2.6: Absolute Value and Families of Functions
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Ex1) Graph y = | x |
-5 -4 -3 -2 -1
-1
-2
-3
-4
-5
5
4
3
2
1
1 2 3 4 5

Family of Functions: Functions with certain common characteristics. (ex: the absolute value functions)
Parent Function: Simplest function with these characteristics
Translation: Shift horizontally, vertically, or both
Stretch: occurs when multiplying the function by a value greater than 1
Shrink: occurs when multiplying the function by a value between
0 and 1
Reflection: Change y values to their opposites when reflecting over the x -axis

y
 a x
 h
 k
#1: vertical stretch (|a| > 1) or shrink (0 < |a| < 1)
*negative: vertical reflection
-5 -4 -3 -2 -1
-1
-2
-3
-4
-5
5
4
3
2
1
1 2 3 4 5
#2: horizontal translation
(opposite)
Parent function y
#3: vertical
= | translation x |
Key points (-1, 1), (0, 0), (1, 1)

Describe, in order, the sequence of transformations of each function and then graph the function by hand.
Ex2) ( ) x 3 4 f x
 
3 x

2  
1
2 1
2 x
-5 -4 -3 -2 -1
-1
-2
-3
-4
-5
3
2
1
5
4
1 2 3 4 5 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
3
2
1
5
4
1 2 3 4 5 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
3
2
1
5
4
1 2 3 4 5

Write a function rule for each of the graphs below.
5 5
Ex5)
4
Ex6)
4
3 3
2 2
1 1
5
4
3
2
1
-5 -4 -3 -2 -1
-1
-2
-3
-4
-5
1 2 3 4 5 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
1 2 3 4 5 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
Ex7)
1 2 3 4 5

2.6: Absolute Value and Families of Functions
HW: 1-13 odd, 17-27 odd, 30-34 even, 35-37 all
Our feelings are our most genuine paths to knowledge.