Graph y=|2x-3|

$y = | 2 x - 3 |$

Step 1

Find the absolute value vertex. In this case, the vertex for $y = | 2 x - 3 |$ is $(\frac{3}{2}, 0)$ .

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.

Add $3$ to 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 $1$ .

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$y = | 3 - 3 |$

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 $(\frac{3}{2}, 0)$ .

$(\frac{3}{2}, 0)$

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

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

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

Simplify the result.

Multiply $2$ by $0$ .

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

Subtract $3$ from $0$ .

$f (0) = | - 3 |$

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

$f (0) = 3$

The final answer is $3$ .

$y = 3$

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

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

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

Simplify the result.

Multiply $2$ by $1$ .

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

Subtract $3$ from $2$ .

$f (1) = | - 1 |$

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

$f (1) = 1$

The final answer is $1$ .

$y = 1$

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

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

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

Simplify the result.

Multiply $2$ by $3$ .

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

Subtract $3$ from $6$ .

$f (3) = | 3 |$

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

$f (3) = 3$

The final answer is $3$ .

$y = 3$

The absolute value can be graphed using the points around the vertex $(\frac{3}{2}, 0), (0, 3), (1, 1), (2, 1), (3, 3)$

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

Step 4

Graph y=|2x-3|

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