Graph y=(3/4)|x|

$y = (\frac{3}{4}) | x |$

Step 1

Find the absolute value vertex. In this case, the vertex for $y = \frac{3 | x |}{4}$ is $(0, 0)$ .

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

$x = 0$

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

$y = \frac{3 | 0 |}{4}$

Simplify $\frac{3 | 0 |}{4}$ .

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

$y = \frac{3 \cdot 0}{4}$

Multiply $3$ by $0$ .

$y = \frac{0}{4}$

Divide $0$ by $4$ .

$y = 0$

The absolute value vertex is $(0, 0)$ .

$(0, 0)$

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 $- 2$ into $f (x) = \frac{3 | x |}{4}$ . In this case, the point is $(- 2, \frac{3}{2})$ .

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

$f (- 2) = \frac{3 | - 2 |}{4}$

Simplify the result.

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

$f (- 2) = \frac{3 \cdot 2}{4}$

Multiply $3$ by $2$ .

$f (- 2) = \frac{6}{4}$

Cancel the common factor of $6$ and $4$ .

Factor $2$ out of $6$ .

$f (- 2) = \frac{2 (3)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$f (- 2) = \frac{2 \cdot 3}{2 \cdot 2}$

Cancel the common factor.

$f (- 2) = \frac{2 \cdot 3}{2 \cdot 2}$

Rewrite the expression.

$f (- 2) = \frac{3}{2}$

The final answer is $\frac{3}{2}$ .

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

Substitute the $x$ value $- 1$ into $f (x) = \frac{3 | x |}{4}$ . In this case, the point is $(- 1, \frac{3}{4})$ .

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

$f (- 1) = \frac{3 | - 1 |}{4}$

Simplify the result.

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

$f (- 1) = \frac{3 \cdot 1}{4}$

Multiply $3$ by $1$ .

$f (- 1) = \frac{3}{4}$

The final answer is $\frac{3}{4}$ .

$y = \frac{3}{4}$

Substitute the $x$ value $2$ into $f (x) = \frac{3 | x |}{4}$ . In this case, the point is $(2, \frac{3}{2})$ .

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

$f (2) = \frac{3 | 2 |}{4}$

Simplify the result.

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

$f (2) = \frac{3 \cdot 2}{4}$

Multiply $3$ by $2$ .

$f (2) = \frac{6}{4}$

Cancel the common factor of $6$ and $4$ .

Factor $2$ out of $6$ .

$f (2) = \frac{2 (3)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$f (2) = \frac{2 \cdot 3}{2 \cdot 2}$

Cancel the common factor.

$f (2) = \frac{2 \cdot 3}{2 \cdot 2}$

Rewrite the expression.

$f (2) = \frac{3}{2}$

The final answer is $\frac{3}{2}$ .

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

The absolute value can be graphed using the points around the vertex $(0, 0), (- 2, 1.5), (- 1, 0.75), (1, 0.75), (2, 1.5)$

$\begin{array}{c} x & y \\ - 2 & 1.5 \\ - 1 & 0.75 \\ 0 & 0 \\ 1 & 0.75 \\ 2 & 1.5 \end{array}$

Step 4

Graph y=(3/4)|x|

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