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

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.

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

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)

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.

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

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

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