Graphing Absolute Value Functions

Graphing Absolute Value Functions

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

Example 1: y = |x|

Graph the given function. 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

Graph the given function. 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 = 2x - 3.

Case 2: x < 0

Then |x| = -x.

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

So y = |x| can be written
as a piecewise function:

y = 2x - 3 (x ≥ 0)
= -2x - 3 (x < 0).

Graph the piecewise function
on a coordinate plane.

Draw y = 2x - 3
on the right side of x = 0.

Draw y = -2x - 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 = |2x - 3|

Graph the given function. y = |2x - 3|

|2x - 3| can be either
2x - 3 or -(2x - 3).

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

Case 1: 2x - 3 ≥ 0

This is the case when x ≥ 3/2.

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

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

Case 2: 2x - 3 < 0

This is the case when x < 3/2.

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

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

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

So y = -2x + 3.

So y = |2x - 3| can be written
as a piecewise function:

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

Graph the piecewise function
on a coordinate plane.

Draw y = 2x - 3
on the right side of x = 3/2.

Draw y = -2x + 3
on the left side of x = 3/2.

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

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

Graph the given function. |y| = 2x - 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| = 2x - 3 is
y = 2x - 3.

Case 2: y < 0

Then |y| = -y.

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

Multiply -1 on both sides.

Then y = -2x + 3.

So |y| = 2x - 3 can be written
as a piecewise function:

y = 2x - 3 (y ≥ 0)
= -2x + 3 (y < 0).

Graph the piecewise function
on a coordinate plane.

Draw y = 2x - 3
above y = 0.

Draw y = -2x + 3
below x = 0.

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

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

Graph the given function. |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 = 2x - 3.

So draw y = 2x - 3
on the quadrant I:
where x ≥ 0 and y ≥ 0.

Draw the image of y = 2x - 3
under the reflection in the y-axis
on the quadrant II.

Draw the image of y = 2x - 3
under the reflection in the origin
on the quadrant III.

Draw the image of y = 2x - 3
under the reflection in the x-axis
on the quadrant IV.

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