Precalculus Examples

$y = | 9 x - 3 |$

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

$9 x - 3 = 0$

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

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Add $3$ to both sides of the equation.

$9 x = 3$

Divide each term by $9$ and simplify.

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

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

Reduce the expression by cancelling the common factors.

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Cancel the common factor.

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

Divide $x$ by $1$ .

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

Reduce the expression by cancelling the common factors.

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Factor $3$ out of $3$ .

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

Cancel the common factors.

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Factor $3$ out of $9$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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Simplify each term.

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Cancel the common factor of $3$ .

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Write $9$ as a fraction with denominator $1$ .

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

Factor out the greatest common factor $3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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Multiply $\frac{3}{1}$ and $\frac{1}{1}$ .

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

Divide $3$ by $1$ .

$y = | 3 - 3 |$

Simplify by subtracting numbers.

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Subtract $3$ from $3$ .

$y = | 0 |$

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

$y = 0$

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

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

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