2.6 Families of Functions Learning goals Analyze transformations of functions

parent function : the simplest form in a set of functions that form a family Each function in the family is a transformation of the parent function. Translation – one type of transformation, Shifts the graph of the parent function horizontally, vertically, or both without changing shape or orientation.

Vertical Translations

Ex 1 Translate up 5 units.

What is the graph of translated up 2 units? Ex 2 How are and related? What is the graph of translated up 2 units?

Horizontal Translations

Ex 3 Describe the translation.

Ex 4 Write the equation for the given graph

Ex 5 Write the equation for the given graph

reflections: Flips a graph across a line, such as the x- or y-axis. f(-x) reflects in the y-axis The x value changes sign -f(x) reflects in the x-axis The y value changes sign

Ex 6 Given the following table, find f(-x) and –f(x) and sketch. y 1 3 4 5 f(– x) x y – f(x) x y

Ex 7 If looks like Graph Graph

Vertical Stretch : multiplies all y-values by a factor of a > 1 Vertical Compression (shrink): reduces all y-values by the same factor between 0 and 1; 0 < a <1

Ex 8 Given the following table, find 3f(x) and ⅓f(x) and sketch. y -5 2 -2 -3 3 1 5 3f(x) x y -5 -2 3 5 ⅓f(x) x y -5 -2 3 5

Ex 9 Write the equation for the following statement The graph of g(x) is the graph of reflected in the y-axis, left 7 units, and shifted down 3 units.

Ex 10 What transformations change the graph of

Transformation f(x) Vertical Shift f(x)+c Horizontal Shift f(x-c) Reflection across x-axis -f(x) Reflection across y-axis f(-x) Vertical stretch cf(x) if c>1, stretch if 0<c<1, shrink