1. Vertex Form Quadratic Function in Vertex Form:
The graph of y = a(x – h)2 + k is the graph of y = ax2 translated h units to the right and k units up.
y = a(x – h)2 + k

2. Vertex Form Quadratic Function in Vertex Form: Vertex: (h, k)
Axis of Symmetry: x = h
y = a(x – h)2 + k

3. Exercise 1
What is the y-intercept of the graph of y = a(x – h)2 + k?

4. Exercise 2
What are the x-intercepts of the graph of y = a(x – h)2 + k?

5. Graphing Quadratic Functions, II
To graph y = a(x – h)2 + k use SRT transformations.
Scaling: a times the y-coordinate
Reflecting: over the x-axis if a < 0
Translating: Left/Right h units and Up/Down k units
Remember that the up/down movement is in the direction of the sign of k, but for left/right, it’s the opposite direction as the sign of h.

6. Exercise 3
Graph

7. Exercise 4
Graph

8. 4.2: Quad Graphs in Vertex and Intercept Forms
Objective 1b
4.2: Quad Graphs in Vertex and Intercept Forms
You will be able to graph a quadratic function in intercept form
𝑦=−0.42 𝑥−16.9 𝑥−26.9
A bouncing ball captured with a stroboscopic flash at 25 images per second.
Michael Maggs

9. Intercept Form Quadratic Function in Intercept Form:
x-intercepts: p and q
Vertex: Average of p and q
Axis of Symmetry:
y = a(x – p)(x – q)

10. Exercise 5
What is the y-intercept of the graph of y = a(x – p)(x – q)?

11. Graphing Quadratic Functions, III
To graph y = a(x – p)(x – q):
Use the a-value to determine
Plot the x-intercepts p and q
Find and plot the vertex (x = average of p and q)
Graph the axis of symmetry
Find and plot a couple of other function values and their reflections
 or 

12. Exercise 6
Graph

13. Exercise 7
Graph

14. 4.2: Quad Graphs in Vertex and Intercept Forms
𝑦=−0.05 𝑥 𝑥−1.9
𝑦=−0.05 𝑥−
𝑦=−0.05 𝑥−1.6 𝑥−23.1
You will be able to rewrite your quadratic function in standard, vertex, or intercept form
Objective 2
Eigo Sato

15. Exercise 8a
Write each function in standard form.

16. Protip #1: Standard Form
To write your quadratic equation in standard form is really easy. Just FOIL, distribute, or expand as necessary, and then combine like terms. Finally, make it look like this:
y = ax2 + bx + c

17. Exercise 8b
Write each function in standard form.

18. Exercise 9a
Write each function in intercept form.

19. Protip #2: Intercept Form
To write your equation in intercept form, you have to do some factoring:
Factor out GCF. That includes a −1 if the leading term is negative
Factor the trinomial that is left over

20. Protip #2: Intercept Form
To write your equation in intercept form, you have to do some factoring:
Each factor has to look like (x ± p). This means you have to divide out any coefficient of x from each factor. Whatever you divide out becomes part of the a-value.
Divide out a 2
Divide out a 3

21. Protip #2: Intercept Form
To write your equation in intercept form, you have to do some factoring:
Each factor has to look like (x ± p). This means you have to divide out any coefficient of x from each factor. Whatever you divide out becomes part of the a-value.

22. Exercise 9b
Write each function in intercept form.

23. Exercise 10a
Write each function in vertex form.

24. Protip #3: Vertex Form
To put a quadratic equation in vertex form, you have to complete the square on only one side of the equation.
Separate the constant term from the variable terms
Factor out the leading coefficient from the variable terms

25. Protip #3: Vertex Form
To put a quadratic equation in vertex form, you have to complete the square on only one side of the equation.
Complete the square with the variable terms
Whatever you add on the inside of the parenthesis, subtract on the outside of the parenthesis. 9 36

26. Protip #3: Vertex Form
Remember that what you add on the inside of the parenthesis is actually multiplied by the leading coefficient that you factored out in Step 2. Make sure to multiply by this number before subtracting on the outside of the parenthesis 9 36

27. Exercise 10b
Write each function in vertex form.

28. Solving Quadratic Equations
Objectives:
To solve a quadratic equation by factoring

29. Warm-Up
If the product of 𝐴 and 𝐵 equals zero, what must be true about 𝐴 or 𝐵?
Zero Product Property
𝐴∙𝐵=0
If the product of two expressions is zero, then at least of the expressions equal zero.
Maybe that’s zero
Or maybe this one’s zero
(Or maybe they’re both zero)

30. Vocabulary 𝑦-intercepts 𝑥-intercepts Parabola Roots (of a function)
Zeros (of a function)

31. Objective 1
You will be able to solve a quadratic equation by factoring

32. x-intercepts = points where parabola crosses x-axis
Exercise 1
Use your calculator to graph the equation y = x2 – x – 6.
Where does the parabola cross the x-axis?
x-intercepts = points where parabola crosses x-axis

33. x-intercepts = points where parabola crosses x-axis
The graph of a quadratic function is a parabola.
x-intercepts: where the parabola intersects the x-axis
x-intercepts = points where parabola crosses x-axis

34. Exercise 2
How many y-intercepts can the graph of the quadratic function y = ax2 + bx + c have?
Only one since it’s a function!
How many x-intercepts can the graph of the quadratic function y = ax2 + bx + c have?
Two, one, or none!

35. Plug in zero for x and solve for y
Exercise 3
If you wanted to find the y-intercept of a quadratic function, what would you do?
Plug in zero for x and solve for y

36. Exercise 4
Find the y-intercept of y = x2 – 6x – 7.

37. Plug in zero for y and solve for x
Exercise 5
If you wanted to find the x-intercept of a quadratic function, what would you do?
Plug in zero for y and solve for x

38. The answer is to use the Zero Product Property
Exercise 6
Find the x-intercept(s) of y = x2 – 6x – 7. The problem here is, how do you solve 0 = x2 – 6x – 7 since you can’t just get x by itself?
The answer is to use the Zero Product Property

39. Solving Quadratic Equations
The standard form of a quadratic equation in one variable is ax2 + bx + c = 0, where a is not zero.
Solving a quadratic equation in standard form is the same thing as finding the x-intercepts of y = ax2 + bx + c.

40. Solving Quadratic Equations
The standard form of a quadratic equation in one variable is ax2 + bx + c = 0, where a is not zero.
We can use the zero product property to solve certain quadratic equations in standard form if we can write ax2 + bx + c as a product of two expressions. To do that, we have to factor!

41. Set each factor equal to zero
Exercise 7
Find the x-intercept(s) of y = x2 – 6x – 7.
Let y = 0
Factor
Set each factor equal to zero
x-intercepts: (7, 0) and (−1, 0)

42. Solving Quadratic Equations
To solve a quadratic equation, try applying the zero product property.
Factor your quadratic
Step 1
Set each factor equal to zero and solve
Step 2

43. Exercise 8
Solve 0 = x2 – x – 42.

44. Same Thing as…
In solving a quadratic function, you must find the x-values that make the function equal to zero.
Solving Quadratics
Same as the roots of the quadratic equation
Same as finding the 𝑥-values of the 𝑥-intercepts
Same as the zeroes of the quadratic function

45. How Many Solutions?
There can be 2, 1, or 0 solutions to a quadratic equation, depending upon where it is in the coordinate plane.

46. Exercise 9 Find the roots of each equation. x2 + 2x = 0

47. Exercise 10
Explain why you cannot use the zero product property to solve every quadratic equation.

48. Exercise 11 Find the zeros of each function. y = 16x2 – 4
y = 9x2 + 12x + 4
y = 5x2 + 16x + 3
y = 2x2 + x + 3

49. Exercise 12 Solve the equation. 4x2 – 17x – 15 = 0
3x2 + 22x + 60 = -14x – 48

50. Example 13
Find the value(s) of x.

51. Exercise 14
Find the value of x if the area of the triangle is 42 square units.

52. Exercise 15
Use substitution to solve the system of equations.

53. Solving Quadratic Equations
Objectives:
To solve a quadratic equation by factoring

54. Warm-Up
Solve for x.
x2 – 10x + 25 = 35

55. Perfect Squares?
The previous exercise was an interesting example, but not every trinomial is a perfect square trinomial. However, we can cleverly rearrange the terms to make any trinomial into a perfect square.
This is called completing the square.

56. Objective 1
You will be able to solve quadratic functions by completing the square

57. Algebra Tiles Activity!
In this activity, we will be using algebra tiles to learn how to complete the square. Don’t worry; unlike most algebra tile activities, this one is actually quite useful. But first, what is an algebra tile?
Geometric representation of x2: x x2 x 1 x: x
Unit Square: 1 x 1 1

58. Algebra Tiles Activity!
You’ll need some tiles, so cut them out.
Use your tiles to represent x2 + 6x.
What you want to accomplish is to make your polynomial model into a square by rearranging your tiles and adding the correct number of unit squares. x2 x x x x x x

59. Algebra Tiles Activity!
To create this square, put half of your x’s on one side of the square and the other half of your x’s on the other side of the square. Now fill in the missing bits with unit squares.
x + 3 x2 x x x
= (x + 3)2
x + 3 x x
= x2 + 6x + 9 x 1 1 1 1 1 1 1 1 1

60. Algebra Tiles Activity!
Now, let’s try again, this time with the following expressions.
Based on your investigation, how could you complete the square given x2 + bx?

61. Algebra Tiles Activity!
In general then, to complete the square, you take half of the middle term and then square it. The number you get becomes the constant term.
x + 3 x2 x x x
= (x + 3)2
x + 3 x x
= x2 + 6x + 9 x 1 1 1
half 1 1 1
square 1 1 1 3

62. Algebra Tiles Activity!
Notice also that the number you get from taking half of the middle term becomes the constant of the binomial that gets squared.
x + 3 x2 x x x
= (x + 3)2
x + 3 x x
= x2 + 6x + 9 x 1 1 1
half 1 1 1
square 1 1 1 3

63. Completing The Square

64. Exercise 1a
Find the value of c that makes x2 – 26x + c a perfect square trinomial. The write the expression as the square of a binomial.

65. Exercise 1b
Find the value of c that makes x2 + 7x + c a perfect square trinomial. The write the expression as the square of a binomial.

66. Exercise 1c
Find the value of c that the expression a perfect square trinomial. The write the expression as the square of a binomial.
x2 + 14x + c
x2 – 22x + c
x2 – 9x + c

67. Exercise 2 Solve by finding square roots. x2 + 6x + 9 = 36

68. Solving Quadratics
Completing the square allows you to solve almost any quadratic equation, regardless of whether it is a perfect square. This is basically an application of the Addition Property of Equality.

69. Solving Quadratics
Solving a Quadratic Equation by Completing the Square:
Use addition/subtraction to write your equation in the form x2 + bx = c.
Complete the square on the quadratic side of the equation and add this number to both sides of the equation.
Factor and take the square root.

70. Exercise 3a
Solve 3x2 – 36x +150 = 0 by completing the square.

71. Exercise 3b Solve the equation by completing the square.
x2 + 6x + 4 = 0
x2 – 10x + 8 = 0
2x2 – 4x – 14 = 0
3x2 + 12x – 18 = 0

72. Exercise 4a
Solve x2 – 9x + 20 = 0 by completing the square.

73. Exercise 4b Solve the equation by completing the square.
x2 + 3x − 5 = 0
x2 – 7x + 13 = 0

74. Exercise 5a
Solve 2x2 – 12x + 7 = 0 by completing the square.

75. Exercise 5b Solve the equation by completing the square.
3x2 + 24x + 41 = 0
x2 – x + 6 = 0

76. Complete The Square Objectives:
To solve quadratic equations by completing the square
x + 3 x2 x x x
= (x + 3)2
x + 3 x x
= x2 + 6x + 9 x 1 1 1
half 1 1 1
square 1 1 1 3

77. Vocabulary Real Number Complex Number Imaginary Unit Imaginary Number
Pure Imaginary Number
Absolute Value
Complex Plane
Complex Conjugates

78. 4.6: Complex Numbers
12/1/2018
Objective 1
You will be able to simplify square roots of negative numbers
Electronic circuits use imaginary numbers

79. Exercise 1 Use a graphing calculator to graph y = x2 + 1.
What are the x-intercepts?

80. Exercise 2 Solve the quadratic equation x2 + 1 = 0.
The problem here is that −1 is not a real number since there is no real number that you can square to get −1.
This does not mean there is no solution; it’s more complex than that.

81. Imaginary Unit
The imaginary unit i can be used to find the square roots of negative numbers. 𝑖= −1 𝑖 2 =−1

82. Exercise 3
A pattern exists as a result of raising i, an imaginary number, to n, an integer greater than or equal to 1. in
Solution i1 −1 i2 i3
− −1 i4 1 i5 i6
According to the table, what is the value of i raised to the 16th power?

83. Taking Powers of i
As the previous example shows, there are only 4 possible values for in.
All other powers of 4 just repeat this pattern.
So, how do you think you would evaluate i101?

84. Taking Powers of i
As the previous example shows, there are only 4 possible values for in.
Divide the exponent by 4, then use the remainder to find the result:
Remainder
Result 1 i 2 −1 3 −i

85. Exercise 4
Evaluate each of the following.
i54
i120
i89
i39

86. Any multiple of 100 is divisible by 4.
Protip: Divisible by 4
How can you tell if a number is divisible by 4?
Any multiple of 100 is divisible by 4.
Note that 100𝑛 is multiple of 4, where 𝑛∈ℤ
Since this is an integer with no remainder, any multiple of 100 is divisible by 4.
100𝑛 4 =
100 4 𝑛=
25𝑛

87. Protip: Divisible by 4 How can you tell if a number is divisible by 4?
If the last two digits of a number are divisible by 4, the whole number is divisible by 4.
132 4 = = =
25+8= 33
Since this is an integer with no remainder, as long as the last two digits are divisible by 4, the whole number is divisible by 4.

88. Protip: Divisible by 4 How can you tell if a number is divisible by 4?
If the last two digits of a number are divisible by 4, the whole number is divisible by 4.
228 4 = = =
50+7= 57
Since this is an integer with no remainder, as long as the last two digits are divisible by 4, the whole number is divisible by 4.

89. Protip: Divisible by 4 How can you tell if a number is divisible by 4?
If the last two digits of a number are divisible by 4, the whole number is divisible by 4.
316 4 = = =
75+4= 79
Since this is an integer with no remainder, as long as the last two digits are divisible by 4, the whole number is divisible by 4.

90. Exercise 5
Simplify each of the following.
−36
−13 𝑖

91. Negative Square Roots The Square Root of a Negative Number Property
Example
If 𝑟 is a positive number, then −𝑟 =𝑖 𝑟
−5 =𝑖 5
𝑖 𝑟 2 =−𝑟
𝑖 = 𝑖 2 ∙5=−5

92. Exercise 6a Find the roots of each quadratic equation. x2 + 1 = 0

93. Exercise 6b Solve each equation. x2 = −13 x2 + 11 = 3 3x2 – 7 = −31

94. 4.6: Complex Numbers
12/1/2018
Objective 2
You will be able to plot complex numbers in the complex plane and find their absolute value
Stephen J. Brooks
Algebraic numbers in the complex plane, colored by degree: red = 1, green = 2, blue = 3, yellow = quartic
Bottom: 0, 1, 2; Top: i

95. Complex Numbers A complex number in standard form is written
All real numbers are complex numbers
This happens when b = 0
For imaginary numbers, b ≠ 0.
For a pure imaginary number, a = 0.
𝑎,𝑏∈ℝ
Real Part
Imaginary Part

96. Exercise 7
Draw a Venn Diagram that represents the set of complex numbers and includes real, imaginary, and pure imaginary numbers.
ℂ: The set of all complex numbers

97. Complex Plane All complex numbers are essentially 2-dimensional.
When you graph a real number, it appears on a 1-D number line

98. Complex Plane All complex numbers are essentially 2-dimensional.
However, complex numbers have both a real and an imaginary part.
Imaginary
Axis
Real Axis

99. Exercise 8 Plot the complex numbers on the same complex plane. 4 + 2i

100. Exercise 9
Plot the complex number 4 + 3i in the complex plane. How could we find its distance from the origin? In other words, how could we find its absolute value?

101. Absolute Value Absolute Value of a Complex Number
The absolute value of a complex number z = a + bi, denoted |z|, is a nonnegative real number defined as

102. Exercise 10a Find the absolute value of each complex number. 5 – 12i

103. Exercise 10a Find the absolute value of each complex number. 4 – i

104. 4.6: Complex Numbers
12/1/2018
Objective 3
You will be able to add, subtract, multiply, and divide complex numbers
Buddhabrot: A certain way to color the Mandelbrot Set

105. Adding and Subtracting
To add or subtract two complex numbers, simply add or subtract their real and imaginary parts separately.
Sum
(a + bi) + (c + di) = (a + c) + (b + d)i
Difference
(a + bi) – (c + di) = (a – c) + (b – d)i

106. Exercise 11a
Write the expression as a complex number in standard form.
(12 – 11i) + (-8 + 3i)
(15 – 9i) – (24 – 9i)
35 – (13 + 4i) + i

107. Exercise 11b
Write the expression as a complex number in standard form.
(9 – i) + (−6 + 7i)
(3 + 7i) – (8 – 2i)
−4 – (1 + i) – (5 + 9i)

108. Multiplying
To multiply complex numbers, you have to use a combination of the distributive property and properties of the imaginary unit.

109. Exercise 12a
Write the expression as a complex number in standard from.
−5i(8 – 9i)
(-8 + 2i)(4 – 7i)

110. Exercise 12b
Write the expression as a complex number in standard from.
i(9 – i)
(3 + i)(5 – i)

111. Exercise 13 Multiply and classify the product. (5i)(−5i)

112. Dividing
To “divide” complex numbers, you have to multiply by a complex conjugate
The complex numbers a + bi and a – bi are complex conjugates
The product of complex conjugates is always a real number

113. Dividing
To “divide” complex numbers, you have to multiply by a complex conjugate
The complex numbers a + bi and a – bi are complex conjugates
The product of complex conjugates is always a real number

114. Exercise 14a
Write the quotient in standard form.

115. Exercise 14b
Write each quotient in standard form.