Calculus Examples

$f (x) = | 2 x + 4 |$

Step 1

Find the absolute value vertex. In this case, the vertex for $y = | 2 x + 4 |$ is $(- 2, 0)$ .

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Step 1.1

To find the $x$ coordinate of the vertex, set the inside of the absolute value $2 x + 4$ equal to $0$ . In this case, $2 x + 4 = 0$ .

$2 x + 4 = 0$

Step 1.2

Solve the equation $2 x + 4 = 0$ to find the $x$ coordinate for the absolute value vertex.

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Step 1.2.1

Subtract $4$ from both sides of the equation.

$2 x = - 4$

Step 1.2.2

Divide each term in $2 x = - 4$ by $2$ and simplify.

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Step 1.2.2.1

Divide each term in $2 x = - 4$ by $2$ .

$\frac{2 x}{2} = \frac{- 4}{2}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 4}{2}$

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{2}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $- 4$ by $2$ .

$x = - 2$

Step 1.3

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

$y = | 2 (- 2) + 4 |$

Step 1.4

Simplify $| 2 (- 2) + 4 |$ .

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Step 1.4.1

Multiply $2$ by $- 2$ .

$y = | - 4 + 4 |$

Step 1.4.2

Add $- 4$ and $4$ .

$y = | 0 |$

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 $(- 2, 0)$ .

$(- 2, 0)$

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.

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Step 3.1

Substitute the $x$ value $- 4$ into $f (x) = | 2 x + 4 |$ . In this case, the point is $(- 4, 4)$ .

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Step 3.1.1

Replace the variable $x$ with $- 4$ in the expression.

$f (- 4) = | 2 (- 4) + 4 |$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Multiply $2$ by $- 4$ .

$f (- 4) = | - 8 + 4 |$

Step 3.1.2.2

Add $- 8$ and $4$ .

$f (- 4) = | - 4 |$

Step 3.1.2.3

The absolute value is the distance between a number and zero. The distance between $- 4$ and $0$ is $4$ .

$f (- 4) = 4$

Step 3.1.2.4

The final answer is $4$ .

$y = 4$

Step 3.2

Substitute the $x$ value $- 3$ into $f (x) = | 2 x + 4 |$ . In this case, the point is $(- 3, 2)$ .

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Step 3.2.1

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

$f (- 3) = | 2 (- 3) + 4 |$

Step 3.2.2

Simplify the result.

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Step 3.2.2.1

Multiply $2$ by $- 3$ .

$f (- 3) = | - 6 + 4 |$

Step 3.2.2.2

Add $- 6$ and $4$ .

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

Step 3.2.2.3

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

$f (- 3) = 2$

Step 3.2.2.4

The final answer is $2$ .

$y = 2$

Step 3.3

Substitute the $x$ value $0$ into $f (x) = | 2 x + 4 |$ . In this case, the point is $(0, 4)$ .

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Step 3.3.1

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

$f (0) = | 2 (0) + 4 |$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Multiply $2$ by $0$ .

$f (0) = | 0 + 4 |$

Step 3.3.2.2

Add $0$ and $4$ .

$f (0) = | 4 |$

Step 3.3.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$f (0) = 4$

Step 3.3.2.4

The final answer is $4$ .

$y = 4$

Step 3.4

The absolute value can be graphed using the points around the vertex $(- 2, 0), (- 4, 4), (- 3, 2), (- 1, 2), (0, 4)$

$\begin{array}{c} x & y \\ - 4 & 4 \\ - 3 & 2 \\ - 2 & 0 \\ - 1 & 2 \\ 0 & 4 \end{array}$

Step 4