2. Functions from Tables and Graphs

3. Determining Functions From Graphs To be a function, the graph must pass the vertical line test. When a vertical line passes through the graph, it should only touch one point at a time

4. Example

10. Describing Graphs of Linear Functions Positive Slope = IncreasingNegative Slope = Decreasing

11. Linear and Proportional Relationships All graphs of linear proportional relationships are functions because they form a straight line. Proportional: Straight line through (0, 0)Linear: Straight line Function

12. Nonlinear Relationships Many nonlinear relationships are functions, but a graph or table may be needed to be sure. Function Not a Function

13. Notes- Functions from Tables A function is when each input (x-value) corresponds to exactly one output (y- value) In other words, when you substitute (x) into an equation there is only one possible answer (y)

14. Identifying Functions From a Table Every x input can have only one corresponding output.

15. Example xy 12 26 34 42 58 xy 12 26 14 42 58 FunctionNot a function

16. Example xy 12 26 34 42 58 xy 12 26 14 42 58 FunctionNot a function Two different outputs for the same input

17. Try- is this a function? xy 110 28 36 28 52 xy 18 24 30 44 58 xy 12 26 34 42 30

18. Lets Graph One to See Why Each x Must Have a Unique y xy 00 24 34 42 31

19. Lets Graph One to See Why xy 00 24 34 42 31

20. For the following set of points, determine if the relationship is a function 1)(-2, 3); (4, 2); (-3, 2); (4, 0) 2)(1, 4); (-3, 5); (1, 4); (-2, 5); (3, 5) 3)(-5, 4); (4, -5); (-4, 5); (5, 4)

21. Determine if the following is a function y = 2x y = 3x + 4

22. Nope

23. Determine if the following is a function by completing the table and graphing y = x² - 2 xy 0 1 2 -2

24. Determine if the following is a function by completing the table and graphing y = x³ - 3 xy 0 1 2 -2

25. Writing the Rule for a Function

26. Writing the Rule You need to look at the inputs and outputs in the table to find a way to get from x to y that works for all points. May be addition, subtraction, multiplication, or a combination Write in the form y = mx + b

27. Find the rule y = x + 3

28. Find the Rule

30. y = 3x

31. Find the Rule What does the changing of signs tell us about the rule?

34. Find the Rule y = 2x + 2 How does is the value x = 0 helpful?

35. Find the Rule A good trick is to find the difference or change in x and y. That tells us what we are multiplying by

36. Find the Rule 1 3 When the x value increases by 1, the y value increases by 3. This tells us that x is being multiplied by 3 y = 3x ± ___

43. Find the Rule

47. Closure – Get up and find a new partner Write the rule for the following:

48. Writing the Rule Given Two Points

49. Rate of Change

50. Find the rate of change The linear function goes through the points (2, 4) and (4, 8)

51. Find the rate of change The linear function goes through the points (-3, 2) and (6, -1)

52. Find the rate of change The linear function goes through the points (-3, -5) and (-1, 3)

53. Writing a Linear Function From Two Points This is a skill we need to revisit. – Find the rate of change (slope) – Find the y intercept (initial position) by substituting one coordinate pair into y = mx + b

54. Write the equation of a linear function that goes through the points (-1, 1) and (1, 5)

55. Write the equation of a linear function that goes through the points (-4, 1) and (4, -3)

56. Are You Serious Right Now?

57. Carnival Amber and Mark went to the carnival on the same day. There is a flat fee to enter, and all games are the same price. Mark played 7 games and spent $12 (7, 12) and Amber played 11 games and spent $16 (11, 16). What is the rate of change (how much is each game)? How much was the entrance into the carnival?

58. Kayaking Max and Ryder both rented kayaks and equipment on the same day from the same company for a different length of time. The company charges a flat fee to rent equipment and an hourly rate for the kayaks. Max rented the kayak for 3 hours and paid $52. Ryder rented the kayak for 7 hours and paid $112. What is the rate of change (cost for one hour kayak rental)? How much was the equipment rental?

59. Cell Phone Bill Marcy recently signed up for a cell phone plan and has no idea how much she is paying per minute, but knows that her bill consists of a monthly fee and a cost per minute. She looked at her bills from the last two months and found that she used 500 minutes and paid $75 one month (500, 75) and she used 750 minutes and paid $100 the other month (750, 100). What is the rate of change (cost per minute)? What is the monthly fee?

60. Comparing Rate of Change

61. Rate of Change

62. Initial Value