Graph y=-|1/4x|-3

$y = - | \frac{1}{4} x | - 3$

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

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.

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

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

Simplify each term.

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

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

Divide $0$ by $4$ .

$y = - 0 - 3$

Multiply $- 1$ by $0$ .

$y = 0 - 3$

Subtract $3$ from $0$ .

$y = - 3$

The absolute value vertex is $(0, - 3)$ .

$(0, - 3)$

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 $- 2$ into $f (x) = - \frac{| x |}{4} - 3$ . In this case, the point is $(- 2, - \frac{7}{2})$ .

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

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

Simplify the result.

Simplify each term.

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

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

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2$ .

$f (- 2) = - \frac{2 (1)}{4} - 3$

Cancel the common factors.

Factor $2$ out of $4$ .

$f (- 2) = - \frac{2 \cdot 1}{2 \cdot 2} - 3$

Cancel the common factor.

$f (- 2) = - \frac{2 \cdot 1}{2 \cdot 2} - 3$

Rewrite the expression.

$f (- 2) = - \frac{1}{2} - 3$

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- 2) = - \frac{1}{2} - 3 \cdot \frac{2}{2}$

Combine $- 3$ and $\frac{2}{2}$ .

$f (- 2) = - \frac{1}{2} + \frac{- 3 \cdot 2}{2}$

Combine the numerators over the common denominator.

$f (- 2) = \frac{- 1 - 3 \cdot 2}{2}$

Simplify the numerator.

Multiply $- 3$ by $2$ .

$f (- 2) = \frac{- 1 - 6}{2}$

Subtract $6$ from $- 1$ .

$f (- 2) = \frac{- 7}{2}$

Move the negative in front of the fraction.

$f (- 2) = - \frac{7}{2}$

The final answer is $- \frac{7}{2}$ .

$y = - \frac{7}{2}$

Substitute the $x$ value $- 1$ into $f (x) = - \frac{| x |}{4} - 3$ . In this case, the point is $(- 1, - \frac{13}{4})$ .

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

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

Simplify the result.

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

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

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Combine $- 3$ and $\frac{4}{4}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 3$ by $4$ .

$f (- 1) = \frac{- 1 - 12}{4}$

Subtract $12$ from $- 1$ .

$f (- 1) = \frac{- 13}{4}$

Move the negative in front of the fraction.

$f (- 1) = - \frac{13}{4}$

The final answer is $- \frac{13}{4}$ .

$y = - \frac{13}{4}$

Substitute the $x$ value $2$ into $f (x) = - \frac{| x |}{4} - 3$ . In this case, the point is $(2, - \frac{7}{2})$ .

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

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

Simplify the result.

Simplify each term.

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

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

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2$ .

$f (2) = - \frac{2 (1)}{4} - 3$

Cancel the common factors.

Factor $2$ out of $4$ .

$f (2) = - \frac{2 \cdot 1}{2 \cdot 2} - 3$

Cancel the common factor.

$f (2) = - \frac{2 \cdot 1}{2 \cdot 2} - 3$

Rewrite the expression.

$f (2) = - \frac{1}{2} - 3$

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (2) = - \frac{1}{2} - 3 \cdot \frac{2}{2}$

Combine $- 3$ and $\frac{2}{2}$ .

$f (2) = - \frac{1}{2} + \frac{- 3 \cdot 2}{2}$

Combine the numerators over the common denominator.

$f (2) = \frac{- 1 - 3 \cdot 2}{2}$

Simplify the numerator.

Multiply $- 3$ by $2$ .

$f (2) = \frac{- 1 - 6}{2}$

Subtract $6$ from $- 1$ .

$f (2) = \frac{- 7}{2}$

Move the negative in front of the fraction.

$f (2) = - \frac{7}{2}$

The final answer is $- \frac{7}{2}$ .

$y = - \frac{7}{2}$

The absolute value can be graphed using the points around the vertex $(0, - 3), (- 2, - 3.5), (- 1, - 3.25), (1, - 3.25), (2, - 3.5)$

$\begin{array}{c} x & y \\ - 2 & - 3.5 \\ - 1 & - 3.25 \\ 0 & - 3 \\ 1 & - 3.25 \\ 2 & - 3.5 \end{array}$

Graph y=-|1/4x|-3

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