Graph Quadratic Functions

1. SECTION 6-6
Graph Quadratic Equations

2. ESSENTIAL QUESTIONS
• How do you identify points given the graph of a quadratic
function?
• How do you graph simple quadratic functions?
• Where you’ll see this:
• Sports, physics, agriculture

3. QUADRATIC FUNCTION

4. QUADRATIC FUNCTION
A nonlinear function with a highest exponent of 2 on the
independent variable

5. QUADRATIC FUNCTION
A nonlinear function with a highest exponent of 2 on the
independent variable
The graph will be a parabola

6. QUADRATIC FUNCTION
A nonlinear function with a highest exponent of 2 on the
independent variable
The graph will be a parabola

7. EXAMPLE 1
Make a table and graph.
2
y = x +2
x y

8. EXAMPLE 1
Make a table and graph.
2
y = x +2
x y
-2

9. EXAMPLE 1
Make a table and graph.
2
y = x +2
x y
-2 6

10. EXAMPLE 1
Make a table and graph.
2
y = x +2
x y
-2 6 (-2, 6)

11. EXAMPLE 1
Make a table and graph.
2
y = x +2
x y
-2 6 (-2, 6)
-1

12. EXAMPLE 1
Make a table and graph.
2
y = x +2
x y
-2 6 (-2, 6)
-1 3

13. EXAMPLE 1
Make a table and graph.
2
y = x +2
x y
-2 6 (-2, 6)
-1 3 (-1, 3)

14. EXAMPLE 1
Make a table and graph.
2
y = x +2
x y
-2 6 (-2, 6)
-1 3 (-1, 3)
0

15. EXAMPLE 1
Make a table and graph.
2
y = x +2
x y
-2 6 (-2, 6)
-1 3 (-1, 3)
0 2

16. EXAMPLE 1
Make a table and graph.
2
y = x +2
x y
-2 6 (-2, 6)
-1 3 (-1, 3)
0 2 (0, 2)

17. EXAMPLE 1
Make a table and graph.
2
y = x +2
x y
-2 6 (-2, 6)
-1 3 (-1, 3)
0 2 (0, 2)
1

18. EXAMPLE 1
Make a table and graph.
2
y = x +2
x y
-2 6 (-2, 6)
-1 3 (-1, 3)
0 2 (0, 2)
1 3

19. EXAMPLE 1
Make a table and graph.
2
y = x +2
x y
-2 6 (-2, 6)
-1 3 (-1, 3)
0 2 (0, 2)
1 3 (1, 3)

20. EXAMPLE 1
Make a table and graph.
2
y = x +2
x y
-2 6 (-2, 6)
-1 3 (-1, 3)
0 2 (0, 2)
1 3 (1, 3)
2

21. EXAMPLE 1
Make a table and graph.
2
y = x +2
x y
-2 6 (-2, 6)
-1 3 (-1, 3)
0 2 (0, 2)
1 3 (1, 3)
2 6

22. EXAMPLE 1
Make a table and graph.
2
y = x +2
x y
-2 6 (-2, 6)
-1 3 (-1, 3)
0 2 (0, 2)
1 3 (1, 3)
2 6 (2, 6)

23. EXAMPLE 1
Make a table and graph.
2
y = x +2
x y y
-2 6 (-2, 6)
-1 3 (-1, 3)
0 2 (0, 2)
1 3 (1, 3) x
2 6 (2, 6)

24. EXAMPLE 1
Make a table and graph.
2
y = x +2
x y y
-2 6 (-2, 6)
-1 3 (-1, 3)
0 2 (0, 2)
1 3 (1, 3) x
2 6 (2, 6)

25. EXAMPLE 1
Make a table and graph.
2
y = x +2
x y y
-2 6 (-2, 6)
-1 3 (-1, 3)
0 2 (0, 2)
1 3 (1, 3) x
2 6 (2, 6)

26. EXAMPLE 1
Make a table and graph.
2
y = x +2
x y y
-2 6 (-2, 6)
-1 3 (-1, 3)
0 2 (0, 2)
1 3 (1, 3) x
2 6 (2, 6)

27. EXAMPLE 1
Make a table and graph.
2
y = x +2
x y y
-2 6 (-2, 6)
-1 3 (-1, 3)
0 2 (0, 2)
1 3 (1, 3) x
2 6 (2, 6)

28. EXAMPLE 1
Make a table and graph.
2
y = x +2
x y y
-2 6 (-2, 6)
-1 3 (-1, 3)
0 2 (0, 2)
1 3 (1, 3) x
2 6 (2, 6)

29. EXAMPLE 1
Make a table and graph.
2
y = x +2
x y y
-2 6 (-2, 6)
-1 3 (-1, 3)
0 2 (0, 2)
1 3 (1, 3) x
2 6 (2, 6)

30. EXAMPLE 1
Make a table and graph.
2
y = x +2
x y y
-2 6 (-2, 6)
-1 3 (-1, 3)
0 2 (0, 2)
1 3 (1, 3) x
2 6 (2, 6)

31. EXAMPLE 1
Make a table and graph.
2
y = x +2
x y y
-2 6 (-2, 6)
-1 3 (-1, 3)
0 2 (0, 2)
1 3 (1, 3) x
2 6 (2, 6)

32. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7
x y

33. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7
x y
-3

34. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7
x y
-3 2

35. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7
x y
-3 2
-2

36. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7
x y
-3 2
-2 -5

37. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7
x y
-3 2
-2 -5
-1

38. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7
x y
-3 2
-2 -5
-1 -8

39. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7
x y
-3 2
-2 -5
-1 -8
0

40. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7
x y
-3 2
-2 -5
-1 -8
0 -7

41. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7
x y
-3 2
-2 -5
-1 -8
0 -7
1

42. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7
x y
-3 2
-2 -5
-1 -8
0 -7
1 -2

43. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7
x y
-3 2
-2 -5
-1 -8
0 -7
1 -2
2

44. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7
x y
-3 2
-2 -5
-1 -8
0 -7
1 -2
2 7

45. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7
x y
-3 2
-2 -5
-1 -8
0 -7
1 -2
2 7
3

46. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7
x y
-3 2
-2 -5
-1 -8
0 -7
1 -2
2 7
3 20

47. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7
x y
y
-3 2
-2 -5
-1 -8
0 -7 x
1 -2
2 7
3 20

48. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7
x y
y
-3 2
-2 -5
-1 -8
0 -7 x
1 -2
2 7
3 20

49. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7
x y
y
-3 2
-2 -5
-1 -8
0 -7 x
1 -2
2 7
3 20

50. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7
x y
y
-3 2
-2 -5
-1 -8
0 -7 x
1 -2
2 7
3 20

51. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7
x y
y
-3 2
-2 -5
-1 -8
0 -7 x
1 -2
2 7
3 20

52. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7
x y
y
-3 2
-2 -5
-1 -8
0 -7 x
1 -2
2 7
3 20

53. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7
x y
y
-3 2
-2 -5
-1 -8
0 -7 x
1 -2
2 7
3 20

54. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7
x y
y
-3 2
-2 -5
-1 -8
0 -7 x
1 -2
2 7
3 20

55. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7 y
x

56. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7 y
x
D = {x: x is all real numbers}

57. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7 y
y=
x
D = {x: x is all real numbers}

58. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7 y
y= 2nd TABLE
x
D = {x: x is all real numbers}

59. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7 y
y= 2nd TABLE GRAPH
x
D = {x: x is all real numbers}

60. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7 y
y= 2nd TABLE GRAPH
x
3:minimum
D = {x: x is all real numbers}

61. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7 y
y= 2nd TABLE GRAPH
x
3:minimum Left and Right Bound
D = {x: x is all real numbers}

62. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7 y
y= 2nd TABLE GRAPH
x
3:minimum Left and Right Bound
D = {x: x is all real numbers}

63. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7 y
y= 2nd TABLE GRAPH
x
3:minimum Left and Right Bound
D = {x: x is all real numbers}
R = {y: y ≥ -8.125}

64. EXAMPLE 2
Graph, then state the domain and range.
2
y = 2x + 3x − 7 y
y= 2nd TABLE GRAPH
x
3:minimum Left and Right Bound
D = {x: x is all real numbers}
R = {y: y ≥ -8.125}

65. EXAMPLE 3
A stationary supply wholesale chain found that pens that sell
for x dollars have a profit of y in thousands of dollars,
2
x y modeled by y = −x + 8x − 4.
a. Make a table and draw a graph

66. EXAMPLE 3
A stationary supply wholesale chain found that pens that sell
for x dollars have a profit of y in thousands of dollars,
2
x y modeled by y = −x + 8x − 4.
a. Make a table and draw a graph
1

67. EXAMPLE 3
A stationary supply wholesale chain found that pens that sell
for x dollars have a profit of y in thousands of dollars,
2
x y modeled by y = −x + 8x − 4.
a. Make a table and draw a graph
1 3

68. EXAMPLE 3
A stationary supply wholesale chain found that pens that sell
for x dollars have a profit of y in thousands of dollars,
2
x y modeled by y = −x + 8x − 4.
a. Make a table and draw a graph
1 3
2

69. EXAMPLE 3
A stationary supply wholesale chain found that pens that sell
for x dollars have a profit of y in thousands of dollars,
2
x y modeled by y = −x + 8x − 4.
a. Make a table and draw a graph
1 3
2 8

70. EXAMPLE 3
A stationary supply wholesale chain found that pens that sell
for x dollars have a profit of y in thousands of dollars,
2
x y modeled by y = −x + 8x − 4.
a. Make a table and draw a graph
1 3
2 8
3

71. EXAMPLE 3
A stationary supply wholesale chain found that pens that sell
for x dollars have a profit of y in thousands of dollars,
2
x y modeled by y = −x + 8x − 4.
a. Make a table and draw a graph
1 3
2 8
3 11

72. EXAMPLE 3
A stationary supply wholesale chain found that pens that sell
for x dollars have a profit of y in thousands of dollars,
2
x y modeled by y = −x + 8x − 4.
a. Make a table and draw a graph
1 3
2 8
3 11
4

73. EXAMPLE 3
A stationary supply wholesale chain found that pens that sell
for x dollars have a profit of y in thousands of dollars,
2
x y modeled by y = −x + 8x − 4.
a. Make a table and draw a graph
1 3
2 8
3 11
4 12

74. EXAMPLE 3
A stationary supply wholesale chain found that pens that sell
for x dollars have a profit of y in thousands of dollars,
2
x y modeled by y = −x + 8x − 4.
a. Make a table and draw a graph
1 3
2 8
3 11
4 12
5

75. EXAMPLE 3
A stationary supply wholesale chain found that pens that sell
for x dollars have a profit of y in thousands of dollars,
2
x y modeled by y = −x + 8x − 4.
a. Make a table and draw a graph
1 3
2 8
3 11
4 12
5 11

76. EXAMPLE 3
A stationary supply wholesale chain found that pens that sell
for x dollars have a profit of y in thousands of dollars,
2
x y modeled by y = −x + 8x − 4.
a. Make a table and draw a graph
1 3
2 8
3 11
4 12
5 11
6

77. EXAMPLE 3
A stationary supply wholesale chain found that pens that sell
for x dollars have a profit of y in thousands of dollars,
2
x y modeled by y = −x + 8x − 4.
a. Make a table and draw a graph
1 3
2 8
3 11
4 12
5 11
6 8

78. EXAMPLE 3
A stationary supply wholesale chain found that pens that sell
for x dollars have a profit of y in thousands of dollars,
2
x y modeled by y = −x + 8x − 4.
a. Make a table and draw a graph
1 3
2 8
3 11
4 12
5 11
6 8
7

79. EXAMPLE 3
A stationary supply wholesale chain found that pens that sell
for x dollars have a profit of y in thousands of dollars,
2
x y modeled by y = −x + 8x − 4.
a. Make a table and draw a graph
1 3
2 8
3 11
4 12
5 11
6 8
7 3

80. EXAMPLE 3
A stationary supply wholesale chain found that pens that sell
for x dollars have a profit of y in thousands of dollars,
2
x y modeled by y = −x + 8x − 4.
a. Make a table and draw a graph
1 3 y
2 8
3 11
4 12
5 11
6 8
x
7 3

81. EXAMPLE 3
A stationary supply wholesale chain found that pens that sell
for x dollars have a profit of y in thousands of dollars,
2
x y modeled by y = −x + 8x − 4.
a. Make a table and draw a graph
1 3 y
2 8
3 11
4 12
5 11
6 8
x
7 3

82. EXAMPLE 3
A stationary supply wholesale chain found that pens that sell
for x dollars have a profit of y in thousands of dollars,
2
x y modeled by y = −x + 8x − 4.
a. Make a table and draw a graph
1 3 y
2 8
3 11
4 12
5 11
6 8
x
7 3

83. EXAMPLE 3
A stationary supply wholesale chain found that pens that sell
for x dollars have a profit of y in thousands of dollars,
2
x y modeled by y = −x + 8x − 4.
a. Make a table and draw a graph
1 3 y
2 8
3 11
4 12
5 11
6 8
x
7 3

84. EXAMPLE 3
A stationary supply wholesale chain found that pens that sell
for x dollars have a profit of y in thousands of dollars,
2
x y modeled by y = −x + 8x − 4.
a. Make a table and draw a graph
1 3 y
2 8
3 11
4 12
5 11
6 8
x
7 3

85. EXAMPLE 3
A stationary supply wholesale chain found that pens that sell
for x dollars have a profit of y in thousands of dollars,
2
x y modeled by y = −x + 8x − 4.
a. Make a table and draw a graph
1 3 y
2 8
3 11
4 12
5 11
6 8
x
7 3

86. EXAMPLE 3
A stationary supply wholesale chain found that pens that sell
for x dollars have a profit of y in thousands of dollars,
2
x y modeled by y = −x + 8x − 4.
a. Make a table and draw a graph
1 3 y
2 8
3 11
4 12
5 11
6 8
x
7 3

87. EXAMPLE 3
A stationary supply wholesale chain found that pens that sell
for x dollars have a profit of y in thousands of dollars,
2
x y modeled by y = −x + 8x − 4.
a. Make a table and draw a graph
1 3 y
2 8
3 11
4 12
5 11
6 8
x
7 3

88. EXAMPLE 3
A stationary supply wholesale chain found that pens that sell
for x dollars have a profit of y in thousands of dollars,
2
x y modeled by y = −x + 8x − 4.
a. Make a table and draw a graph
1 3 y
2 8
3 11
4 12
5 11
6 8
x
7 3

89. EXAMPLE 3
b. If the pens sell for $2.00 each, what is the expected
profit?
c. What is the price of the pens that would yield the
maximum profit? What is the maximum profit?

90. EXAMPLE 3
b. If the pens sell for $2.00 each, what is the expected
profit?
When x is replaced with 2, we get an output of 8
c. What is the price of the pens that would yield the
maximum profit? What is the maximum profit?

91. EXAMPLE 3
b. If the pens sell for $2.00 each, what is the expected
profit?
When x is replaced with 2, we get an output of 8
There will be an $8000 profit
c. What is the price of the pens that would yield the
maximum profit? What is the maximum profit?

92. EXAMPLE 3
b. If the pens sell for $2.00 each, what is the expected
profit?
When x is replaced with 2, we get an output of 8
There will be an $8000 profit
c. What is the price of the pens that would yield the
maximum profit? What is the maximum profit?
$4 sale price per pen gives a maximum profit of $12000

93. EXAMPLE 4
Consider the equation x = y2, which is the equivalent of
x y y=± x.
a. Make a table and graph.

94. EXAMPLE 4
Consider the equation x = y2, which is the equivalent of
x y y=± x.
25 a. Make a table and graph.

95. EXAMPLE 4
Consider the equation x = y2, which is the equivalent of
x y y=± x.
25 ±5 a. Make a table and graph.

96. EXAMPLE 4
Consider the equation x = y2, which is the equivalent of
x y y=± x.
25 ±5 a. Make a table and graph.
16

97. EXAMPLE 4
Consider the equation x = y2, which is the equivalent of
x y y=± x.
25 ±5 a. Make a table and graph.
16 ±4

98. EXAMPLE 4
Consider the equation x = y2, which is the equivalent of
x y y=± x.
25 ±5 a. Make a table and graph.
16 ±4
9

99. EXAMPLE 4
Consider the equation x = y2, which is the equivalent of
x y y=± x.
25 ±5 a. Make a table and graph.
16 ±4
9 ±3

100. EXAMPLE 4
Consider the equation x = y2, which is the equivalent of
x y y=± x.
25 ±5 a. Make a table and graph.
16 ±4
9 ±3
4

101. EXAMPLE 4
Consider the equation x = y2, which is the equivalent of
x y y=± x.
25 ±5 a. Make a table and graph.
16 ±4
9 ±3
4 ±2

102. EXAMPLE 4
Consider the equation x = y2, which is the equivalent of
x y y=± x.
25 ±5 a. Make a table and graph.
16 ±4
9 ±3
4 ±2
1

103. EXAMPLE 4
Consider the equation x = y2, which is the equivalent of
x y y=± x.
25 ±5 a. Make a table and graph.
16 ±4
9 ±3
4 ±2
1 ±1

104. EXAMPLE 4
Consider the equation x = y2, which is the equivalent of
x y y=± x.
25 ±5 a. Make a table and graph.
16 ±4
9 ±3
4 ±2
1 ±1
0

105. EXAMPLE 4
Consider the equation x = y2, which is the equivalent of
x y y=± x.
25 ±5 a. Make a table and graph.
16 ±4 y
9 ±3
4 ±2
x
1 ±1
0
10 20 30

106. EXAMPLE 4
Consider the equation x = y2, which is the equivalent of
x y y=± x.
25 ±5 a. Make a table and graph.
16 ±4 y
9 ±3
4 ±2
x
1 ±1
0 0
10 20 30

107. EXAMPLE 4
Consider the equation x = y2, which is the equivalent of
x y y=± x.
25 ±5 a. Make a table and graph.
16 ±4 y
9 ±3
4 ±2
x
1 ±1
0 0
10 20 30

108. EXAMPLE 4
Consider the equation x = y2, which is the equivalent of
x y y=± x.
25 ±5 a. Make a table and graph.
16 ±4 y
9 ±3
4 ±2
x
1 ±1
0 0
10 20 30

109. EXAMPLE 4
Consider the equation x = y2, which is the equivalent of
x y y=± x.
25 ±5 a. Make a table and graph.
16 ±4 y
9 ±3
4 ±2
x
1 ±1
0 0
10 20 30

110. EXAMPLE 4
Consider the equation x = y2, which is the equivalent of
x y y=± x.
25 ±5 a. Make a table and graph.
16 ±4 y
9 ±3
4 ±2
x
1 ±1
0 0
10 20 30

111. EXAMPLE 4
Consider the equation x = y2, which is the equivalent of
x y y=± x.
25 ±5 a. Make a table and graph.
16 ±4 y
9 ±3
4 ±2
x
1 ±1
0 0
10 20 30

112. EXAMPLE 4
Consider the equation x = y2, which is the equivalent of
x y y=± x.
25 ±5 a. Make a table and graph.
16 ±4 y
9 ±3
4 ±2
x
1 ±1
0 0
10 20 30

113. EXAMPLE 4
Consider the equation x = y2, which is the equivalent of
x y y=± x.
25 ±5 a. Make a table and graph.
16 ±4 y
9 ±3
4 ±2
x
1 ±1
0 0
10 20 30

114. EXAMPLE 4
Consider the equation x = y2, which is the equivalent of
x y y=± x.
25 ±5 a. Make a table and graph.
16 ±4 y
9 ±3
4 ±2
x
1 ±1
0 0
10 20 30

115. EXAMPLE 4
Consider the equation x = y2, which is the equivalent of
x y y=± x.
25 ±5 a. Make a table and graph.
16 ±4 y
9 ±3
4 ±2
x
1 ±1
0 0
10 20 30

116. EXAMPLE 4
Consider the equation x = y2, which is the equivalent of
x y y=± x.
25 ±5 a. Make a table and graph.
16 ±4 y
9 ±3
4 ±2
x
1 ±1
0 0
10 20 30

117. EXAMPLE 4
Consider the equation x = y2, which is the equivalent of
x y y=± x.
25 ±5 a. Make a table and graph.
16 ±4 y
9 ±3
4 ±2
x
1 ±1
0 0
10 20 30

118. EXAMPLE 4
Consider the equation x = y2, which is the equivalent of
x y y=± x.
25 ±5 a. Make a table and graph.
16 ±4 y
9 ±3
4 ±2
x
1 ±1
0 0
10 20 30

119. EXAMPLE 4
b. Is this a function? How do you know?
c. What are the domain and range of the equation?

120. EXAMPLE 4
b. Is this a function? How do you know?
No, it does not pass the Vertical-Line Test
c. What are the domain and range of the equation?

121. EXAMPLE 4
b. Is this a function? How do you know?
No, it does not pass the Vertical-Line Test
c. What are the domain and range of the equation?
y
x

122. EXAMPLE 4
b. Is this a function? How do you know?
No, it does not pass the Vertical-Line Test
c. What are the domain and range of the equation?
y
D = {x: x ≥ 0}
x

123. EXAMPLE 4
b. Is this a function? How do you know?
No, it does not pass the Vertical-Line Test
c. What are the domain and range of the equation?
y
D = {x: x ≥ 0}
R = {y: y is all real numbers} x

124. HOMEWORK

125. HOMEWORK
p. 269 #1-29 odd
“In theory, there is no difference between theory and
practice. In practice there is.” - Yogi Berra