2-6 Families of
Functions
Identify transformations by analyzing
functions.
Some Vocabulary
O A parent function is the simplest form of a family of
functions.
O Ex: we have the family of linear functions and the parent
function is y = x
O Each function in the family is a transformation of the
parent function.
O One type of transformation is a translation.
O Shifts the graph vertically, horizontally, or both without
changing the shape or orientation.
Translations
O How are the graphs of y = x and y = x – 2 related?
Vertical translation
O
O What is the equation of the graph of 𝑦 = 𝑥 2 − 1
translated up 5 units?
O 𝑦 = 𝑥2 + 4
Translations
O The graph shows the projected altitude 𝑓(𝑥) of an
airplane. If the plane leaves 2 hours late, what function
represents the transformation?
O Notice the graph shows a horizontal translation.
O 𝑓(𝑥 − 2)
Reflection
O Flips the graph across a line, such as the x- or y-axis.
O Each point on the reflected graph is the same distance
from the line as the original graph.
O For the function 𝑓(𝑥)
O 𝑓(−𝑥) represents a reflection across the y-axis
O −𝑓(𝑥) represents a reflection across the x-axis.
Reflecting a Function
Algebraically
O Let 𝑔(𝑥) be the reflection of 𝑓 𝑥 = 3𝑥 + 3 in the y-axis.
O
O
O
O
O
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
Stretch and Compress
O A vertical stretch multiplies all y-values of a function by
the same factor greater than 1.
O A vertical compression multiplies all y-values by the
same factor between 0 and 1.
O If 𝑔 𝑥 = 3𝑓(𝑥 + 2), name the transformations taking
place.
O Vertical stretch
O Horizontal translation to the left
Assignment
O Odds p.104 #11-15, 21-35