Trigonometry Examples

$2 y - 1 = - 4 | x - 3 | - 1$

Step 1

Solve for $y$ .

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Step 1.1

Move all terms not containing $y$ to the right side of the equation.

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Step 1.1.1

Add $1$ to both sides of the equation.

$2 y = - 4 | x - 3 | - 1 + 1$

Step 1.1.2

Combine the opposite terms in $- 4 | x - 3 | - 1 + 1$ .

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Step 1.1.2.1

Add $- 1$ and $1$ .

$2 y = - 4 | x - 3 | + 0$

Step 1.1.2.2

Add $- 4 | x - 3 |$ and $0$ .

$2 y = - 4 | x - 3 |$

Step 1.2

Divide each term in $2 y = - 4 | x - 3 |$ by $2$ and simplify.

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Step 1.2.1

Divide each term in $2 y = - 4 | x - 3 |$ by $2$ .

$\frac{2 y}{2} = \frac{- 4 | x - 3 |}{2}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{- 4 | x - 3 |}{2}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 4 | x - 3 |}{2}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

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

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Step 1.2.3.1.1

Factor $2$ out of $- 4 | x - 3 |$ .

$y = \frac{2 (- 2 | x - 3 |)}{2}$

Step 1.2.3.1.2

Cancel the common factors.

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Step 1.2.3.1.2.1

Factor $2$ out of $2$ .

$y = \frac{2 (- 2 | x - 3 |)}{2 (1)}$

Step 1.2.3.1.2.2

Cancel the common factor.

$y = \frac{2 (- 2 | x - 3 |)}{2 \cdot 1}$

Step 1.2.3.1.2.3

Rewrite the expression.

$y = \frac{- 2 | x - 3 |}{1}$

Step 1.2.3.1.2.4

Divide $- 2 | x - 3 |$ by $1$ .

$y = - 2 | x - 3 |$

Step 2

Find the absolute value vertex. In this case, the vertex for $y = - 2 | x - 3 |$ is $(3, 0)$ .

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Step 2.1

To find the $x$ coordinate of the vertex, set the inside of the absolute value $x - 3$ equal to $0$ . In this case, $x - 3 = 0$ .

$x - 3 = 0$

Step 2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.3

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

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

Step 2.4

Simplify $- 2 | (3) - 3 |$ .

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Step 2.4.1

Subtract $3$ from $3$ .

$y = - 2 | 0 |$

Step 2.4.2

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

$y = - 2 \cdot 0$

Step 2.4.3

Multiply $- 2$ by $0$ .

$y = 0$

Step 2.5

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

$(3, 0)$

Step 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:

${x | x \in ℝ}$

Step 4

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 4.1

Substitute the $x$ value $1$ into $f (x) = - 2 | x - 3 |$ . In this case, the point is $(1, - 4)$ .

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Step 4.1.1

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

$f (1) = - 2 | (1) - 3 |$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Subtract $3$ from $1$ .

$f (1) = - 2 | - 2 |$

Step 4.1.2.2

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

$f (1) = - 2 \cdot 2$

Step 4.1.2.3

Multiply $- 2$ by $2$ .

$f (1) = - 4$

Step 4.1.2.4

The final answer is $- 4$ .

$y = - 4$

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Subtract $3$ from $2$ .

$f (2) = - 2 | - 1 |$

Step 4.2.2.2

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

$f (2) = - 2 \cdot 1$

Step 4.2.2.3

Multiply $- 2$ by $1$ .

$f (2) = - 2$

Step 4.2.2.4

The final answer is $- 2$ .

$y = - 2$

Step 4.3

Substitute the $x$ value $5$ into $f (x) = - 2 | x - 3 |$ . In this case, the point is $(5, - 4)$ .

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Step 4.3.1

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

$f (5) = - 2 | (5) - 3 |$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Subtract $3$ from $5$ .

$f (5) = - 2 | 2 |$

Step 4.3.2.2

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

$f (5) = - 2 \cdot 2$

Step 4.3.2.3

Multiply $- 2$ by $2$ .

$f (5) = - 4$

Step 4.3.2.4

The final answer is $- 4$ .

$y = - 4$

Step 4.4

The absolute value can be graphed using the points around the vertex $(3, 0), (1, - 4), (2, - 2), (4, - 2), (5, - 4)$

$\begin{array}{c} x & y \\ 1 & - 4 \\ 2 & - 2 \\ 3 & 0 \\ 4 & - 2 \\ 5 & - 4 \end{array}$

Step 5