and $3$ .

$f (- 3) = 4$

The final answer is $4$ .

$y = 4$

Substitute the $x$ value $0$ into $f (x) = | x + 2 | + 3$ . In this case, the point is $(0, 5)$ .

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

$f (0) = | (0) + 2 | + 3$

Simplify the result.

Simplify each term.

Add $0$ and $2$ .

$f (0) = | 2 | + 3$

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

$f (0) = 2 + 3$

Add $2$ and $3$ .

$f (0) = 5$

The final answer is $5$ .

$y = 5$

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

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

Step 4

Graph y=|x+2|+3

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