1. 2-6: Families of Functions
Essential Question: What ways can the graph of the parent function y = |x| be transformed?

2. 2-6: Families of Functions
A parent function is a function with a certain shape that has the simplest rule for that shape.
Other equations will resemble the parent function in both formula as well as shape.
The parent function for an absolute value graph is y = |x|
A translation is a shift of the parent function either horizontally (left/right), vertically (up/down) or both.
It results in a graph with the same size and shape, but in a different location
The translations we will cover here today apply to both y = |x| as well as y = -|x|

3. 2-6: Families of Functions
Vertical translations
For a positive number k, y = |x| + k, is a vertical translation
If k is a positive number, shift the graph up k units
If k is a negative number, shift the graph down k units
Vertical translations work as expected
parent function: y = |x|
y = |x| - 3
vertical shift down 3 units
y = |x| + 2
vertical shift up 2 units

4. 2-6: Families of Functions
Your Turn:
Describe the translation:
y = |x| + 1
y = |x| - 2
Write an equation for the translation of:
y = |x| up 8 units
y = |x| down ½ unit
y = |x| shifted up 1 unit
y = |x| shifted down 2 units
y = |x| + 8
y = |x| - ½

5. 2-6: Families of Functions
Horizontal translations
For a positive number k, y = |x + h|, is a horizontal translation (horizontal change is on the inside)
If h is a positive number, shift the graph left h units
If h is a negative number, shift the graph right h units
Horizontal translations work opposite of expected
“HI HO”
parent function: y = |x|
y = |x + 3|
horizontal shift left 3 units
y = |x – 2|
horizontal shift right 2 units

6. 2-6: Families of Functions
Your Turn:
Describe the translation:
y = |x + 3|
y = |x – 1|
Write an equation for the translation of:
y = |x| right 2 units
y = |x| left 4 units
y = |x| shifted left 3 units
y = |x| shifted right 1 unit
y = |x – 2|
y = |x + 4|

7. 2-6: Families of Functions
Putting it all together
Describe the translation of f(x) = |x|
y = |x + 2| + 4
y = |x – 6| + 5
Horizontal shift 2 units left
Vertical shift 4 units up
Horizontal shift 6 units right
Vertical shift 5 units up

8. 2-6: Families of Functions
Assignment
Page 97 – 98
Problems 1 – 14 & 29 – 34
All problems
Instead of graphing (1 – 4, 8 – 11, 29 – 34), simply list the transformations that occur (e.g. “vertical shift down 2 units”) like what we did earlier