1. SECTION 1.6 TRANSFORMATION OF FUNCTIONS

2. Graphs Of Common Functions AKA Mother Functions

6. Reciprocal Function

7. VERTICAL SHIFTS Let f be a function and c be a positive real number… The graph of y = f (x) - c is the graph of y = f (x) shifted c units vertically downward. The graph of y = f (x) + c is the graph of y = f (x) shifted c units vertically upward.

8. Vertical Shifts

9. EXAMPLE 1 Use the graph f (x) = |x| to graph f (x) = |x| + 2 Begin with the graph of f(x) = |x| Now to graph f(x) = |x| + 2, move every y- coordinate UP 2 spaces.

10. Example Use the graph of f(x)=|x| to obtain g(x)=|x| - 4 Down 4 spaces!!!

11. HORIZONTAL SHIFTS Let f be a function and c be a positive real number… The graph of y = f (x – c) is the graph of y = f (x) shifted to the right c units. The graph of y = f (x + c) is the graph of y = f (x) shifted to the left c units.

12. EXAMPLE Use the graph f (x) = x 2 to graph f (x) = (x + 1) 2 Begin with the graph of f(x) = x 2 Now to graph f(x) = (x + 1 )2, move every x- coordinate left 1 space.

13. Example Use the graph of f(x) = x 2 to obtain g(x) = (x-3) 2 Right 3 spaces!!!

14. Combining Horizontal and Vertical Shifts Begin with the graph of f(x) = x 2 Then graph h(x) = (x+1) 2 Then graph h(x) = (x+1) 2 -3

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

16. REFLECTION ABOUT THE X-AXIS

17. Reflections about the x-axis

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

20. Example

21. On the graph paper pick 6 problems from 53-66 to recreate Pick 4 problems from 67-80 to recreate Pick 4 problems from 81-94 to recreate Pick 5 problems from 95-106 to recreate Pick 5 problems from 107-118 to recreate. GRAB A CALCULATOR FROM THE BACK AND ONE OF EACH PAPER UP FRONT

22. Follow directions on the paper (you got it Friday) about how to log in. Now follow the directions on the Transformations Packet and complete all the work. When finished, let Miss Ettore know and she will give you the Exit Cards. ALL Exit Cards must be completed and submitted! GRAB A CALCULATOR FROM THE BACK AND THE PACKET UP FRONT

23. VERTICALLY STRETCHING AND SHRINKING GRAPHS If c > 1, the graph of y = c f (x) is the graph of y = f (x) vertically STRETCHED by multiplying each of its y-coordinates by c. If 0 < c < 1, the graph of y = c f (x) is the graph of y = f (x) vertically SHRUNK by multiplying each of its y-coordinates by c.

24. 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.

25. Vertically Shrinking Graph of f(x)=x 3 This is vertical shrunk  each y coordinate is multiplied by 1/2 to shrink the graph. Graph of f(x)=1/2 x 3

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

27. Example Use the graph of f(x) = |x| to graph g(x) = 1/4|x|

29. Summary of Transformations

30. ORDER OF TRANSFORMATIONS A function involving more than one transformation should be graphing by performing transformations in the following order: 1.Horizontal Shifting  f (x + c) or f (x – c) 2.Stretching or Shrinking  c f (x), stretch if c > 1, shrink if 0 < c < 1 3.Reflecting  x-axis if –f (x), y-axis if f (-x) 4.Vertical Shifting  f(x) + c or f(x) – c

31. A Sequence of Transformations: Graphing g(x) = 2 (x + 3) 2 - 1 2. Move the graph to the left 3 units. 1. Graph the original. 3. Stretch the graph vertically by 2. 4. Shift down 1 unit.

32. Example

35. GROUP CHALLENGE!! REVIEW TIME

36. (a) (b) (c) (d)

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

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

39. Horizontal Shrinking f(x) Graph f(2x)  Horizontal shrink so we divide each x-coordinate by 2 You can see how The graph shrunk horizontally

40. Horizontal Stretching f(x) Graph f(1/2x)  Horizontal stretch so we divide each x-coordinate (AKA multiply it by 2) You can see how The graph stretched horizontally

41. Example