Precalculus Examples

$y = | x - 4 | - 4$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = | x - 4 | - 4$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $| x - 4 | - 4 = 0$ .

$| x - 4 | - 4 = 0$

Step 1.2.2

Add $4$ to both sides of the equation.

$| x - 4 | = 4$

Step 1.2.3

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x - 4 = \pm 4$

Step 1.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x - 4 = 4$

Step 1.2.4.2

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

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Step 1.2.4.2.1

Add $4$ to both sides of the equation.

$x = 4 + 4$

Step 1.2.4.2.2

Add $4$ and $4$ .

$x = 8$

Step 1.2.4.3

Next, use the negative value of the $\pm$ to find the second solution.

$x - 4 = - 4$

Step 1.2.4.4

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

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Step 1.2.4.4.1

Add $4$ to both sides of the equation.

$x = - 4 + 4$

Step 1.2.4.4.2

Add $- 4$ and $4$ .

$x = 0$

Step 1.2.4.5

The complete solution is the result of both the positive and negative portions of the solution.

$x = 8, 0$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(8, 0), (0, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = | (0) - 4 | - 4$

Step 2.2

Simplify $| (0) - 4 | - 4$ .

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Subtract $4$ from $0$ .

$y = | - 4 | - 4$

Step 2.2.1.2

The absolute value is the distance between a number and zero. The distance between $- 4$ and $0$ is $4$ .

$y = 4 - 4$

Step 2.2.2

Subtract $4$ from $4$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 0)$

Step 3

List the intersections.

x-intercept(s): $(8, 0), (0, 0)$

y-intercept(s): $(0, 0)$

Step 4