The document discusses various ways to transform graphs of functions including vertical and horizontal shifts, reflections, stretching and shrinking. It provides examples of transforming graphs by applying these techniques sequentially in a specified order: first horizontal shifting, then stretching or shrinking, then reflecting, and finally vertical shifting. The summary provides examples of transforming an original function f(x) into another function g(x) by applying these transformations.

1. Section 2.5 Transformation of Functions

2. Graphs of Common Functions

6. Reciprocal Function

7. Vertical Shifts

9. Vertical Shifts

11. Example Use the graph of f(x)=|x| to obtain g(x)=|x|-2

12. Horizontal Shifts

14. Horizontal Shifts

15. Example Use the graph of f(x)=x 2 to obtain g(x)=(x+1) 2

16. Combining Horizontal and Vertical Shifts

17. Example Use the graph of f(x)=x 2 to obtain g(x)=(x+1) 2 +2

18. Reflections of Graphs

20. Reflections about the x-axis

22. Example Use the graph of f(x)=x 3 to obtain the graph of g(x)= (-x) 3 .

23. Example

24. Vertical Stretching and Shrinking

26. Vertically Shrinking

27. Vertically Stretching Graph of f(x)=x 3 Graph of g(x)=3x 3 This is vertical stretching – each y coordinate is multiplied by 3 to stretch the graph.

28. Example Use the graph of f(x)=|x| to graph g(x)= 2|x|

29. Horizontal Stretching and Shrinking

31. Horizontal Shrinking

32. Horizontal Stretching

33. Example

34. Sequences of Transformations

36. Summary of Transformations

37. A Sequence of Transformations Move the graph to the left 3 units Starting graph . Stretch the graph vertically by 2. Shift down 1 unit.

38. Example

39. Example

40. Example

41. (a) (b) (c) (d)

42. (a) (b) (c) (d) Write the equation of the given graph g(x). The original function was f(x) =x 2 g(x)

43. (a) (b) (c) (d) Write the equation of the given graph g(x). The original function was f(x) =|x| g(x)