# Graphing Absolute Value Functions

How to graph the given absolute value functions on the coordinate plane: examples and their solutions.

## Example 1: *y* = |*x*|

|*x*| can be either*x* (*x* ≥ 0)

or -*x* (*x* < 0).

Solve |*x*| for these two cases.

Absolute value equation (One variable)

Case 1: *x* ≥ 0

Then |*x*| = *x*.

So *y* = |*x*| is*y* = *x*.

Case 2: *x* < 0

Then |*x*| = -*x*.

So *y* = |*x*| is*y* = -*x*.

So *y* = |*x*| can be written

as a piecewise function:*y* = *x* (*x* ≥ 0)

= -*x* (*x* ≥ 0).

Graph the piecewise function

on a coordinate plane.

Draw *y* = *x*

on the right side of *x* = 0.

Draw *y* = -*x*

on the left side of *x* = 0.

Graphing piecewise functions

Then this is the graph of *y* = |*x*|.

## Example 2: Graph *y* = 2|*x*| - 3

|*x*| can be either*x* (*x* ≥ 0)

or -*x* (*x* < 0).

Solve |*x*| for these two cases.

Case 1: *x* ≥ 0

Then |*x*| = *x*.

So *y* = 2|*x*| - 3 is*y* = 2*x* - 3.

Case 2: *x* < 0

Then |*x*| = -*x*.

So *y* = 2|*x*| - 3 is*y* = -2*x* - 3.

So *y* = |*x*| can be written

as a piecewise function:*y* = 2*x* - 3 (*x* ≥ 0)

= -2*x* - 3 (*x* < 0).

Graph the piecewise function

on a coordinate plane.

Draw *y* = 2*x* - 3

on the right side of *x* = 0.

Draw *y* = -2*x* - 3

on the left side of *x* = 0.

Then this is the graph of *y* = 2|*x*| - 3.

Slope-intercept form

## Example 3: Graph *y* = |2*x* - 3|

|2*x* - 3| can be either

2*x* - 3 or -(2*x* - 3).

Solve |2*x* - 3| for these two cases.

Case 1: 2*x* - 3 ≥ 0

This is the case when *x* ≥ 3/2.

Then |2*x* - 3| = 2*x* - 3.

So *y* = |2*x* - 3| is*y* = 2*x* - 3.

Case 2: 2*x* - 3 < 0

This is the case when *x* < 3/2.

Then |2*x* - 3| = -(2*x* - 3).

So *y* = |2*x* - 3| is*y* = -(2*x* - 3).

-(2*x* - 3) = 2*x* + 3

So *y* = -2*x* + 3.

So *y* = |2*x* - 3| can be written

as a piecewise function:*y* = 2*x* - 3 (*x* ≥ 3/2)

= -2*x* + 3 (*x* < 0).

Graph the piecewise function

on a coordinate plane.

Draw *y* = 2*x* - 3

on the right side of *x* = 3/2.

Draw *y* = -2*x* + 3

on the left side of *x* = 3/2.

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

## Example 4: Graph |*y*| = 2*x* - 3

|*y*| can be either*y* (*y* ≥ 0)

or -*y* (*y* < 0).

Solve |*y*| for these two cases.

Case 1: *y* ≥ 0

Then |*y*| = *y*.

So |*y*| = 2*x* - 3 is*y* = 2*x* - 3.

Case 2: *y* < 0

Then |*y*| = -*y*.

So |*y*| = 2*x* - 3 is

-*y* = 2*x* - 3.

Multiply -1 on both sides.

Then *y* = -2*x* + 3.

So |*y*| = 2*x* - 3 can be written

as a piecewise function:*y* = 2*x* - 3 (*y* ≥ 0)

= -2*x* + 3 (*y* < 0).

Graph the piecewise function

on a coordinate plane.

Draw *y* = 2*x* - 3

above *y* = 0.

Draw *y* = -2*x* + 3

below *x* = 0.

Then this is the graph of |*y*| = 2*x* - 3.

## Example 5: Graph |*y*| = 2|*x*| - 3

There are two absolute value signs:

|*x*| and |*y*|.

So there are four cases to solve.

(2 × 2 = 4)

So, instead of solving for each case,

let's graph |*y*| = 2|*x*| - 3 by differently.

When *x* ≥ 0 and *y* ≥ 0,

|*y*| = 2|*x*| - 3 is*y* = 2*x* - 3.

So draw *y* = 2*x* - 3

on the quadrant I:

where *x* ≥ 0 and *y* ≥ 0.

Draw the image of *y* = 2*x* - 3

under the reflection in the *y*-axis

on the quadrant II.

Draw the image of *y* = 2*x* - 3

under the reflection in the origin

on the quadrant III.

Draw the image of *y* = 2*x* - 3

under the reflection in the *x*-axis

on the quadrant IV.

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