Graph y=9-|2x+3|

$y = 9 - | 2 x + 3 |$

Find the absolute value vertex. In this case, the vertex for $y = 9 - | 2 x + 3 |$ is $(- \frac{3}{2}, 9)$ .

To find the $x$ coordinate of the vertex, set the inside of the absolute value $2 x + 3$ equal to $0$ . In this case, $2 x + 3 = 0$ .

$2 x + 3 = 0$

Solve the equation $2 x + 3 = 0$ to find the $x$ coordinate for the absolute value vertex.

Subtract $3$ from both sides of the equation.

$2 x = - 3$

Divide each term in $2 x = - 3$ by $2$ and simplify.

Divide each term in $2 x = - 3$ by $2$ .

$\frac{2 x}{2} = \frac{- 3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 3}{2}$

Divide $x$ by $1$ .

$x = \frac{- 3}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{3}{2}$

Replace the variable $x$ with $- \frac{3}{2}$ in the expression.

$y = 9 - | 2 (- \frac{3}{2}) + 3 |$

Simplify $9 - | 2 (- \frac{3}{2}) + 3 |$ .

Simplify each term.

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$y = 9 - | 2 (\frac{- 3}{2}) + 3 |$

Cancel the common factor.

$y = 9 - | 2 (\frac{- 3}{2}) + 3 |$

Rewrite the expression.

$y = 9 - | - 3 + 3 |$

Add $- 3$ and $3$ .

$y = 9 - | 0 |$

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

$y = 9 - 0$

Multiply $- 1$ by $0$ .

$y = 9 + 0$

Add $9$ and $0$ .

$y = 9$

The absolute value vertex is $(- \frac{3}{2}, 9)$ .

$(- \frac{3}{2}, 9)$

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 $- 4$ into $f (x) = 9 - | 2 x + 3 |$ . In this case, the point is $(- 4, 4)$ .

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

$f (- 4) = 9 - | 2 (- 4) + 3 |$

Simplify the result.

Simplify each term.

Multiply $2$ by $- 4$ .

$f (- 4) = 9 - | - 8 + 3 |$

Add $- 8$ and $3$ .

$f (- 4) = 9 - | - 5 |$

The absolute value is the distance between a number and zero. The distance between $- 5$ and $0$ is $5$ .

$f (- 4) = 9 - 1 \cdot 5$

Multiply $- 1$ by $5$ .

$f (- 4) = 9 - 5$

Subtract $5$ from $9$ .

$f (- 4) = 4$

The final answer is $4$ .

$y = 4$

Substitute the $x$ value $- 3$ into $f (x) = 9 - | 2 x + 3 |$ . In this case, the point is $(- 3, 6)$ .

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

$f (- 3) = 9 - | 2 (- 3) + 3 |$

Simplify the result.

Simplify each term.

Multiply $2$ by $- 3$ .

$f (- 3) = 9 - | - 6 + 3 |$

Add $- 6$ and $3$ .

$f (- 3) = 9 - | - 3 |$

The absolute value is the distance between a number and zero. The distance between $- 3$ and $0$ is $3$ .

$f (- 3) = 9 - 1 \cdot 3$

Multiply $- 1$ by $3$ .

$f (- 3) = 9 - 3$

Subtract $3$ from $9$ .

$f (- 3) = 6$

The final answer is $6$ .

$y = 6$

Substitute the $x$ value $- 2$ into $f (x) = 9 - | 2 x + 3 |$ . In this case, the point is $(- 2, 8)$ .

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

$f (- 2) = 9 - | 2 (- 2) + 3 |$

Simplify the result.

Simplify each term.

Multiply $2$ by $- 2$ .

$f (- 2) = 9 - | - 4 + 3 |$

Add $- 4$ and $3$ .

$f (- 2) = 9 - | - 1 |$

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

$f (- 2) = 9 - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$f (- 2) = 9 - 1$

Subtract $1$ from $9$ .

$f (- 2) = 8$

The final answer is $8$ .

$y = 8$

The absolute value can be graphed using the points around the vertex $(- \frac{3}{2}, 9), (- 4, 4), (- 3, 6), (- 2, 8), (- 1, 8)$

$\begin{array}{c} x & y \\ - 4 & 4 \\ - 3 & 6 \\ - 2 & 8 \\ - \frac{3}{2} & 9 \\ - 1 & 8 \end{array}$

Graph y=9-|2x+3|

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