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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