Trigonometry Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Solve the equation $\frac{x}{4} + 1 = 0$ to find the $x$ coordinate for the absolute value vertex.

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Step 1.2.1

Subtract $1$ from both sides of the equation.

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

Step 1.2.2

Multiply both sides of the equation by $4$ .

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

Step 1.2.3

Simplify both sides of the equation.

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Step 1.2.3.1

Simplify the left side.

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Step 1.2.3.1.1

Cancel the common factor of $4$ .

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Step 1.2.3.1.1.1

Cancel the common factor.

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

Step 1.2.3.1.1.2

Rewrite the expression.

$x = 4 \cdot - 1$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply $4$ by $- 1$ .

$x = - 4$

Step 1.3

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

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

Step 1.4

Simplify $| \frac{- 4}{4} + 1 | - 4$ .

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Divide $- 4$ by $4$ .

$y = | - 1 + 1 | - 4$

Step 1.4.1.2

Add $- 1$ and $1$ .

$y = | 0 | - 4$

Step 1.4.1.3

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

$y = 0 - 4$

Step 1.4.2

Subtract $4$ from $0$ .

$y = - 4$

Step 1.5

The absolute value vertex is $(- 4, - 4)$ .

$(- 4, - 4)$

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 $- 6$ into $f (x) = | \frac{x}{4} + 1 | - 4$ . In this case, the point is $(- 6, - \frac{7}{2})$ .

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Step 3.1.1

Replace the variable $x$ with $- 6$ in the expression.

$f (- 6) = | \frac{- 6}{4} + 1 | - 4$

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

Simplify each term.

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Step 3.1.2.1.1.1

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

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Step 3.1.2.1.1.1.1

Factor $2$ out of $- 6$ .

$f (- 6) = | \frac{2 (- 3)}{4} + 1 | - 4$

Step 3.1.2.1.1.1.2

Cancel the common factors.

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Step 3.1.2.1.1.1.2.1

Factor $2$ out of $4$ .

$f (- 6) = | \frac{2 \cdot - 3}{2 \cdot 2} + 1 | - 4$

Step 3.1.2.1.1.1.2.2

Cancel the common factor.

$f (- 6) = | \frac{2 \cdot - 3}{2 \cdot 2} + 1 | - 4$

Step 3.1.2.1.1.1.2.3

Rewrite the expression.

$f (- 6) = | \frac{- 3}{2} + 1 | - 4$

Step 3.1.2.1.1.2

Move the negative in front of the fraction.

$f (- 6) = | - \frac{3}{2} + 1 | - 4$

Step 3.1.2.1.2

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

$f (- 6) = | - \frac{3}{2} + \frac{2}{2} | - 4$

Step 3.1.2.1.3

Combine the numerators over the common denominator.

$f (- 6) = | \frac{- 3 + 2}{2} | - 4$

Step 3.1.2.1.4

Add $- 3$ and $2$ .

$f (- 6) = | \frac{- 1}{2} | - 4$

Step 3.1.2.1.5

Move the negative in front of the fraction.

$f (- 6) = | - \frac{1}{2} | - 4$

Step 3.1.2.1.6

$- \frac{1}{2}$ is approximately $- 0.5$ which is negative so negate $- \frac{1}{2}$ and remove the absolute value

$f (- 6) = \frac{1}{2} - 4$

Step 3.1.2.2

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

$f (- 6) = \frac{1}{2} - 4 \cdot \frac{2}{2}$

Step 3.1.2.3

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

$f (- 6) = \frac{1}{2} + \frac{- 4 \cdot 2}{2}$

Step 3.1.2.4

Combine the numerators over the common denominator.

$f (- 6) = \frac{1 - 4 \cdot 2}{2}$

Step 3.1.2.5

Simplify the numerator.

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Step 3.1.2.5.1

Multiply $- 4$ by $2$ .

$f (- 6) = \frac{1 - 8}{2}$

Step 3.1.2.5.2

Subtract $8$ from $1$ .

$f (- 6) = \frac{- 7}{2}$

Step 3.1.2.6

Move the negative in front of the fraction.

$f (- 6) = - \frac{7}{2}$

Step 3.1.2.7

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

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

Step 3.2

Substitute the $x$ value $- 5$ into $f (x) = | \frac{x}{4} + 1 | - 4$ . In this case, the point is $(- 5, - \frac{15}{4})$ .

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Step 3.2.1

Replace the variable $x$ with $- 5$ in the expression.

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

Step 3.2.2

Simplify the result.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Move the negative in front of the fraction.

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

Step 3.2.2.1.2

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

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

Step 3.2.2.1.3

Combine the numerators over the common denominator.

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

Step 3.2.2.1.4

Add $- 5$ and $4$ .

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

Step 3.2.2.1.5

Move the negative in front of the fraction.

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

Step 3.2.2.1.6

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

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

Step 3.2.2.2

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

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

Step 3.2.2.3

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

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

Step 3.2.2.4

Combine the numerators over the common denominator.

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

Step 3.2.2.5

Simplify the numerator.

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Step 3.2.2.5.1

Multiply $- 4$ by $4$ .

$f (- 5) = \frac{1 - 16}{4}$

Step 3.2.2.5.2

Subtract $16$ from $1$ .

$f (- 5) = \frac{- 15}{4}$

Step 3.2.2.6

Move the negative in front of the fraction.

$f (- 5) = - \frac{15}{4}$

Step 3.2.2.7

The final answer is $- \frac{15}{4}$ .

$y = - \frac{15}{4}$

Step 3.3

Substitute the $x$ value $- 2$ into $f (x) = | \frac{x}{4} + 1 | - 4$ . In this case, the point is $(- 2, - \frac{7}{2})$ .

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Step 3.3.1

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

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

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

Simplify each term.

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Step 3.3.2.1.1.1

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

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Step 3.3.2.1.1.1.1

Factor $2$ out of $- 2$ .

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

Step 3.3.2.1.1.1.2

Cancel the common factors.

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Step 3.3.2.1.1.1.2.1

Factor $2$ out of $4$ .

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

Step 3.3.2.1.1.1.2.2

Cancel the common factor.

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

Step 3.3.2.1.1.1.2.3

Rewrite the expression.

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

Step 3.3.2.1.1.2

Move the negative in front of the fraction.

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

Step 3.3.2.1.2

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

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

Step 3.3.2.1.3

Combine the numerators over the common denominator.

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

Step 3.3.2.1.4

Add $- 1$ and $2$ .

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

Step 3.3.2.1.5

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

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

Step 3.3.2.2

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

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

Step 3.3.2.3

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

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

Step 3.3.2.4

Combine the numerators over the common denominator.

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

Step 3.3.2.5

Simplify the numerator.

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Step 3.3.2.5.1

Multiply $- 4$ by $2$ .

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

Step 3.3.2.5.2

Subtract $8$ from $1$ .

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

Step 3.3.2.6

Move the negative in front of the fraction.

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

Step 3.3.2.7

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

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

Step 3.4

The absolute value can be graphed using the points around the vertex $(- 4, - 4), (- 6, - 3.5), (- 5, - 3.75), (- 3, - 3.75), (- 2, - 3.5)$

$\begin{array}{c} x & y \\ - 6 & - 3.5 \\ - 5 & - 3.75 \\ - 4 & - 4 \\ - 3 & - 3.75 \\ - 2 & - 3.5 \end{array}$

Step 4