Features of Graphs of Functions

1. Features of Graphs of Functions

2. Thinking with your fingers

3. Terminology • Increasing – As your x finger moves right, your y finger moves up. • Decreasing – As your x finger moves right, your y finger moves down • Constant – As your x finger moves right, your y finger does not move Because x is the independent variable, increasing/decreasing/constant intervals are described in terms of values of x

4. Thinking with your fingers

5. Interval Notation Increasing on (2,∞) Increasing on (-∞,-2) Decreasing on (-2,2)

6. Extrema and Local Extrema • The maximum of a graph is its highest y value. • The minimum of a graph is its lowest y value. • A local maximum of a graph is a y value where the graph changes from increasing to decreasing. (Stops getting bigger) • A local minimum of a graph is a y value where the graph changes from decreasing to increasing (Stops getting smaller) • These extrema are all y values. But because x is the independent variable, they are often described by their location (x value).

7. Extrema No maximum, or maximum of ∞ Local maximum of ƒ(x)=5.3333 at x=-2 Local minimum of ƒ(x)=-5.3333 at x=2 No minimum, or minimum of -∞

8. Using your graphing utility, graph the function below and determine on which x-interval the graph of h(x) is increasing:

9. Using your graphing utility, graph the function below and determine on which x-interval the graph of h(x) is increasing:

10. Piecewise Functions

11. The Bank Problem • Frances puts $50 in a bank account on Monday morning every week. • Draw a graph of what Frances's bank account looks like over time. Put number of weeks on the horizontal axis, and number of dollars in her account on the vertical axis.

12. Frances’ Bank Account

13. Writing a formula for Frances

14. Piecewise function • Function definition is given over interval “pieces” • Ex: • Means: “When x is between 0 and 2, use the formula “2x+1.” When x is between 2 and 5, use the formula “(x-3)2”

15. Example • Find ƒ(3) • Check the first condition: 0≤3<2? FALSE. • Check the second condition: 2≤3<5? TRUE. • Use (x-3)2 • (3-3)2=02=0 • ƒ(3)=0

16. Example For the interval of x [0,2) draw a graph of 2x+1 For the interval of x [2,5) draw a graph of (x-3)2

17. Example For the interval of x [0,2) draw a graph of 2x+1 For the interval of x [2,5) draw a graph of (x-3)2

18. Consider the piecewise function below: Find h(3). • 7 • -2 • -3 • Both (a) and (b) • None of the above

19. Consider the piecewise function below: Find h(3). • Check first condition: 3>5? FALSE • Check second condition: 3≤5? TRUE • Use x-5 • 3-5=-2 • B) -2

20. Combining Functions (Algebra of Functions)

21. Things to remember • Function notation • ƒ(x)=2x-1 is a function definition • x is a number • ƒ(x) is a number • 2x-1 is a number • ƒ is the action taken to get from x to ƒ(x) • Multiply by 2 and add -1

22. Things to remember • Function notation • ƒ(x)=2x-1 is a function definition • 3 is a number • ƒ(3) is a number • 2*3-1 is a number (it’s 5) • ƒ is the action taken to get from 3 to ƒ(3) • Multiply by 2 and add -1

23. In visual form

24. Numbers can be added and multiplied

25. Numbers can be added and multiplied

26. When I add and multiply the results of functions, I create a new function f(x) f x f(x)+g(x) + g g(x)

27. When I add and multiply the results of functions, I create a new function f(x) f x f(x)+g(x) + g g(x)

28. I can give this new function a name f+g x f(x)+g(x)

29. I can give this new function a name f+g x f(x)+g(x) • The function f+g is the action: • Do f to x. get f(x) • Do g to x. get g(x) • Add f(x)+g(x)

30. Example 2x f x 2x+3x + g 3x

31. Example 2x f x 5x + g 3x

32. Example f+g x 5x

33. Notation • (f+g)(x)=f(x)+g(x) • (f-g)(x)=f(x)-g(x) • (fg)(x)=f(x)*g(x) • (f/g)(x)=f(x)/g(x), g(x)≠0

34. WARNING (fg)(x) and f(g(x)) are not the same thing • (fg)(x) means “do f to x, then do g to x, then multiply the numbers f(x) and g(x).” • f(g(x)) means “do g to x, get the number g(x), then do f to the number g(x)” • No multiplying. • We will talk more about f(g(x)) next time.

35. Example • Find (g/f)(-2)

36. Graphing sums • Function ƒ has a graph. • Function g has a graph • Function (ƒ+g) also has a graph. • Can I find the graph of (ƒ+g) from the graphs of ƒ and g? • Hint: the answer is yes.

37. How to sum function graphs

38. How to sum function graphs

39. How to sum function graphs

40. How to sum function graphs

41. How to sum function graphs

42. How to sum function graphs

43. How to sum function graphs

44. What if one of them is negative?

45. What if one of them is negative?

46. What if one of them is negative?

47. What if one of them is negative?

48. What if one of them is negative?

49. Use as many points as you need

50. Given the function definitions below: Find (ƒ+g)(3). • -11 • 3 • 0 • -21 • None of the above