# Absolute Value Function

See how to graph an absolute value function/inequality.

7 examples and their solutions.

## Absolute Value Function: Graph

### Example

Graph y = |x|.

Solution ### Example: y = f(|x|)

Graph y = 2|x| - 3.

Solution ### Example: |y| = f(x)

Graph |y| = 2x - 3.

Solution |y| = 2x - 3

There's |y|.

Then first draw y = 2x - 3

on y ≥ 0.

There's |y|.

Then first draw y = 2x - 3

on y ≥ 0.

↓

Draw the image of the line

under the reflection in the x-axis.

under the reflection in the x-axis.

↓

Close

### Example: |y| = f(|x|)

Graph |y| = 2|x| - 3.

Solution |y| = 2|x| - 3

There are |x| and |y|.

Then first draw y = 2x - 3

on quadrant I.

(x ≥ 0, y ≥ 0)

There are |x| and |y|.

Then first draw y = 2x - 3

on quadrant I.

(x ≥ 0, y ≥ 0)

↓

↓

Close

### Example: y = |f(x)|

Graph y = |x

Solution ^{2}- 4|.y = |x

This is y = |f(x)|.

Then first draw y = x

^{2}- 4|This is y = |f(x)|.

Then first draw y = x

^{2}- 4.↓

Draw the image of the graph

that is below the x-axis

under the reflection in the x-axis.

And erase the original graph

below the x-axis.

that is below the x-axis

under the reflection in the x-axis.

And erase the original graph

below the x-axis.

↓

Close

## Absolute Value Inequality (Two Variables)

### Example

Graph y > |x| - 2.

Solution The inequality sign, >, does not include '='.

So use a dashed line

to draw y = |x| - 2.

So use a dashed line

to draw y = |x| - 2.

↓

y > |x| - 2

0 > |0| - 2 - [1]

0 > 0 - 2

0 > -2 ( o )

[1]

Pick a point that is not on the graph.

(0, 0) seems to be good.

Put this into y > |x| - 2.

See if this inequality is true.

(0, 0) seems to be good.

Put this into y > |x| - 2.

See if this inequality is true.

↓

0 > -2

This is true.

So color the region

that includes (0, 0).

And don't color the region

that is adjacent to the colored region.

This is true.

So color the region

that includes (0, 0).

And don't color the region

that is adjacent to the colored region.

Close

### Example

Graph |y| ≤ -|x| + 4.

Solution The inequality sign, ≤, does include '='.

So use a solid line

to draw |y| = -|x| + 4.

So use a solid line

to draw |y| = -|x| + 4.

↓

|y| ≤ -|x| + 4

|1| ≤ -|4| + 4 - [1]

1 ≤ -4 + 4

1 ≤ 0 ( x )

[1]

Pick a point that is not on the line.

Let's pick (4, 1).

Put this into |y| ≤ -|x| + 4.

Let's pick (4, 1).

Put this into |y| ≤ -|x| + 4.

↓

1 ≤ 0

This is false.

So color the region

that does not include (4, 1).

And don't color the region

that is adjacent to the colored region.

This is false.

So color the region

that does not include (4, 1).

And don't color the region

that is adjacent to the colored region.

Close