1. 2-6 Families of Functions
Identify transformations by analyzing functions.

2. Some Vocabulary
A parent function is the simplest form of a family of functions.
Ex: we have the family of linear functions and the parent function is y = x
Each function in the family is a transformation of the parent function.
One type of transformation is a translation.
Shifts the graph vertically, horizontally, or both without changing the shape or orientation.

3. Translations How are the graphs of y = x and y = x – 2 related?
Vertical translation
What is the equation of the graph of 𝑦= 𝑥 2 −1 translated up 5 units?
𝑦= 𝑥 2 +4

4. Translations
The graph shows the projected altitude 𝑓(𝑥) of an airplane. If the plane leaves 2 hours late, what function represents the transformation?
Notice the graph shows a horizontal translation.
𝑓(𝑥−2)

5. Reflection Flips the graph across a line, such as the x- or y-axis.
Each point on the reflected graph is the same distance from the line as the original graph.
For the function 𝑓(𝑥)
𝑓(−𝑥) represents a reflection across the y-axis
−𝑓(𝑥) represents a reflection across the x-axis.

6. Reflecting a Function Algebraically
Let 𝑔(𝑥) be the reflection of 𝑓 𝑥 =3𝑥+3 in the y-axis. What is a function rule for 𝑔(𝑥)?
Reflection of 𝑓(𝑥) across the y-axis is 𝑓(−𝑥)
𝑔 𝑥 =𝑓 −𝑥
𝑓 −𝑥 =3 −𝑥 +3
𝑓 −𝑥 =−3𝑥+3
So 𝑔 𝑥 =−3𝑥+3

7. Stretch and Compress
A vertical stretch multiplies all y-values of a function by the same factor greater than 1.
A vertical compression multiplies all y-values by the same factor between 0 and 1.
If 𝑔 𝑥 =3𝑓(𝑥+2), name the transformations taking place.
Vertical stretch
Horizontal translation to the left

8. Assignment
Odds p.104 #11-15, 21-35