The ABSOLUTE VALUE Function
PART A – An Introduction
Which number(s) is 5 units from
0 on the real number line?
|
–5
|
0
5 units
|
5
5 units
The numbers 5 and –5 are both 5 units from 0 on the number line.
(Direction isn’t important – distance cannot be negative!!)
ABSOLUTE VALUE:
f(x) = |x| is the absolute value function
On a number line, this function describes the
distance of any number, x, from the origin (0).
Ex.
Ex 
|5| = 5 and |–5| = 5
Evaluate each of the following:
a)
|–7|
b)
|4|
c)
– |5| – |–3|
d)
|–6 + 3|
e)
8
2

 15
3
R
PART B – Graphing Absolute Value Inequalities
Recall…when graphing inequalities on a real number line:
 or 
use a closed dot 
 or 
use an open dot 
x1
Ex.
x < –2





0
1
2
3
R
“x is all values greater
than or equal to 1”
Ex 




–4
–3
–2

–1
R
“x is all values less than –2”
Graph each absolute value inequality on a real number line:
a)
|x| < 2
b)





–2
–1
0
1
2
|x|  1

R
–2




–1
0
1
2
R
(Special Cases occur when the inequality contains a negative number.)
c)
Ex 
|x|  –3
d)
|x| > –5
Rewrite using absolute value notation:
a)
c)
b)



–6
–3




0
3
6
x  –6 or x  6
R
d)



–10
–5
–8 < x < 8




0
5
10
R
PART C – The Absolute Value Function
The graph of the absolute value function is defined as follows:
y
y = |x|
x, if x  0
f(x) =
–x, if x  0
D={xR}
R={yR|y0}
o
x
Transformations of the absolute value function:
y = a|x – d| + c
 vertical stretch (|a| > 1),
reflection (a < 0) and
compression (0 < |a| < 1)
Ex 
 horizontal translation
(+) move left
(–) move right
 vertical translation
(+) move up
(–) move down
Sketch each of the following functions:
a)
f(x) = |x – 1|
f(x) = |x| + 2
b)
f(x)
0
f(x) = |2x – 6|
f(x)
x
0
x