1. Lesson 2-6
Families of Functions

2. A family of functions is made up of functions with certain common characteristics.
Parent function – simplest function with these characteristics.
The equations of the functions in a family resemble each other

3. Our parent function is y = |x|
A translation shifts a graph horizontally, vertically, or both.
Consider the functions
y = |x|
y = |x| y = |x| – 2
y = |x + 4| y = |x – 4|

4. Horizontal Translation
Vertical Translation
Translation up k units y = |x| + k
Translation down k units y = |x| – k
Horizontal Translation
Translation right h units y = |x – h|
Translation left h units y = |x + h|
Combined Translation
Both a horizontal and vertical shift.
y = |x – 2| + 3

5. A vertical stretch multiplies all values by the same factor greater than 1.
A vertical shrink reduces y-values by a factor between 0 and 1.
Consider the functions
y = |x|
y = 2|x| y = 4|x|
y = ½|x| y = ¼|x|

6. A reflection in the x-axis changes y values to their opposites.
y = |x|
y = 2|x|
y = -2|x|

7. Vertical Stretch Vertical Shrink Reflection
Stretch by a factor of a y = a|x|
Note a must be greater than 1
Vertical Shrink
Shrink by a factor of a y = a|x|
Note a must be between 0 -1
Reflection
Reflect on the x-axis y = -a|x|

8. Summary of how to graph absolute values.
Find the vertex
Use the number out front as the slope
Get points on both the left and right side
Draw the graph
Note this method only works when x is alone inside the absolute value symbols