Precalculus Examples

$y = | 9 x - 3 |$

Step 1

To find the $x$ coordinate of the vertex, set the inside of the absolute value $9 x - 3$ equal to $0$ . In this case, $9 x - 3 = 0$ .

$9 x - 3 = 0$

Step 2

Solve the equation $9 x - 3 = 0$ to find the $x$ coordinate for the absolute value vertex.

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Step 2.1

Add $3$ to both sides of the equation.

$9 x = 3$

Step 2.2

Divide each term in $9 x = 3$ by $9$ and simplify.

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Step 2.2.1

Divide each term in $9 x = 3$ by $9$ .

$\frac{9 x}{9} = \frac{3}{9}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $9$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{3}{9}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{9}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Cancel the common factor of $3$ and $9$ .

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Step 2.2.3.1.1

Factor $3$ out of $3$ .

$x = \frac{3 (1)}{9}$

Step 2.2.3.1.2

Cancel the common factors.

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Step 2.2.3.1.2.1

Factor $3$ out of $9$ .

$x = \frac{3 \cdot 1}{3 \cdot 3}$

Step 2.2.3.1.2.2

Cancel the common factor.

$x = \frac{3 \cdot 1}{3 \cdot 3}$

Step 2.2.3.1.2.3

Rewrite the expression.

$x = \frac{1}{3}$

Step 3

Replace the variable $x$ with $\frac{1}{3}$ in the expression.

$y = | 9 (\frac{1}{3}) - 3 |$

Step 4

Simplify $| 9 (\frac{1}{3}) - 3 |$ .

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Step 4.1

Cancel the common factor of $3$ .

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Step 4.1.1

Factor $3$ out of $9$ .

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

Step 4.1.2

Cancel the common factor.

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

Step 4.1.3

Rewrite the expression.

$y = | 3 - 3 |$

Step 4.2

Subtract $3$ from $3$ .

$y = | 0 |$

Step 4.3

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

$y = 0$

Step 5

The absolute value vertex is $(\frac{1}{3}, 0)$ .

$(\frac{1}{3}, 0)$

Step 6