1. Determining the Key Features of Function Graphs 10 February 2011

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

3. Domain  Reminder: Domain is the set of all possible input or x-values  When we 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. Determining Domain 1. Start at the origin 2. Move along the x-axis until you find the lowest possible x-value. This is your lower bound. 3. Return to the origin 4. Move along the x-axis until you find your highest possible x-value. This is your upper bound.

6. Examples Domain:

7. Example Domain:

8. Determining Domain - Infinity Domain:

9. Examples Domain:

10. Your Turn:  In the purple Precalculus textbooks, complete problems 3, 7, and find the domain of 9 and 10 on pg. 160 3.7. 9.10.

11. Range  The set of all possible output or y- values  When we 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

12. Determining Range  Start at the origin  Move along the y-axis until you find the lowest possible y-value. This is your lower bound.  Return to the origin  Move along the y-axis until you find your highest possible y-value. This is your upper bound.

13. Examples Range:

14. Examples Range:

15. Your Turn:  In the purple Precalculus textbooks, complete problems 4, 8, and find the domain of 9 and 10 on pg. 160 4.8. 9.10.

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

17. Examples x-intercepts:

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

19. Seek and Solve!!!

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

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

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

25. 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

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

27. 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!

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

29. 1. 2. 3. 4.

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

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

32. 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

33. Function Name: Linear  Parent Function: f(x) = x  “Baby” Functions: f(x) = 3x f(x) = x + 6 f(x) = –4x – 2 y x2 2

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

35. “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

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

37. 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)

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

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

40. Examples

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

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

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

44. Examples

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

46. 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!

47. 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) =

48. 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

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

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