2.6 Families of Functions Sets of functions, called families, in what each function is a transformation of a special function called the parent. Linear.

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Presentation transcript:

2.6 Families of Functions Sets of functions, called families, in what each function is a transformation of a special function called the parent. Linear functions form a family of functions. – Each linear function is a transformation of the function y = x. – The function y = x is the parent linear function.

Families of Functions A parent function is the simplest form in a set of functions that form a family. – Each function in the family is a transformation of the parent function. One type of transformation is a translation. – A translation shifts the graph of the parent function horizontally, vertically, or both without changing shape or orientation.

Translations Parent function: f(x) Vertical translation: f(x) ± k Horizontal translation: f(x ± h)

Vertical and Horizontal Translations Example 1 and 2 textbook p. 100

Transformations A vertical stretch multiplies all y-values of a function by the same factor greater than 1. A vertical compression reduces all y-values of a function by the same factor between 0 and 1. For a function f(x) and a constant a, y = af(x) is a vertical stretch when a > 1 and a vertical compression when 0 < a < 1. See example 4 on p. 102.

Reflection When you reflect a graph in the y-axis, the x-values change signs and the y-values stay the same. When you reflect a graph in the x-axis, the x-values stay the same and the y-values change signs. For a function f(x), the reflection in the y-axis is f(-x) and the reflection in the x-axis is –f(x).

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)? g(x) = f(-x) g(x) = 3(-x) + 3 g(x) = -3x + 3

Combining Transformations A. The graph of g(x) is the graph of f(x) = 4x compressed vertically by the factor ½ and then reflected in the y-axis. What is a function rule for g(x)? af(x) ½ (4x) 2x 2(-x) = -2x The function rule is g(x) = -2x.

Combining Transformations B.What transformations change the graph of f(x) to the graph of g(x)? f(x) = 2x²g(x) = 6x² - 1 g(x) = 6x² - 1 = 3(2x²) – 1 = 3(f(x)) – 1 The graph of g(x) is the graph of f(x) stretched vertically by a factor of 3 and then translated down 1 unit.

More Practice!!!!! Homework – Textbook p. 104 #10 – 33.