1. Math 2 Warm Up 2x2 – 4x(3x – 5) 3x(x – 2) (x – 2)(x + 5)
Simplify each expression:
2x2 – 4x(3x – 5)
3x(x – 2)
(x – 2)(x + 5)
(-4x + 3)(2x – 7)
3x(2x – 7) + 6x(4x + 5)
x(1 – x) – (1 – 2x2)
(5x + 3) 2
-7x(5x2 – 4x)
(4x – 5) (-2x2 + 3x – 9)

2. Unit 5: “Quadratic Functions” Lesson 1 - Properties of Quadratics
Objective: To find the vertex & axis of symmetry of a quadratic function then graph the function.
quadratic function – is a function that can be written in
the standard form:
y = ax2 + bx + c, where a ≠ 0.
Examples:
y = 5x2 y = -2x2 + 3x y = x2 – x – 3

3. Properties of Quadratics
parabola – the graph of a quadratic equation. It is in the form of a “U” which opens either upward or downward. vertex – the maximum or minimum point of a parabola.

4. Properties of Quadratics
axis of symmetry – the line passing through the vertex about which the parabola is symmetric (the same on both sides).

5. Properties of Quadratics
Find the coordinates of the vertex, the equation for the axis of symmetry of each parabola. Find the coordinates points corresponding to P and Q.

6. Graphing a Quadratic Equation y = ax2 + bx + c
1) Direction of the parabola? If a is positive, then the graph opens up. If a is negative, then the graph opens down.

7. Graphing a Quadratic Equation y = ax2 + bx + c
2) Find the vertex and axis of symmetry. The x-coordinate of the vertex is 𝐱= −𝐛 𝟐𝐚 (also the equation for the axis of symmetry). Substitute the value of x into the quadratic equation and solve for the y-coordinate. Write vertex as an ordered pair (x , y).

8. Graphing a Quadratic Equation y = ax2 + bx + c
3) Table of Values. Choose two values for x that are one side of the vertex (either right or left). Substitute those values into the quadratic equation to find y values. Graph the two points. Graph the reflection of the two points on the other side of the parabola (same y-values and same distance away from the axis of symmetry).

9. y = 2x2 + 4x + 3 Direction: _____ Vertex: ______ Axis: _______
Find the vertex and axis of symmetry of the following quadratic equation. Then, make a table of values and graph the parabola.
y = 2x2 + 4x + 3 Direction: _____ Vertex: ______ Axis: _______

10. y = – x2 + 3x – 1 Direction: _____ Vertex: ______ Axis: _______
Find the vertex and axis of symmetry of the following quadratic equation. Then, make a table of values and graph the parabola.
y = – x2 + 3x – 1 Direction: _____ Vertex: ______ Axis: _______

11. y = – 𝟏 𝟐 x2 + 2x + 5 Direction: _____ Vertex: ______ Axis: _______
Find the vertex and axis of symmetry of the following quadratic equation. Then, make a table of values and graph the parabola.
y = – 𝟏 𝟐 x2 + 2x + 5 Direction: _____ Vertex: ______ Axis: _______

12. y = 3x2 – 4 Direction: _____ Vertex: ______ Axis: _______
Find the vertex and axis of symmetry of the following quadratic equation. Then, make a table of values and graph the parabola.
y = 3x2 – 4 Direction: _____ Vertex: ______ Axis: _______

13. Apply!
The number of widgets the Woodget Company sells can be modeled by the equation -5p2 + 10p + 100, where p is the selling price of a widget. What price for a widget will maximize the company’s revenue? What is the maximum revenue?

15. End of Day 1

16. where (h, k) is the vertex.
Math 2
Unit 5
Lesson 2
Unit 5:"Quadratic Functions"
Title: Translating Quadratic Functions
Objective: To use the vertex form of a
quadratic function.
y = a(x – h)2 + k
where (h, k) is the vertex.

17. y = 2(x – 1) 2 + 2 Direction: _____ Vertex: ______ Axis: _______
Example 1: Graphing from Vertex Form
y = 2(x – 1) 2 + 2
Direction: _____
Vertex: ______
Axis: _______

18. y = (x + 3) 2 – 1 Direction: _____ Vertex: ______ Axis: _______
Example 2: Graphing from Vertex Form
y = (x + 3) 2 – 1
Direction: _____

Vertex: ______
Axis: _______

19. y = −1 2 (x – 3) 2 – 2 Direction: _____ Vertex: ______ Axis: _______
Example 3: Graphing from Vertex Form
y = −1 2 (x – 3) 2 – 2
Direction: _____

Vertex: ______
Axis: _______

20. Example 4: Write quadratic equation in vertex form.

21. Example 5: Write quadratic equation in vertex form.

22. y = x2 - 4x + 6 Example 6: Converting Standard Form to Vertex Form.
Step 1: Find the Vertex
x = -b =
y =
Step 2: Substitute into Vertex Form:
y = x2 - 4x + 6 2a

23. y = 6x2 – 10 Example 7: Converting Standard Form to Vertex Form.
Step 1: Find the Vertex
x = -b =
y =
Step 2: Substitute into Vertex Form:
y = 6x2 – 10 2a

24. y = 2(x – 1) 2 + 2 Example 8: Converting Vertex Form to Standard Form.
Step 1: Square the Binomial.
Step 2: Simplify to
y = 2(x – 1) 2 + 2

25. Example 9: Converting Vertex Form to Standard Form.
Step 1: Square the Binomial.
Step 2: Simplify to
y = −1 2 (x – 3) 2 – 2

26. Honors Math 2 Assignment:
In the Algebra 2 textbook:
pp #3, 6, 9, 17-20, 25, 27, 31, 34, 52, 54

27. End of Day 2

29. Factoring Quadratic Expressions
Objective: To find common factors and binomial factors
of quadratic expressions.
factor – if two or more polynomials are multiplied together,
then each polynomial is a factor of the product.
(2x + 7)(3x – 5) = 6x2 + 11x – 35
FACTORS PRODUCT
(2x – 5)(3x + 7) = 6x2 – x – 35
“factoring a polynomial” – reverses the multiplication!

30. Finding Greatest Common Factor
greatest common factor (GCF) – the greatest of the common factors of two or more monomials.
𝟕 𝒙 𝟐 + 𝟐𝟏
𝟒 𝒙 𝟐 + 20x − 12
𝟗 𝒙 𝟐 − 24x

32. Finding Binomial Factors
𝒙 𝟐 + 14x + 40

33. Finding Binomial Factors
𝒙 𝟐 + 12x + 32

34. Finding Binomial Factors
𝒙 𝟐 − 11x + 24

35. Finding Binomial Factors
𝒙 𝟐 − 17x + 72

36. Finding Binomial Factors
𝒙 𝟐 − 14x − 32

37. Finding Binomial Factors
𝒙 𝟐 + 3x − 28

38. Finding Binomial Factors
𝟐𝒙 𝟐 + 11x + 12

39. Finding Binomial Factors
𝟔𝒙 𝟐 − 31x + 35

40. Finding Binomial Factors
𝟏𝟐𝒙 𝟐 + 32x − 35

41. Finding Binomial Factors
𝟑𝒙 𝟐 − 16x − 12

42. Finding Binomial Factors*
𝟏𝟎𝒙 𝟐 + 35x − 45

43. Finding Binomial Factors*
𝟗𝒙 𝟐 + 42x + 𝟒𝟗

44. Finding Binomial Factors*
𝟐𝟓𝒙 𝟐 − 90x + 𝟖𝟏

45. Factoring Special Expressions*
𝟒𝒙 𝟐 − 49
𝟐𝟓𝒙 𝟐 − 9
𝟑𝒙 𝟐 − 192
𝟗𝒙 𝟐 − 36

46. Honors Math 2 Assignment
In the Algebra 2 textbook, pp
#1, 5, 6, 7-45 odd, 48, 54

47. End of Day 3

48. Factor.
𝟏𝟎𝒙 𝟐 + 35x − 45
𝟗𝒙 𝟐 − 36
𝟑𝒙 𝟐 − 16x − 12

49. Solving Quadratics Equations: Factoring and Square Roots
Objective: To solve quadratic equations by factoring and by finding the square root.

50. Solve by Factoring
𝒙 𝟐 + 7x − 18 = 0

51. Solve by Factoring
𝟑 𝒙 𝟐 − 20x − 7 = 0

52. Solve by Factoring
𝟖𝒙 𝟐 − 5 = 6x

53. Solve by Factoring
𝟔𝒙 𝟐 =𝟒𝟏𝐱 −𝟔𝟑

54. Solve by Factoring*
𝟒𝒙 𝟐 + 16x = 10x +𝟒𝟎

55. Solve by Factoring*
𝟏𝟔𝒙 𝟐 − 𝟖𝒙 =𝟎

56. Solve Using Square Roots
Quadratic equations in the form 𝒂 𝒙 𝟐 =𝒄 can be solved by finding square roots.
𝟑𝒙 𝟐 = 243

57. Solve Using Square Roots
𝟓𝒙 𝟐 − 200 = 0

58. Solve Using Square Roots*
𝟒𝒙 𝟐 − 25 = 0

59. Honors Math 2 Assignment
In the Algebra 2 textbook,
p. 266
#1-19

60. End of Day 4

61. Complex Numbers

62. Math 2 Warm Up
In the Algebra 2 Practice Workbook,
Practice 5-5 (p. 64)
#1, 10, 13, 19, 25, 31,
40, 46, 55, 61, 71, 73

64. Unit 4, Lesson 5:
Complex Numbers
Objective: To define imaginary and complex numbers and to perform operations on complex numbers

65. Introducing Imaginary Numbers
Find the solutions to the following equation:

66. Introducing Imaginary Numbers
Now find the solutions to this equation:

67. Imaginary numbers offer solutions to this problem!
i1 = i
i2 = -1
i3 = -i
i4 = 1

68. Simplifying Complex Numbers

69. Adding/Subtracting Complex Numbers
(8 + 3i) – (2 + 4i)
7 – (3 + 2i)
(4 - 6i) + (4 + 3i)

70. Multiplying Complex Numbers
(12i)(7i)
(6 - 5i)(4 - 3i)
(4 - 9i)(4 + 3i)
(3 - 7i)(2 - 4i)

71. So, now we can finally find ALL solutions to this equation!

72. Complex Solutions
3x² + 48 = 0
-5x² = 0
8x² + 2 = 0
9x² + 54 = 0

73. Math 2 Assignment #s 1-17 odd, 29-39 odd, 41-46
In the Algebra 2 Textbook,
Pgs
#s 1-17 odd, odd, 41-46

74. End of Day 5

76. Completing the Square
1.) Move the constant to opposite side of the equation as the terms with variables in them. 2.) Take half of the coefficient with the x-term and square it 3.) Add the number found in step 2 to both sides of the equation. 4.) Factor side with variables into a perfect square. 5.) Square root both sides (put + in front of square root on side with only constant) 6.) Solve for x.

77. using completing the square
Solve the following,
using completing the square
1.) x2 – 3x – 28 = ) x2 – 3x = ) x2 + 6x + 9 = 0

78. If a ≠ 1, then divide all the term by “a”.
1.) 2x2 + 6x = ) 3x2 – 12x + 7 = 0 3.) 5x2 + 20x + -50

79. Math 2 Assignment # 15 – 25, 37, 39, 51-53 In the Algebra 2 Textbook,
Pgs
# 15 – 25, 37, 39, 51-53

80. End of Day 6

81. Solve using Completing the square
x2 + 4x = 21
x2 – 8x – 33 = 0
4x2 + 4x = 3

82. Solving Quadratic Equations: Quadratic Formula
Objective: To solve quadratic equations using the Quadratic Formula.
Not every quadratic equation can be solved by factoring or by taking the square root!
𝟐𝒙 𝟐 + 5x − 𝟖 = 0

83. Solve using Quadratic Formula
𝟐𝒙 𝟐 + 5x − 8 = 0

84. Solve using Quadratic Formula
𝟑𝒙 𝟐 + 23x + 40 = 0

85. Solve using Quadratic Formula
𝟗𝒙 𝟐 +𝟔𝐱−𝟏=𝟎

86. Solve using Quadratic Formula*
𝟒𝒙 𝟐 −𝟖𝒙=− 𝟏𝟎

87. Solve using Quadratic Formula
𝟐𝟓𝒙 𝟐 −𝟑𝟎𝒙+𝟏𝟐=𝟎

88. Solve using Quadratic Formula
𝟑𝒙 𝟐 −𝟐𝒙+𝟒=𝟎

89. Solve using Quadratic Formula
𝟐𝒙 𝟐 = -6x – 7

90. Honors Math 2 Assignment
In the Algebra 2 textbook, pp
#1, 2, 22-30

91. Solve
𝒙 𝟐 +𝟒𝒙 = 41
{-8.71, 4.71}
𝟐𝒙 𝟐 = -6x – 7
No Solution

92. End of Day 7

93. Solving Quadratic Equations: Graphing
Objective: To solve quadratic equations and systems that contain a quadratic equation by graphing.
When the graph of a function intersects the x-axis, the y-value of the function is 0.
Therefore, the solutions of the quadratic equation ax2 + bx + c = 0 are the x-intercepts of the graph.
Also known as the “zeros of the function” or the “roots of the function”.

94. Solve Quadratic Equations by Graphing
Solution

95. Solve Quadratic Equations by Graphing
Step 1: Quadratic equation must equal 0! ax2 + bx + c = 0
Step 2: Press [Y=]. Enter the quadratic equation in Y1. Enter 0 in Y2. Press [Graph]. MAKE SURE BOTH X-INTERCEPTS ARE ON SCREEN! ZOOM IF NEEDED!
Step 3: Find the intersection of ax2 + bx + c and Press [2nd] [Trace]. Select [5: Intersection].
Press [Enter] 2 times for 1st and 2nd curve.
Move cursor to one of the x-intercepts then press
[Enter] for the 3rd time.
Repeat Step 3 for the second x-intercept!

96. Solve by Graphing
𝒙 𝟐 + 6x + 4 = 0

97. Solve by Graphing
𝟐𝒙 𝟐 + 4x – 7 = 0

98. Solve by Graphing
𝟑𝒙 𝟐 + 5x = 20

99. Solve by Graphing
𝟓𝒙 𝟐 +𝟕 = 19x

100. Solve by Graphing
𝒙 𝟐 = -2x + 7

101. Solve by Graphing
−𝟑𝒙 𝟐 + 2x – 6 = 0

102. Solve by Graphing
𝒙 𝟐 + 𝟖𝒙 + 16 = 0

103. End of Day 8

104. Solving Systems of Equations

105. Solve a System with a Quadratic Equation
𝒚= 𝒙 𝟐 + x − 𝟐
𝒚=−𝒙+𝟑

106. Solve a System with a Quadratic Equation
𝒚=𝟐𝒙 𝟐 + x
𝒚= 𝟒 𝟑 𝒙+𝟒

107. Solve a System with a Quadratic Equation
𝒚= 𝒙 𝟐 + 𝟒𝒙 + 𝟕
𝒚=−𝟐𝒙

108. Solve a System with a Quadratic Equation
𝒚= 𝒙 𝟐 −𝟔x + 𝟏𝟎
𝒚=𝟏

109. Solve a System with Quadratic Equations
𝒚= 𝒙 𝟐 − 𝟔𝒙+𝟓
𝒚=−𝟐𝒙 𝟐 +𝟓𝒙

110. Solve a System with Quadratic Equations
𝒚= 𝒙 𝟐 + 𝟕𝒙
𝐲= 𝟏 𝟒 𝒙 𝟐 −𝟓𝒙−𝟗
𝒚= 𝒙 𝟐 − 𝟔𝒙+𝟓
𝒚=−𝟐𝒙 𝟐 +𝟓𝒙

111. Honors Math 2 Assignment
In the Algebra 2 textbook, pp
#20-31, 35, 54-56
Solve each quadratic equation or system by graphing.

112. Modeling Data with Quadratic Equations
Objective: To model a set of data with a quadratic function.
Graph: Graph:
(-3, 7), (-2, 2), (0, -2) (-1, -8), (2, 1), (3, 8)
(3, 7), (1, -1), (2, 2)

113. End of Day 9

114. Finding a Quadratic Model
1) Turn on plot:
Press [2nd] [Y=], [ENTER], Highlight “On”,
Press [ENTER]  
2) Turn on diagnostic:
Press [2nd] [0] (for catalog),
Scroll down to find DiagonsticOn.
Press [ENTER] to select.
Press [ENTER] again to activate.

115. Finding a Quadratic Model
3) Enter data values: Press [STAT], [ENTER] (for EDIT), Enter x-values (independent) in L1 Enter y-values (dependent) in L2 Clear Lists (if needed): Highlight L1 or L2 (at top) Press [CLEAR], [ENTER].

116. Finding a Quadratic Model
4) Graph scatter plot:
Press [ZOOM], 9 (zoomstat)
5) Find quadratic equation to fit data:
Press [STAT], over to CALC,
For Quadratic Model - Press 5: QuadReg
Press [ENTER] 4 times, then Calculate.
Write quadratic equation using the values of a, b, and c rounded to the nearest thousandths if needed.
Write down the R2 value!

117. Find a quadratic equation to model the values in the table.X Y -1 -8 2 1 3 8

118. 𝑹 𝟐 is a measure of the “goodness-of-fit” of a regression model.
the value of R2 is between 0 and 1 (0 ≤ R2 ≤ 1)
R2  = 1 means all the data points “fit” the model (lie exactly on the graph with no scatter) – “knowing x lets you predict y perfectly!”
R2 = 0 means none of the data points “fit” the model – “knowing x does not help predict y!”
An R2 value closer to 1 means the better the regression model “fits” the data.

119. Find a quadratic equation to model the values in the table.X Y 2 3 13 4 29

120. Find a quadratic equation to model the values in the table.X Y -5
-18 -4 2
-14

121. Find a quadratic equation to model the values in the table.X Y -2 27 1 10 5
-10 7 12

122. Apply!
The table shows data about the wavelength (in meters) and the wave speed (in meters per second) of the deep water ocean waves. Model the data with a quadratic function then use the model to estimate:
the wave speed of a deep water wave that has a wavelength of 6 meters.
the wavelength of a deep water wave with a speed of 50 meters per second.
Wavelength
(m)
Wave Speed (m/s) 3 6 5 16 7 31 8 40

123. Apply!
The table at the right shows the height of a column of water as it drains from its container. Model the data with a quadratic function then use the model to estimate:
the water level at 35 seconds.
the waver level at 80 seconds.
the water level at 3 minutes.
the elapsed time for the water level to reach 20 mm.

124. Honors Math 2 Assignment
In the Algebra 2 textbook, pp
#16-22, 30, 31, 38
Write down the R² value for each equation!

125. End of Day 10

126. Unit 5 Test Review: “Quadratics”
Quadratic Function
Standard form: 𝒚=𝒂 𝒙 𝟐 +𝒃𝒙+𝒄
Vertex form: 𝒚=𝒂 (𝒙−𝒉) 𝟐 + 𝒌
Change Forms!
Direction - parabola opens up or down?
Vertex (𝒙= −𝒃 𝟐𝒂 , substitute x to find y) or (h, k )
Vertex – is a Maximum or Minimum?
Axis of Symmetry 𝒙= −𝒃 𝟐𝒂 or x = h
y-intercept (0, c) or (0, substitute 0 to find y)
Graph (at least 5 points – vertex and 2 points on each side of axis of symmetry)

127. Unit 5 Test Review: “Quadratics”
Solve Quadratic Equations by:
Factoring – Zero Product Property
Square Root – Don’t forget ±
Quadratic Formula 𝒙= −𝒃 ± 𝒃 𝟐 − 𝟒𝒂𝒄 𝟐𝒂
Use Discriminant for Number & Types of Solutions
Graphing – Find Intersection on Calculator
Solve System with Quadratic by Graphing
Quadratic Model for a Set of Data
Quadratic Regression Model: 𝒚=𝒂 𝒙 𝟐 +𝒃𝒙+𝒄
Find R² value and what it means
Predict Values (x or y) using Quadratic Model

128. Math 2 Assignment pp. 293-295 #2-11, 12ce, 13-38, 70-72
In the Algebra 2 textbook, pp
#2-11, 12ce, 13-38, 70-72

129. Math 2 homework
In the Algebra 2 textbook,
p. 296
#6, 14, 24, 25, 27
28, 30, 33, 38, 39