Pre-Algebra Examples

$| x - \frac{1}{4} |$

Step 1

Find the absolute value vertex. In this case, the vertex for $y = | x - \frac{1}{4} |$ is $(\frac{1}{4}, 0)$ .

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Step 1.1

To find the $x$ coordinate of the vertex, set the inside of the absolute value $x - \frac{1}{4}$ equal to $0$ . In this case, $x - \frac{1}{4} = 0$ .

$x - \frac{1}{4} = 0$

Step 1.2

Add $\frac{1}{4}$ to both sides of the equation.

$x = \frac{1}{4}$

Step 1.3

Replace the variable $x$ with $\frac{1}{4}$ in the expression.

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

Step 1.4

Simplify $| (\frac{1}{4}) - \frac{1}{4} |$ .

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Step 1.4.1

Combine the numerators over the common denominator.

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

Step 1.4.2

Simplify the expression.

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Step 1.4.2.1

Subtract $1$ from $1$ .

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

Step 1.4.2.2

Divide $0$ by $4$ .

$y = | 0 |$

Step 1.4.3

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

$y = 0$

Step 1.5

The absolute value vertex is $(\frac{1}{4}, 0)$ .

$(\frac{1}{4}, 0)$

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.

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Step 3.1

Substitute the $x$ value $- 2$ into $f (x) = | x - \frac{1}{4} |$ . In this case, the point is $(- 2, \frac{9}{4})$ .

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Step 3.1.1

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

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

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 3.1.2.2

Combine $- 2$ and $\frac{4}{4}$ .

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

Step 3.1.2.3

Combine the numerators over the common denominator.

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

Step 3.1.2.4

Simplify the numerator.

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Step 3.1.2.4.1

Multiply $- 2$ by $4$ .

$f (- 2) = | \frac{- 8 - 1}{4} |$

Step 3.1.2.4.2

Subtract $1$ from $- 8$ .

$f (- 2) = | \frac{- 9}{4} |$

Step 3.1.2.5

Move the negative in front of the fraction.

$f (- 2) = | - \frac{9}{4} |$

Step 3.1.2.6

$- \frac{9}{4}$ is approximately $- 2.25$ which is negative so negate $- \frac{9}{4}$ and remove the absolute value

$f (- 2) = \frac{9}{4}$

Step 3.1.2.7

The final answer is $\frac{9}{4}$ .

$y = \frac{9}{4}$

Step 3.2

Substitute the $x$ value $- 1$ into $f (x) = | x - \frac{1}{4} |$ . In this case, the point is $(- 1, \frac{5}{4})$ .

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Step 3.2.1

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

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

Step 3.2.2

Simplify the result.

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Step 3.2.2.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 3.2.2.2

Combine $- 1$ and $\frac{4}{4}$ .

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

Step 3.2.2.3

Combine the numerators over the common denominator.

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

Step 3.2.2.4

Simplify the numerator.

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Step 3.2.2.4.1

Multiply $- 1$ by $4$ .

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

Step 3.2.2.4.2

Subtract $1$ from $- 4$ .

$f (- 1) = | \frac{- 5}{4} |$

Step 3.2.2.5

Move the negative in front of the fraction.

$f (- 1) = | - \frac{5}{4} |$

Step 3.2.2.6

$- \frac{5}{4}$ is approximately $- 1.25$ which is negative so negate $- \frac{5}{4}$ and remove the absolute value

$f (- 1) = \frac{5}{4}$

Step 3.2.2.7

The final answer is $\frac{5}{4}$ .

$y = \frac{5}{4}$

Step 3.3

Substitute the $x$ value $0$ into $f (x) = | x - \frac{1}{4} |$ . In this case, the point is $(0, \frac{1}{4})$ .

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Step 3.3.1

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

$f (0) = | (0) - \frac{1}{4} |$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Subtract $\frac{1}{4}$ from $0$ .

$f (0) = | - \frac{1}{4} |$

Step 3.3.2.2

$- \frac{1}{4}$ is approximately $- 0.25$ which is negative so negate $- \frac{1}{4}$ and remove the absolute value

$f (0) = \frac{1}{4}$

Step 3.3.2.3

The final answer is $\frac{1}{4}$ .

$y = \frac{1}{4}$

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

Simplify the result.

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Step 3.4.2.1

Write $1$ as a fraction with a common denominator.

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

Step 3.4.2.2

Combine the numerators over the common denominator.

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

Step 3.4.2.3

Subtract $1$ from $4$ .

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

Step 3.4.2.4

$\frac{3}{4}$ is approximately $0.75$ which is positive so remove the absolute value

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

Step 3.4.2.5

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

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

Step 3.5

The absolute value can be graphed using the points around the vertex $(\frac{1}{4}, 0), (- 2, 2.25), (- 1, 1.25), (0, 0.25), (1, 0.75)$

$\begin{array}{c} x & y \\ - 2 & 2.25 \\ - 1 & 1.25 \\ 0 & 0.25 \\ \frac{1}{4} & 0 \\ 1 & 0.75 \end{array}$

Step 4