1. Determining the Key Features of Function Graphs

2. The Key Features of Function Graphs - Preview  Domain and Range  x-intercepts and y-intercepts  Intervals of increasing, decreasing, and constant behavior  Parent Equations  Maxima and Minima

3. Domain  Domain is the set of all possible input or x-values  To find the domain of the graph we look at the x-axis of the graph

4. Determining Domain - Symbols  Open Circle →  Exclusive  ( )  Closed Circle →  Inclusive  [ ]

5. Identifying the Domain 1. Start at the far left of the graph. 2. Move along the x-axis until you find the lowest possible x-value of the graph. This is your lower bound. Note if you have a open circle or a closed circle. 3. Keep moving along the x-axis until you find your highest possible x-value of the graph. This is your upper bound. Note if you have a open circle or a closed circle.

6. Examples Domain:

7. Example Domain:

8. Determining Domain - Infinity Domain:

9. Examples Domain:

10. Your Turn:  Complete problems 3, 7, and find the domain of 9 and 10 on pg. 160 from the Xeroxed sheets 3.7. 9.10.

11. 3.7. 9.10.

12. 11.12.

13. Range  The set of all possible output or y- values  To find the range of the graph we look at the y-axis of the graph  We also use open and closed circles for the range

14. ***Identifying the Range 1. Start at the bottom of the graph. 2. Move along the y-axis until you find the lowest possible y-value of the graph. This is your lower bound. Note if you have a open circle or a closed circle. 3. Keep moving along the y-axis until you find your highest possible y-value of the graph. This is your upper bound. Note if you have a open circle or a closed circle.

15. Examples Range:

16. Examples Range:

17. Alternative Way to Identify the Range – This slide isn’t in your notes!

18. Your Turn:  Complete problems 4, 8, and find the range of 9 and 10 on pg. 160 from the Xeroxed sheets 4.8. 9.10.

19. 4.8. 9.10.

20. 11.12.

21. Challenge – Not in your notes! Identify the domain and range

22. X-Intercepts  Where the graph crosses the x-axis  Has many names: x-intercept Roots Zeros

23. Examples x-intercepts:

24. Y-Intercepts  Where the graph crosses the y-axis y-intercepts:

25. Seek and Solve!!!

26. Types of Function Behavior  3 types: Increasing Decreasing Constant  When determining the type of behavior, we always move from left to right on the graph

27. Roller Coasters!!! Fujiyama in Japan

28. Types of Behavior – Increasing  As x increases, y also increases  Direct Relationship

29. Types of Behavior – Constant  As x increases, y stays the same

30. Types of Behavior – Decreasing  As x increases, y decreases  Inverse Relationship

31. Identifying Intervals of Behavior  We use interval notation  The interval measures x-values. The type of behavior describes y-values. Increasing: [0, 4) The y-values are increasing when the x-values are between 0 inclusive and 4 exclusive

32. Identifying Intervals of Behavior  Increasing:  Constant:  Decreasing: x 1 1 y

33. Identifying Intervals of Behavior, cont.  Increasing:  Constant:  Decreasing: -3 y x Don’t get distracted by the arrows! Even though both of the arrows point “up”, the graph isn’t increasing at both ends of the graph!

34. Your Turn:  Complete problems 1 – 4 on The Key Features of Function Graphs – Part II handout.

35. 1. 2. 3. 4.

36. What do you think of when you hear the word parent?

37. Parent Function  The most basic form of a type of function  Determines the general shape of the graph

38. Basic Types of Parent Functions 1. Linear 2. Absolute Value 3. Greatest Integer 4. Quadratic 5. Cubic 6. Square Root 7. Cube Root 8. Reciprocal

39. Parent Function Flipbook

40. Function Name: Linear  Parent Function: f(x) = x  “Baby” Functions: y x2 2

41. Greatest Integer Function  f(x) = [[x]]  f(x) = int(x)  Rounding function Always round down

42. “Baby” Functions  Look and behave similarly to their parent functions  To get a “baby” functions, add, subtract, multiply, and/or divide parent equations by (generally) constants f(x) = x 2 f(x) = 5x 2 – 14 f(x) = f(x) = f(x) = x 3 f(x) = -2x 3 + 4x 2 – x + 2

43. “Baby” Functions, cont.  f(x) = |x| 

44. Your Turn:  Create your own “baby” functions in your parent functions book.

45. Identifying Parent Functions  From Equations: Identify the most important operation 1. Special Operation (absolute value, greatest integer) 2. Division by x 3. Highest Exponent (this includes square roots and cube roots)

46. Examples 1. f(x) = x 3 + 4x – 3 2. f(x) = -2| x | + 11 3.

47. Identifying Parent Equations  From Graphs: Try to match graphs to the closest parent function graph

48. Examples

49. Your Turn:  Complete problems 5 – 12 on The Key Features of Function Graphs handout

50. Maximum (Maxima) and Minimum (Minima) Points Peaks (or hills) are your maximum points Valleys are your minimum points

51. Identifying Minimum and Maximum Points  Write the answers as points  You can have any combination of min and max points  Minimum:  Maximum:

52. Examples

53. Your Turn:  Complete problems 1 – 6 on The Key Features of Function Graphs – Part III handout.

54. 1. 2. 3. 4. 5. 6.

55. Reminder: Find f(#) and Find f(x) = x  Find f(#) Find the value of f(x) when x equals #. Solve for f(x) or y!  Find f(x) = # Find the value of x when f(x) equals #. Solve for x!

56. Evaluating Graphs of Functions – Find f(#) 1. Draw a (vertical) line at x = # 2. The intersection points are points where the graph = f(#)  f(1) =  f(–2) =

57. Evaluating Graphs of Functions – Find f(x) = # 1. Draw a (horizontal) line at y = # 2. The intersection points are points where the graph is f(x) = #  f(x) = –2  f(x) = 2

58. Example 1. Find f(1) 2. Find f(–0.5) 3. Find f(x) = 0 4. Find f(x) = –5

59. Your Turn:  Complete Parts A – D for problems 7 – 14 on The Key Features of Function Graphs – Part III handout.

60. 7. 8. 9. 10.

61. 11. 12. 13. 14.