# Absolute Value Function: Graph

How to graph an absolute value function on a coordinate plane: 5 examples and their solutions.

## Example 1: y = |x|

### Example

### Solution

See y = |x|.

There's |x|.

Absolute Value

So draw y = x

at x > 0.

(the right side of x = 0).

There's |x|.

So draw the image of the graph

under the reflection in the y-axis.

This is the graph of y = |x|.

## Example 2: y = f(|x|)

### Example

### Solution

See y = 2|x| - 3.

There's |x|.

So draw y = 2x - 3

at x > 0.

(the right side of x = 0).

Slope-Intercept Form

There's |x|.

So draw the image of the graph

under the reflection in the y-axis.

This is the graph of y = 2|x| - 3.

## Example 3: |y| = f(x)

### Example

### Solution

See |y| = 2x - 3.

There's |y|.

So draw y = 2x - 3

at y > 0.

(the upper side of y = 0).

There's |y|.

So draw the image of the graph

under the reflection in the x-axis.

This is the graph of |y| = 2x - 3.

## Example 4: |y| = f(|x|)

### Example

### Solution

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

There are |x| and |y|.

So draw y = 2x - 3

at the quadrant I.

(x > 0 and y > 0).

There are |x| and |y|.

So draw the images of the graph

under the reflection

in the x-axis,

in the y-axis,

and in the origin.

So draw the images of the graph

under the reflection

in the x-axis,

in the y-axis,

and in the origin.

This is the graph of |y| = 2|x| - 3.

## Example 5: y = |f(x)|

### Example

### Solution

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

This is y = |f(x)|.

Then draw, y = f(x),

y = x^{2} - 4.

At the region below the x-axis,

lightly draw the graph.

Quadratic Function: Vertex Form

There's |f(x)|.

Then draw the image of the graph

that is below the x-axis

under the reflection in the x-axis.

Remove the graph below the x-axis.

Then this is the graph of y = |x^{2} - 4|.