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 = |x2 - 4|.
Solution y = |x2 - 4|
This is y = |f(x)|.
Then first draw y = x2 - 4.
This is y = |f(x)|.
Then first draw y = x2 - 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 line.
(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