Precalculus Examples

$y = | \frac{1}{3} x + 1 |$

Combine $\frac{1}{3}$ and $x$ .

$y = | \frac{x}{3} + 1 |$

To find the $x$ coordinate of the vertex, set the inside of the absolute value $\frac{x}{3} + 1$ equal to $0$ . In this case, $\frac{x}{3} + 1 = 0$ .

$\frac{x}{3} + 1 = 0$

Solve the equation $\frac{x}{3} + 1 = 0$ to find the $x$ coordinate for the absolute value vertex.

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Subtract $1$ from both sides of the equation.

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

Multiply both sides of the equation by $3$ .

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

Simplify both sides of the equation.

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Simplify the left side.

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Cancel the common factor of $3$ .

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Write $3$ as a fraction with denominator $1$ .

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

Factor out the greatest common factor $3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

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

Divide $x$ by $1$ .

$x = 3 \cdot - 1$

Multiply $3$ by $- 1$ .

$x = - 3$

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

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

Simplify $| \frac{- 3}{3} + 1 |$ .

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Divide $- 3$ by $3$ .

$y = | - 1 + 1 |$

Add $- 1$ and $1$ .

$y = | 0 |$

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 $(- 3, 0)$ .

$(- 3, 0)$

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