Linear and Nonlinear Functions

1. SECTION 6-5
Linear and Nonlinear Functions

2. Essential Questions
How 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. Vocabulary
1. Function:
2. Function Notation:
3. Domain:
4. Range:
5. Continuous:
6. Linear Function:
7. Vertical-Line Test:

4. Vocabulary
1. 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. Vocabulary
1. 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
functions
3. Domain:
4. Range:
5. Continuous:
6. Linear Function:
7. Vertical-Line Test:

6. Vocabulary
1. 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
functions
3. Domain: Any possible value for the independent variable (usually x)
4. Range:
5. Continuous:
6. Linear Function:
7. Vertical-Line Test:

7. Vocabulary
1. 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
functions
3. 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. Vocabulary
1. 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
functions
3. 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 connected
6. Linear Function:
7. Vertical-Line Test:

9. Vocabulary
1. 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
functions
3. 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 connected
6. Linear Function: A function that will give a straight line; any line other
than a vertical line
7. Vertical-Line Test:

10. Vocabulary
1. 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
functions
3. 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 connected
6. Linear Function: A function that will give a straight line; any line other
than a vertical line
7. Vertical-Line Test: Tests whether a graph represents a function or not;
can only touch a graph once

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. 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. 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. 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. 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. 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. 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. 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. 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. 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.
b. Make a table and graph the data.

21. 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.
b. Make a table and graph the data.
c

22. 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.
b. Make a table and graph the data.
c s

23. 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.
b. Make a table and graph the data.
c s s = 2c + 5

24. 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.
b. Make a table and graph the data.
c s s = 2c + 5
1

25. 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.
b. Make a table and graph the data.
c s s = 2c + 5
1 7

26. 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.
b. Make a table and graph the data.
c s s = 2c + 5
1 7 (1, 7)

27. 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.
b. Make a table and graph the data.
c s s = 2c + 5
1 7 (1, 7)
2

28. 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.
b. Make a table and graph the data.
c s s = 2c + 5
1 7 (1, 7)
2 9

29. 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.
b. Make a table and graph the data.
c s s = 2c + 5
1 7 (1, 7)
2 9 (2, 9)

30. 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.
b. Make a table and graph the data.
c s s = 2c + 5
1 7 (1, 7)
2 9 (2, 9)
3

31. 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.
b. Make a table and graph the data.
c s s = 2c + 5
1 7 (1, 7)
2 9 (2, 9)
3 11

32. 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.
b. Make a table and graph the data.
c s s = 2c + 5
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)

33. 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.
b. Make a table and graph the data.
c s s = 2c + 5
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4

34. 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.
b. Make a table and graph the data.
c s s = 2c + 5
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4 13

35. 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.
b. Make a table and graph the data.
c s s = 2c + 5
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4 13 (4, 13)

36. 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.
b. Make a table and graph the data.
c s s = 2c + 5
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4 13 (4, 13)
5

37. 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.
b. Make a table and graph the data.
c s s = 2c + 5
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4 13 (4, 13)
5 15

38. 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.
b. Make a table and graph the data.
c s s = 2c + 5
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4 13 (4, 13)
5 15 (5, 15)

39. 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.
b. Make a table and graph the data.
c s s = 2c + 5
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4 13 (4, 13)
5 15 (5, 15)

40. 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.
b. Make a table and graph the data.
c s s = 2c + 5
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4 13 (4, 13)
5 15 (5, 15)

41. 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.
b. Make a table and graph the data.
c s s = 2c + 5
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4 13 (4, 13)
5 15 (5, 15)
c

42. 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.
b. Make a table and graph the data.
c s s = 2c + 5 s
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4 13 (4, 13)
5 15 (5, 15)
c

43. 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.
b. Make a table and graph the data.
c s s = 2c + 5 s
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4 13 (4, 13)
5 15 (5, 15)
c

44. 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.
b. Make a table and graph the data.
c s s = 2c + 5 s
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4 13 (4, 13)
5 15 (5, 15)
c

45. 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.
b. Make a table and graph the data.
c s s = 2c + 5 s
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4 13 (4, 13)
5 15 (5, 15)
c

46. 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.
b. Make a table and graph the data.
c s s = 2c + 5 s
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4 13 (4, 13)
5 15 (5, 15)
c

47. 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.
b. Make a table and graph the data.
c s s = 2c + 5 s
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4 13 (4, 13)
5 15 (5, 15)
c

48. 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.
b. Make a table and graph the data.
c s s = 2c + 5 s
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4 13 (4, 13)
5 15 (5, 15)
c

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. 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. 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. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1

53. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1

54. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x

55. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y

56. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2

57. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1

58. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)

59. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1

60. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0

61. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)

62. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0

63. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0 1

64. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0 1 (0, 1)

65. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0 1 (0, 1)
1

66. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0 1 (0, 1)
1 2

67. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0 1 (0, 1)
1 2 (1, 2)

68. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0 1 (0, 1)
1 2 (1, 2)
2

69. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0 1 (0, 1)
1 2 (1, 2)
2 3

70. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0 1 (0, 1)
1 2 (1, 2)
2 3 (2, 3)

71. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0 1 (0, 1)
1 2 (1, 2)
2 3 (2, 3)

72. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0 1 (0, 1)
1 2 (1, 2)
2 3 (2, 3)

73. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0 1 (0, 1)
1 2 (1, 2)
2 3 (2, 3)

74. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0 1 (0, 1)
1 2 (1, 2)
2 3 (2, 3)

75. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0 1 (0, 1)
1 2 (1, 2)
2 3 (2, 3)

76. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0 1 (0, 1)
1 2 (1, 2)
2 3 (2, 3)

77. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0 1 (0, 1)
1 2 (1, 2)
2 3 (2, 3)

78. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1

79. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
Domain:

80. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
Domain: {x : x is all real numbers}

81. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
Domain: {x : x is all real numbers}
Range:

82. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
Domain: {x : x is all real numbers}
Range: {y : y ≥ 0}

83. Problem Set

84. Problem Set
p. 267 #1-23 odd
"That is what learning is.You suddenly understand something you've
understood all your life, but in a new way." - Doris Lessing