Unit 1

1. Unit 1 Characteristics and Applications of Functions

2. Parent Function Checklist Unit 1: Characteristics and Applications of Functions

3. Parent Function Checklist

4. Parent Function Checklist

5. Parent Function Checklist

6. Parent Function Checklist

7. Parent Function Checklist

8. Function Vocabulary Unit 1: Characteristics and Applications of Functions

9. Increasing • Picture/Example • Common Language: Goes up from left to right. • Technical Language: f(x) is increasing on an interval when, for any a and b in the interval, if a > b, then f(a) > f(b).

10. Decreasing • Picture/Example • Common Language: Goes down from left to right. • Technical Language: f(x) is decreasing on an interval when, for any a and b in the interval, if a > b, then f(a) < f(b).

11. Maximum • Picture/Example • Common Language: Relative “high point” • Technical Language: A function f(x) reaches a maximum value at x = a if f(x) is increasing when x < a and decreasing when x > a. The maximum value of the function is f(a).

12. Minimum • Picture/Example • Common Language: Relative “low point” • Technical Language: A function f(x) reaches a minimum value at x = a if f(x) is decreasing when x < a and increasing when x > a. The minimum value of the function is f(a).

13. Asymptote • Picture/Example • Common Language: A boundary line • Technical Language: A line that a function approaches for extreme values of either x or y.

14. Odd Function • Picture/Example • Common Language: A function that is symmetric with respect to the origin. • Technical Language: A function is odd iff f(-x) = -f(x).

15. Even Function • Picture/Example • Common Language: A function that has symmetry with respect to the y-axis • Technical Language: A function is even iff f(-x)=f(x)

16. End Behavior • Picture/Example • Common Language: Whether the graph (f(x)) goes up, goes down, or flattens out on the extreme left and right. • Technical Language: As x-values approach ∞ or -∞, the function values can approach a number (f(x)n) or can increase or decrease without bound (f(x)±∞).

17. Heart Medicine Unit 1: Characteristics and Applications of Functions

18. In the function editor of your calculator enter:

19. Table

20. Graph

21. 1) Use a graphing calculator to find the maximum rate at which the patient’s heart was beating. After how many minutes did this occur? • 79.267 beats per minute • 1.87 minutes after the medicine was given

22. 2) Describe how the patient’s heart rate behaved after reaching this maximum. • The heart rate starts decreasing, but levels off. • The heart rate never drops below a certain level (asymptote).

23. 3) According to this model, what would be the patient’s heart rate 3 hours after the medicine was given? After 4 hours? • 3 hours = 180 minutes  h(180) ≈ 60.4 bpm • 4 hours = 240 minutes  h(240) ≈ 60.3 bpm

24. 4) This function has a horizontal asymptote. Where does it occur? How can it’s presence be confirmed using a graphing calculator? • Asymptote: h(x)=60 • Scroll down the table and look at large values of x or trace the graph and look at large values of x. • The end behavior of the function is: As x  ∞, f(x)  60 and as x  -∞, f(x)  60

25. End Behavior Unit 1: Characteristics and Applications of Functions

26. End Behavior

27. End Behavior

28. End Behavior

29. End Behavior

30. End Behavior

31. End Behavior

32. Piecewise-Defined Functions Unit 1: Characteristics and Applications of Functions

34. Evaluate the function at the given values by first determining which formula to use.

35. Define a piecewise function based on the description provided.

36. Graph the given piecewise functions on the grids provided.

37. Continuity Unit 1: Characteristics and Applications of Functions

38. Continuity-Uninterrupted in time or space.

39. 1) Complete the table and answer the questions that follow.

40. 2) Graph each function using a “decimal” window (zoom 4) to observe the different ways in which functions can lack continuity.

41. 3) Graph each function to determine where each discontinuity occurs. Classify each type.