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# Int Math 2 Section 6-5 1011

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Linear and Nonlinear Functions

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• ### Int Math 2 Section 6-5 1011

1. 1. SECTION 6-5 Linear and Nonlinear Functions
2. 2. Essential QuestionsHow do you graph linear and nonlinear functions?How do you identify the domain and range of a function?Where you’ll see this: Machinery, travel, temperature
3. 3. Vocabulary1. Function:2. Function Notation:3. Domain:4. Range:5. Continuous:6. Linear Function:7. Vertical-Line Test:
4. 4. Vocabulary1. Function: A relationship where each x-value (independent variable) matches with only one y-value (dependent variable)2. Function Notation:3. Domain:4. Range:5. Continuous:6. Linear Function:7. Vertical-Line Test:
5. 5. Vocabulary1. Function: A relationship where each x-value (independent variable) matches with only one y-value (dependent variable)2. Function Notation: f(x), reads “function of x”; tells us the independent variable is inside the parentheses; allows for working with multiple functions3. Domain:4. Range:5. Continuous:6. Linear Function:7. Vertical-Line Test:
6. 6. Vocabulary1. Function: A relationship where each x-value (independent variable) matches with only one y-value (dependent variable)2. Function Notation: f(x), reads “function of x”; tells us the independent variable is inside the parentheses; allows for working with multiple functions3. Domain: Any possible value for the independent variable (usually x)4. Range:5. Continuous:6. Linear Function:7. Vertical-Line Test:
7. 7. Vocabulary1. Function: A relationship where each x-value (independent variable) matches with only one y-value (dependent variable)2. Function Notation: f(x), reads “function of x”; tells us the independent variable is inside the parentheses; allows for working with multiple functions3. Domain: Any possible value for the independent variable (usually x)4. Range: Any possible value for the dependent variable (usually y)5. Continuous:6. Linear Function:7. Vertical-Line Test:
8. 8. Vocabulary1. Function: A relationship where each x-value (independent variable) matches with only one y-value (dependent variable)2. Function Notation: f(x), reads “function of x”; tells us the independent variable is inside the parentheses; allows for working with multiple functions3. Domain: Any possible value for the independent variable (usually x)4. Range: Any possible value for the dependent variable (usually y)5. Continuous: A graph where all points are connected6. Linear Function:7. Vertical-Line Test:
9. 9. Vocabulary1. Function: A relationship where each x-value (independent variable) matches with only one y-value (dependent variable)2. Function Notation: f(x), reads “function of x”; tells us the independent variable is inside the parentheses; allows for working with multiple functions3. Domain: Any possible value for the independent variable (usually x)4. Range: Any possible value for the dependent variable (usually y)5. Continuous: A graph where all points are connected6. Linear Function: A function that will give a straight line; any line other than a vertical line7. Vertical-Line Test:
10. 10. Vocabulary1. Function: A relationship where each x-value (independent variable) matches with only one y-value (dependent variable)2. Function Notation: f(x), reads “function of x”; tells us the independent variable is inside the parentheses; allows for working with multiple functions3. Domain: Any possible value for the independent variable (usually x)4. Range: Any possible value for the dependent variable (usually y)5. Continuous: A graph where all points are connected6. Linear Function: A function that will give a straight line; any line other than a vertical line7. Vertical-Line Test: Tests whether a graph represents a function or not; can only touch a graph once
11. 11. Example 1 To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot.a. Write a function where c is the number of cups of coffee being made and s is the total number of spoonfuls of coffee used.
12. 12. Example 1 To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot.a. Write a function where c is the number of cups of coffee being made and s is the total number of spoonfuls of coffee used. Which is the independent variable?
13. 13. Example 1 To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. a. Write a function where c is the number of cups of coffee being made and s is the total number of spoonfuls of coffee used. Which is the independent variable?The number of cups determines how many spoonfuls, so c is independent
14. 14. Example 1 To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. a. Write a function where c is the number of cups of coffee being made and s is the total number of spoonfuls of coffee used. Which is the independent variable?The number of cups determines how many spoonfuls, so c is independent f(c) =
15. 15. Example 1 To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. a. Write a function where c is the number of cups of coffee being made and s is the total number of spoonfuls of coffee used. Which is the independent variable?The number of cups determines how many spoonfuls, so c is independent f(c) = 2c
16. 16. Example 1 To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. a. Write a function where c is the number of cups of coffee being made and s is the total number of spoonfuls of coffee used. Which is the independent variable?The number of cups determines how many spoonfuls, so c is independent f(c) = 2c + 5
17. 17. Example 1 To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. a. Write a function where c is the number of cups of coffee being made and s is the total number of spoonfuls of coffee used. Which is the independent variable?The number of cups determines how many spoonfuls, so c is independent f(c) = 2c + 5 s = 2c + 5
18. 18. Example 1 To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. a. Write a function where c is the number of cups of coffee being made and s is the total number of spoonfuls of coffee used. Which is the independent variable?The number of cups determines how many spoonfuls, so c is independent f(c) = 2c + 5 Function s = 2c + 5
19. 19. Example 1 To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. a. Write a function where c is the number of cups of coffee being made and s is the total number of spoonfuls of coffee used. Which is the independent variable?The number of cups determines how many spoonfuls, so c is independent f(c) = 2c + 5 Function s = 2c + 5 Equation
20. 20. Example 1To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. b. Make a table and graph the data.
21. 21. Example 1To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. b. Make a table and graph the data.c
22. 22. Example 1To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. b. Make a table and graph the data.c s
23. 23. Example 1To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. b. Make a table and graph the data.c s s = 2c + 5
24. 24. Example 1To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. b. Make a table and graph the data.c s s = 2c + 51
25. 25. Example 1To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. b. Make a table and graph the data.c s s = 2c + 51 7
26. 26. Example 1To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. b. Make a table and graph the data.c s s = 2c + 51 7 (1, 7)
27. 27. Example 1To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. b. Make a table and graph the data.c s s = 2c + 51 7 (1, 7)2
28. 28. Example 1To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. b. Make a table and graph the data.c s s = 2c + 51 7 (1, 7)2 9
29. 29. Example 1To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. b. Make a table and graph the data.c s s = 2c + 51 7 (1, 7)2 9 (2, 9)
30. 30. Example 1To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. b. Make a table and graph the data.c s s = 2c + 51 7 (1, 7)2 9 (2, 9)3
31. 31. Example 1To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. b. Make a table and graph the data.c s s = 2c + 51 7 (1, 7)2 9 (2, 9)3 11
32. 32. Example 1To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. b. Make a table and graph the data.c s s = 2c + 51 7 (1, 7)2 9 (2, 9)3 11 (3, 11)
33. 33. Example 1To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. b. Make a table and graph the data.c s s = 2c + 51 7 (1, 7)2 9 (2, 9)3 11 (3, 11)4
34. 34. Example 1To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. b. Make a table and graph the data.c s s = 2c + 51 7 (1, 7)2 9 (2, 9)3 11 (3, 11)4 13
35. 35. Example 1To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. b. Make a table and graph the data.c s s = 2c + 51 7 (1, 7)2 9 (2, 9)3 11 (3, 11)4 13 (4, 13)
36. 36. Example 1To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. b. Make a table and graph the data.c s s = 2c + 51 7 (1, 7)2 9 (2, 9)3 11 (3, 11)4 13 (4, 13)5
37. 37. Example 1To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. b. Make a table and graph the data.c s s = 2c + 51 7 (1, 7)2 9 (2, 9)3 11 (3, 11)4 13 (4, 13)5 15
38. 38. Example 1To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. b. Make a table and graph the data.c s s = 2c + 51 7 (1, 7)2 9 (2, 9)3 11 (3, 11)4 13 (4, 13)5 15 (5, 15)
39. 39. Example 1To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. b. Make a table and graph the data.c s s = 2c + 51 7 (1, 7)2 9 (2, 9)3 11 (3, 11)4 13 (4, 13)5 15 (5, 15)
40. 40. Example 1To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. b. Make a table and graph the data.c s s = 2c + 51 7 (1, 7)2 9 (2, 9)3 11 (3, 11)4 13 (4, 13)5 15 (5, 15)
41. 41. Example 1To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. b. Make a table and graph the data.c s s = 2c + 51 7 (1, 7)2 9 (2, 9)3 11 (3, 11)4 13 (4, 13)5 15 (5, 15) c
42. 42. Example 1To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. b. Make a table and graph the data.c s s = 2c + 5 s1 7 (1, 7)2 9 (2, 9)3 11 (3, 11)4 13 (4, 13)5 15 (5, 15) c
43. 43. Example 1To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. b. Make a table and graph the data.c s s = 2c + 5 s1 7 (1, 7)2 9 (2, 9)3 11 (3, 11)4 13 (4, 13)5 15 (5, 15) c
44. 44. Example 1To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. b. Make a table and graph the data.c s s = 2c + 5 s1 7 (1, 7)2 9 (2, 9)3 11 (3, 11)4 13 (4, 13)5 15 (5, 15) c
45. 45. Example 1To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. b. Make a table and graph the data.c s s = 2c + 5 s1 7 (1, 7)2 9 (2, 9)3 11 (3, 11)4 13 (4, 13)5 15 (5, 15) c
46. 46. Example 1To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. b. Make a table and graph the data.c s s = 2c + 5 s1 7 (1, 7)2 9 (2, 9)3 11 (3, 11)4 13 (4, 13)5 15 (5, 15) c
47. 47. Example 1To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. b. Make a table and graph the data.c s s = 2c + 5 s1 7 (1, 7)2 9 (2, 9)3 11 (3, 11)4 13 (4, 13)5 15 (5, 15) c
48. 48. Example 1To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot. b. Make a table and graph the data.c s s = 2c + 5 s1 7 (1, 7)2 9 (2, 9)3 11 (3, 11)4 13 (4, 13)5 15 (5, 15) c
49. 49. Example 1 To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot.c. An ofﬁce cafeteria has a coffee urn with the ability to make 16 to 35 cups. Determine the domain and range of the function as applied to this urn.
50. 50. Example 1 To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot.c. An ofﬁce cafeteria has a coffee urn with the ability to make 16 to 35 cups. Determine the domain and range of the function as applied to this urn. Domain: {c : c = 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35}
51. 51. Example 1 To make coffee in a large coffee urn, one recipe calls for two spoonfuls for each cup plus 5 spoonfuls for the pot.c. An ofﬁce cafeteria has a coffee urn with the ability to make 16 to 35 cups. Determine the domain and range of the function as applied to this urn. Domain: {c : c = 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35} Range: {s : s = 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75}
52. 52. Example 2Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1
53. 53. Example 2Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1
54. 54. Example 2Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1x
55. 55. Example 2Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1x y
56. 56. Example 2Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1x y-2
57. 57. Example 2Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1x y-2 1
58. 58. Example 2Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1x y-2 1 (-2, 1)
59. 59. Example 2Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1x y-2 1 (-2, 1)-1
60. 60. Example 2Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1x y-2 1 (-2, 1)-1 0
61. 61. Example 2Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1x y-2 1 (-2, 1)-1 0 (-1, 0)
62. 62. Example 2Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1x y-2 1 (-2, 1)-1 0 (-1, 0)0
63. 63. Example 2Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1x y-2 1 (-2, 1)-1 0 (-1, 0)0 1
64. 64. Example 2Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1x y-2 1 (-2, 1)-1 0 (-1, 0)0 1 (0, 1)
65. 65. Example 2Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1x y-2 1 (-2, 1)-1 0 (-1, 0)0 1 (0, 1)1
66. 66. Example 2Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1x y-2 1 (-2, 1)-1 0 (-1, 0)0 1 (0, 1)1 2
67. 67. Example 2Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1x y-2 1 (-2, 1)-1 0 (-1, 0)0 1 (0, 1)1 2 (1, 2)
68. 68. Example 2Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1x y-2 1 (-2, 1)-1 0 (-1, 0)0 1 (0, 1)1 2 (1, 2)2
69. 69. Example 2Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1x y-2 1 (-2, 1)-1 0 (-1, 0)0 1 (0, 1)1 2 (1, 2)2 3
70. 70. Example 2Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1x y-2 1 (-2, 1)-1 0 (-1, 0)0 1 (0, 1)1 2 (1, 2)2 3 (2, 3)
71. 71. Example 2Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1x y-2 1 (-2, 1)-1 0 (-1, 0)0 1 (0, 1)1 2 (1, 2)2 3 (2, 3)
72. 72. Example 2Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1x y-2 1 (-2, 1)-1 0 (-1, 0)0 1 (0, 1)1 2 (1, 2)2 3 (2, 3)
73. 73. Example 2Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1x y-2 1 (-2, 1)-1 0 (-1, 0)0 1 (0, 1)1 2 (1, 2)2 3 (2, 3)
74. 74. Example 2Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1x y-2 1 (-2, 1)-1 0 (-1, 0)0 1 (0, 1)1 2 (1, 2)2 3 (2, 3)
75. 75. Example 2Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1x y-2 1 (-2, 1)-1 0 (-1, 0)0 1 (0, 1)1 2 (1, 2)2 3 (2, 3)
76. 76. Example 2Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1x y-2 1 (-2, 1)-1 0 (-1, 0)0 1 (0, 1)1 2 (1, 2)2 3 (2, 3)
77. 77. Example 2Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1x y-2 1 (-2, 1)-1 0 (-1, 0)0 1 (0, 1)1 2 (1, 2)2 3 (2, 3)
78. 78. Example 2Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1
79. 79. Example 2 Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1Domain:
80. 80. Example 2 Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1Domain: {x : x is all real numbers}
81. 81. Example 2 Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1Domain: {x : x is all real numbers} Range:
82. 82. Example 2 Graph the following relation, stating whether it is a function and listing the domain and range. y = x +1Domain: {x : x is all real numbers} Range: {y : y ≥ 0}
83. 83. Problem Set
84. 84. Problem Set p. 267 #1-23 odd"That is what learning is.You suddenly understand something youve understood all your life, but in a new way." - Doris Lessing