1. FUNCTIONS When you see make sure you write down the notes, examples, graphs and all other information in your notes (pencil paper…no technology notes).

2. How do you know if a relationship is a linear function? For example: Joe paid $6 for admission and $4 per ride at the fair!!

3. In this lesson you will learn how to recognize a linear function by examining the four representations of a function.

4. Let’s Review A function relation can be represented in 4 different ways. NumericGraphic Algebraic y = 3x + 6 xy -20 3 06 19 212 Verbal This function multiplies every input by 3 and adds 6.

5. Let’s Review A Common Mistake Four different functions !?!? y = -2x – 1 xy -23 1 0 1-3 Multiply x by -2 and subtract 1 No, the SAME function! Four different ways to represent it.

6. Let’s Review Core Lesson $6 for admission and $4 per ride!! Number of rides (x) Total cost (y) 0 1 2 4 3 6 10 18 14 22 526 Constant rate of change = 4 Initial value=6

7. Let’s Review Core Lesson Graph is a straight line. Not Linear!Linear! y = 4x + 6 y = x 3 + 6

8. Let’s Review Core Lesson Rate of change is constant between any two rows. xy 06 110 214 318 422 xy 06 17 214 333 470 Linear! Not Linear!

9. Let’s Review Core Lesson Independent variable (input) exponent is 1. Linear! Not Linear! y = 4x + 6 y = x 3 + 6

10. Let’s Review Core Lesson Words like “per,” “each,” and “every” $6 for admission and $4 per ride!! y = 4x + 6 The volume of a perfect cube plus 6 extra cubic units. y = x 3 + 6

11. In this lesson you have learned how to recognize a linear function by examining the four representations of a function.

12. Let’s Review Guided Practice Which representation is not linear? You must pay $1 for a copy card and copies are $0.03 per sheet. y = 0.03x + 1 xy 01 11.3 22.6 35.2 410.4

13. Let’s Review Quick Quiz A linear relation has a rate of change.  Changing  Constant  Zero  Shifting The graph at left is:  linear, initial value = 1  not linear, initial value = 1  linear, rate of change = 1  not linear, rate of change = 1

14. In this lesson you will learn how to describe the rate of change in a linear function by using the four representations.

15. Let’s Review A linear function has a constant rate of change. y = 3x + 4 y = mx + b xy 04 17 210 313 416

16. Let’s Review Core Lesson When can a function relation be classified as linear? xy -220 15 010 15 20 y = -5x + 10 When it has a constant rate of change.

17. Let’s Review A Common Mistake xy 04 17 416 622 725 3 3 9 6 Not LinearLinear !_ 3 = 3 _ 1 = 3 _ 2 = 3 _ 1 = 3

18. Let’s Review Core Lesson Is y = 5 + 2x linear? Yes! xy -21 3 05 17 29 y = 5 + 2x 2 2 2 2 _ 1 = 2

19. Let’s Review Core Lesson Is y = 5 + 2x 2 linear? No! xy -213 7 05 17 213 y = 5 + 2x 2 -6 6 -2 2 _ 1 = -2 _ 1 = -6 _ 1 = 2 _ 1 = 6

20. In this lesson you learned how to describe the rate of change in a linear function by using the four representations.

21. Let’s Review Guided Practice What is the constant rate of change for each of the function representations? xy 2 09 116 223 330 y = -10 – 3.5x

22. Let’s Review Quick Quiz What is the constant rate of change for the relation at right? What is the constant rate of change for the relation at left? xy -222 016 210 44 6-2

23. How do you figure out if a parking garage represents a function relation?

24. In this lesson you will learn how to identify function properties by examining the input and output of real world examples.

25. Let’s Review A function can also be called: - a relation rule relationship correspondence

26. Let’s Review A function relation takes any input and yields exactly one output. 4 3 2 1 Inputs 12 9 6 3 Outputs Function

27. Let’s Review A Common Mistake Multiple inputs have the same output. Not a function! 12341234 4

28. Let’s Review Core Lesson Let’s look at 3 real world examples. Input (x)Output (y) 123123 246246 #1

29. Let’s Review Core Lesson #2 Input (x)Output (y)..across town 1212 234234

30. Let’s Review Core Lesson #3 123456123456 3 6 9 12 12 12 123456123456 3 6 9 12 Input (x)Output (y)

31. Let’s Review Core Lesson Input (x)Output (y) 123123 246246 Input (x)Output (y) 1212 234234 123456123456 3 6 9 12 Input (x)Output (y) Function One-to-one Constant rate of change = 2 Function One-to-one Constant rate of change = 2 Not a function No other properties apply Not a function No other properties apply Function Not one-to- one No constant rate of change Function Not one-to- one No constant rate of change

32. Let’s Review Core Lesson InputOutput 123123 246246 123456123456 3 6 9 12 InputOutput Not 1-1 1-1

33. In this lesson you have learned how to identify function properties by examining the input and output of real world examples

34. Let’s Review Guided Practice Pick the relation that is one-to-one. 24682468 12341234 12341234 24682468 24682468 12341234 24682468 12341234 A)B)C)D)

35. Let’s Review Extension Activities How could you make the parking garage from the lesson become a one-to-one function?

36. Let’s Review Quick Quiz 1) Pick all of the one-to-one function relations. 12341234 24682468 12341234 24682468 12341234 24682468 12341234 24682468

37. Let’s Review Quick Quiz  A one-to-one function relation is shown at right. If the input is 6, what is the output? XY 4 3 2 10.5 1 1.5 2 A) 2.5 B) 3 C) 3.5 D) 4

38. How do you know which equation is linear? y = 3x + 4 ?

39. In this lesson you will learn how to identify a linear function relation by analyzing characteristics of a linear function.

40. Let’s Review Linear = “Straight” (graph is a straight line)

41. Let’s Review A Common Mistake Only 1 slope and y-intercept for a linear function. y = 5x + 3x – 4 Is the slope 5?Is the slope 3?Neither, it is 8.Combine all like terms. y = 8x – 4

42. Let’s Review Core Lesson Quadratic Functions Linear Functions Cubic Functions Other types Polynomials (many terms)

43. Let’s Review Core Lesson The graph is a straight line 1 3

44. Let’s Review Core Lesson The equation can give the slope and y-intercept y = 3x + 4 y = mx + b

45. Let’s Review Core Lesson 7 – 4 1 – 0 The table shows a constant rate of change xy 04 17 416 622 725 Constant rate of change =33 1 – 07 – 4 4 – 116 – 7 4 – 1 6 – 422 – 16 6 – 4 3

46. In this lesson you have learned how to identify a linear function relation by analyzing characteristics of a linear function.

47. Let’s Review Guided Practice Which equation is linear? y = x y = 7 – x 2 y = 2x + 5 y = 3 x – 1

48. Let’s Review Extension Activities A local school is selling raffle tickets for $2 each and 3 tickets for $5. Is this a linear relation? Make a table, graph, and/or equation to help visualize and discuss your answer.

49. Let’s Review Quick Quiz Which choice gives the constant rate of change for y = -7x + 3 ? A) 3 B) 7 C) -3 D) -7

50. Let’s Review Quick Quiz Which graph shows a linear relation?  Black  Green  Pink  Red

51. How do you determine which graph is not a function?

52. In this lesson you will learn identify a function by analyzing its graph.

53. Let’s Review Is a function 1 2 Is one-to-one 3 Has a constant rate of change Given a relation you can determine if it:

54. Let’s Review A Common Mistake InputOutput 123123 246246 No, input goes first! Correct!

55. Let’s Review Core Lesson InputOutput 1 2 31 2 3 2 4 62 4 6 Function One-to- one Constant rate of change = 2 Function One-to- one Constant rate of change = 2

56. Let’s Review Core Lesson InputOutput 1 1 31 1 3 2 4 62 4 6 Not a function (input 1 has outputs 2 and 4) Not a function (input 1 has outputs 2 and 4)

57. Let’s Review Core Lesson InputOutput 123456123456 3 6 9 12 Function Not one-to- one (inputs 4, 5, 6 share same output) Function Not one-to- one (inputs 4, 5, 6 share same output)

58. Let’s Review Core Lesson Function Not a functionFunction One-to-one Not one-to-one

59. In this lesson you have learned how to identify a function by analyzing a graph

60. Let’s Review Guided Practice Which graph is not a function?

61. Let’s Review Extension Activities Plot each relation on separate coordinate grids. Decide which is the function and non- function, and discuss the reasons with a friend. InputOutput 123456123456 333333333333 InputOutput 111111111111 012345012345

62. Let’s Review Quick Quiz Which graphs are functions?

63. Let’s Review Quick Quiz InputOutput 12341234 24242424 Which mapping matches the graph? InputOutput 12341234 86428642 InputOutput 12341234 24682468 InputOutput 12341234 24442444