Pre-Algebra Examples

$f (x) = \frac{1}{2} \cdot | x | + \frac{4}{5}$

Step 1

Find the absolute value vertex. In this case, the vertex for $y = \frac{| x |}{2} + \frac{4}{5}$ is $(0, \frac{4}{5})$ .

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Step 1.1

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

$x = 0$

Step 1.2

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

$y = \frac{| 0 |}{2} + \frac{4}{5}$

Step 1.3

Simplify $\frac{| 0 |}{2} + \frac{4}{5}$ .

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

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

$y = \frac{0}{2} + \frac{4}{5}$

Step 1.3.1.2

Divide $0$ by $2$ .

$y = 0 + \frac{4}{5}$

Step 1.3.2

Add $0$ and $\frac{4}{5}$ .

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

Step 1.4

The absolute value vertex is $(0, \frac{4}{5})$ .

$(0, \frac{4}{5})$

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 $f (x) = \frac{| x |}{2} + \frac{4}{5}$ . In this case, the point is $(- 2, \frac{9}{5})$ .

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Step 3.1.1

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

$f (- 2) = \frac{| - 2 |}{2} + \frac{4}{5}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

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

$f (- 2) = \frac{2}{2} + \frac{4}{5}$

Step 3.1.2.1.2

Divide $2$ by $2$ .

$f (- 2) = 1 + \frac{4}{5}$

Step 3.1.2.2

Simplify the expression.

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Step 3.1.2.2.1

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

$f (- 2) = \frac{5}{5} + \frac{4}{5}$

Step 3.1.2.2.2

Combine the numerators over the common denominator.

$f (- 2) = \frac{5 + 4}{5}$

Step 3.1.2.2.3

Add $5$ and $4$ .

$f (- 2) = \frac{9}{5}$

Step 3.1.2.3

The final answer is $\frac{9}{5}$ .

$y = \frac{9}{5}$

Step 3.2

Substitute the $x$ value $- 1$ into $f (x) = \frac{| x |}{2} + \frac{4}{5}$ . In this case, the point is $(- 1, \frac{13}{10})$ .

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Step 3.2.1

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

$f (- 1) = \frac{| - 1 |}{2} + \frac{4}{5}$

Step 3.2.2

Simplify the result.

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Step 3.2.2.1

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

$f (- 1) = \frac{1}{2} + \frac{4}{5}$

Step 3.2.2.2

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (- 1) = \frac{1}{2} \cdot \frac{5}{5} + \frac{4}{5}$

Step 3.2.2.3

To write $\frac{4}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- 1) = \frac{1}{2} \cdot \frac{5}{5} + \frac{4}{5} \cdot \frac{2}{2}$

Step 3.2.2.4

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.2.2.4.1

Multiply $\frac{1}{2}$ by $\frac{5}{5}$ .

$f (- 1) = \frac{5}{2 \cdot 5} + \frac{4}{5} \cdot \frac{2}{2}$

Step 3.2.2.4.2

Multiply $2$ by $5$ .

$f (- 1) = \frac{5}{10} + \frac{4}{5} \cdot \frac{2}{2}$

Step 3.2.2.4.3

Multiply $\frac{4}{5}$ by $\frac{2}{2}$ .

$f (- 1) = \frac{5}{10} + \frac{4 \cdot 2}{5 \cdot 2}$

Step 3.2.2.4.4

Multiply $5$ by $2$ .

$f (- 1) = \frac{5}{10} + \frac{4 \cdot 2}{10}$

Step 3.2.2.5

Combine the numerators over the common denominator.

$f (- 1) = \frac{5 + 4 \cdot 2}{10}$

Step 3.2.2.6

Simplify the numerator.

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Step 3.2.2.6.1

Multiply $4$ by $2$ .

$f (- 1) = \frac{5 + 8}{10}$

Step 3.2.2.6.2

Add $5$ and $8$ .

$f (- 1) = \frac{13}{10}$

Step 3.2.2.7

The final answer is $\frac{13}{10}$ .

$y = \frac{13}{10}$

Step 3.3

Substitute the $x$ value $2$ into $f (x) = \frac{| x |}{2} + \frac{4}{5}$ . In this case, the point is $(2, \frac{9}{5})$ .

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Step 3.3.1

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

$f (2) = \frac{| 2 |}{2} + \frac{4}{5}$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

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

$f (2) = \frac{2}{2} + \frac{4}{5}$

Step 3.3.2.1.2

Divide $2$ by $2$ .

$f (2) = 1 + \frac{4}{5}$

Step 3.3.2.2

Simplify the expression.

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Step 3.3.2.2.1

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

$f (2) = \frac{5}{5} + \frac{4}{5}$

Step 3.3.2.2.2

Combine the numerators over the common denominator.

$f (2) = \frac{5 + 4}{5}$

Step 3.3.2.2.3

Add $5$ and $4$ .

$f (2) = \frac{9}{5}$

Step 3.3.2.3

The final answer is $\frac{9}{5}$ .

$y = \frac{9}{5}$

Step 3.4

The absolute value can be graphed using the points around the vertex $(0, \frac{4}{5}), (- 2, 1.8), (- 1, 1.3), (1, 1.3), (2, 1.8)$

$\begin{array}{c} x & y \\ - 2 & 1.8 \\ - 1 & 1.3 \\ 0 & \frac{4}{5} \\ 1 & 1.3 \\ 2 & 1.8 \end{array}$

Step 4