Precalculus Examples

$| y | = 4 - x$

Step 1

Rewrite the equation as $4 - x = | y |$ .

$4 - x = | y |$

Step 2

Subtract $4$ from both sides of the equation.

$- x = | y | - 4$

Step 3

Divide each term in $- x = | y | - 4$ by $- 1$ and simplify.

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Step 3.1

Divide each term in $- x = | y | - 4$ by $- 1$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

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

Step 3.2.2

Divide $x$ by $1$ .

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Move the negative one from the denominator of $\frac{| y |}{- 1}$ .

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

Step 3.3.1.2

Rewrite $- 1 \cdot | y |$ as $- | y |$ .

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

Step 3.3.1.3

Divide $- 4$ by $- 1$ .

$x = - | y | + 4$

Step 4

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

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Step 4.1

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

$y = 0$

Step 4.2

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

$x = - | 0 | + 4$

Step 4.3

Simplify $- | 0 | + 4$ .

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

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

$x = - 0 + 4$

Step 4.3.1.2

Multiply $- 1$ by $0$ .

$x = 0 + 4$

Step 4.3.2

Add $0$ and $4$ .

$x = 4$

Step 4.4

The absolute value vertex is $(4, 0)$ .

$(4, 0)$

Step 5

The domain is the set of all valid $x$ values. Use the graph to find the domain.

$(- \infty, 4]$

${x | x \leq 4}$

Step 6

For each $x$ value, there is one positive $y$ value and one negative $y$ 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 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Multiply $- 1$ by $2$ .

$f (2) = - 2 + 4$

Step 6.1.2.2

Add $- 2$ and $4$ .

$f (2) = 2$

Step 6.1.2.3

The final answer is $2$ .

$y = 2$

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

Multiply $- 1$ by $2$ .

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

Step 6.2.2.2

Add $- 2$ and $4$ .

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

Step 6.2.2.3

Multiply $- 1$ by $2$ .

$f (2) = - 2$

Step 6.2.2.4

The final answer is $- 2$ .

$y = - 2$

Step 6.3

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

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Step 6.3.1

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

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

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Multiply $- 1$ by $3$ .

$f (3) = - 3 + 4$

Step 6.3.2.2

Add $- 3$ and $4$ .

$f (3) = 1$

Step 6.3.2.3

The final answer is $1$ .

$y = 1$

Step 6.4

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

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Step 6.4.1

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

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

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

Multiply $- 1$ by $3$ .

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

Step 6.4.2.2

Add $- 3$ and $4$ .

$f (3) = - 1 \cdot 1$

Step 6.4.2.3

Multiply $- 1$ by $1$ .

$f (3) = - 1$

Step 6.4.2.4

The final answer is $- 1$ .

$y = - 1$

Step 6.5

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

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

Step 7