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
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}
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