1. 2.6 Families of Functions Learning goals
Analyze transformations of functions

2. 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.

3. Vertical Translations

4. Ex 1 Translate up 5 units.

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

6. Horizontal Translations

7. Ex 3 Describe the translation.

8. Ex 4 Write the equation for the given graph

9. Ex 5 Write the equation for the given graph

10. 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

11. 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

12. Ex 7
If looks like
Graph
Graph

13. 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

14. 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

15. 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.

16. Ex 10 What transformations change the graph of

17. 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