1. Chapter 8 – Quadratic Functions and Equations
Class Notes

2. Identifying Quadratic Functions
Lesson 8.1

3. Identifying Quadratic Functions
A quadratic function does not have constant first differences but constant second differences
Standard form of quadratic functions
Y = 𝑎 𝑥 2 +𝑏𝑥+𝑐

4. Graphing Quadratic Functions by Using a Table of Values
Make a table of values. Choose values of x and use them to find values of y X
Y=2 𝑥 2 -2 8 -1 2 1

5. Graphing Quadratic Functions by Using a Table of Values
Graph the points. Then connect the points with a smooth curve.

6. Identifying the Direction of a Parabola
When a quadratic function is written in the form Y = 𝑎 𝑥 2 +𝑏𝑥+𝑐, the value of a determines the direction a parabola opens.
a > 0 – Parabola opens UPWARD (U-Shaped)
a < 0 – Parabola opens DOWNWARD (Rainbow Shaped)

7. Identifying the Vertex and the Minimum or Maximum
The highest or lowest point on a parabola is the vertex
If a > 0, the parabola opens upward, and the y-value of the vertex is the minimum value of the function
If a < 0, the parabola opens downward, and the y-value of the vertex is the maximum value of the function

8. Finding Domain and Range
Unless a specific domain is given, you may assume that the domain of a quadratic function is all real numbers.

9. Characteristics of Quadratic Functions
Lesson 8.2

10. Finding Zeros of Quadratic Functions From Graphs
A zero of a function is an x-value that makes the function equal to 0.
A zero function is the same as an x-intercept of a function.
A quadratic function may have one, two or no zeros.
Identify the zeros below

11. ANSWER 1.
2 and -1 2. 1 3.
No Zeros

12. Finding the Axis of Symmetry by Using Zeros
A vertical line that divides a parabola into two symmetrical halves is the axis of symmetry
ONE ZERO
If a function has one zero, use the x-coordinates of the vertex to find the axis of symmetry
TWO ZEROS
If a function has two zeros, use the average of the two zeros to find the axis of symmetry

13. Finding the Axis of Symmetry by Using Zeros
Identify the axis of symmetry for each graph

14. ANSWER 1. -3 2. 1

15. Finding the Axis of Symmetry by Using the Formula
If a function has no zeros or they are difficult to identify from a graph, you can use a formula to find the axis of symmetry.

16. Finding the Axis of Symmetry by Using the Formula
Identify the axis of symmetry for each equation by using the formula

17. ANSWER
-1/4

18. Finding the Vertex of a Parabola
Step 1
To find the x-coordinate of the vertex, find the axis of symmetry by using zeros or the formula
Step 2
To find the corresponding y-coordinate, substitute the x-coordinate of the vertex into the function
Step 3
Write the vertex as an ordered pair

19. Finding the Vertex of a Parabola (Example)

20. Finding the Vertex of a Parabola
Find the vertex by using the Axis of Symmetry Formula

21. ANSWER
(2, -14)

22. Additional Practice Workbook Page 425
DUE Tomorrow when class begins (Put in Class Folder upon entering class)
If you complete it in class, place it in your Class Folder on your way out

23. Graphing Quadratic Functions
Lesson 8.3

24. Graphing a Quadratic Function
Recall Standard Form of Quadratic Function
Y = 𝑎 𝑥 2 +𝑏𝑥+𝑐
Remember that when x=o, y=c
The y-intercept of a quadratic function is c.

25. Graphing a Quadratic Function
Step 1:
Find the axis of symmetry (Use Formula from 8.2)
Step 2:
Find the vertex (Substitute your x-coordinate into your function and solve for y
Step 3:
Find your y-intercept (Identify c)

26. Graphing a Quadratic Function
Step 4:
Find two more points on the same side of the axis of symmetry as the point containing the y-intercept (Choose values less than your axis of symmetry)
Substitute x-coordinates
Step 5:
Graph the axis of symmetry, the vertex, the point containing the y-intercept and two other points.
Reflect the points across the axis of symmetry and connect points with smooth curve

27. Graphing a Quadratic Function
Graph quadratic function and label your steps 1-5 on your whiteboard and raise when you are finished

28. ANSWER

29. Additional Practice Workbook page 429
DUE Tomorrow (Place in class folder as you walk into class)

30. Warm Up 2 𝑥 2 +5𝑥+2 Find a Find b Find − 𝑏 2𝑎
Find the Axis of Symmetry
Find the Vertex

31. Transforming Quadratic Functions
Lesson 8.4

32. Comparing Widths of Parabolas
The value of a in a quadratic function determines not only the direction a parabola opens, but also the width of the parabola

33. Comparing Widths of Parabolas
Order the functions in order from most narrow to the widest

34. Comparing Graphs of Quadratic Functions
The value of c makes these graphs look different

35. Comparing Graphs of Quadratic Functions
Two Methods
Comparing the graphs
Comparing the functions

36. Additional Practice Workbook page 435
DUE Friday (Place in class folder as you walk into class)

37. TEST
Get ready and study for test on Lesson 8.1 – 8.4

38. Solving Quadratic Equations by Graphing
Lesson 8.5

39. Solving Quadratic Equations by Graphing

40. Solving Quadratic Equations by Graphing
2𝑥 2 −2=0
Write the Related function
2𝑥 2 −2=𝑦 𝑜𝑟 𝑦= 2𝑥 2 +0𝑥−2
Graph the function
Axis of Symmetry = 0
Vertex = (0,-2)
Two other points = (1,0) and (2,6)
Graph the points and reflect them across the axis of symmetry
Find the zeros
The zeros appear to be -1 and 1

41. Solving Quadratic Equations by Graphing

42. Solving Quadratic Equations by Factoring
Lesson 8.6

43. Using the Zero Product Property

44. Using the Zero Product Property
(x – 3)(x + 7) = 0
Use the zero property
x – 3 = 0 …. x = 3
x + 7 = 0 …. x = -7
The solutions are 3 and -7
Can always check your work by plugging each solution for x into the original equation

45. Solving Quadratic Equations by Factoring
If a quadratic equation is written in standard form, you may need to factor before using the Zero Product Property
𝑥 2 +7𝑥+10=0
𝑥+5 𝑥+2 =0
𝑥+5=0 …𝑥=−5
𝑥+2=0 …𝑥=−2
The solutions are -5 and -2

46. Solving Quadratic Equations by Factoring
−2𝑥 2 =18−12𝑥
−2𝑥 2 +12𝑥−18=0
−2 𝑥 2 −6𝑥+9 =0
−2 𝑥−3 𝑥−3 =0
−2≠0, 𝑥=3

47. Additional Practice
Workbook page 451

48. Solving Quadratic Equations by Using Square Roots
Lesson 8.7

49. Using Square Roots to Solve 𝑥 2 =𝑎

50. Using Square Roots to Solve 𝑥 2 =𝑎
When you take the square root of a positive real number and the sign of the square root is not indicated, you must find both the positive and negative square root. This is indicated by ±√
𝑥 2 =16
𝑥=± 16
𝑥=±4
The solutions are 4 and -4

51. Using Square Roots to Solve 𝑥 2 =𝑎
𝑥 2 =−4
𝑥=± −4
There is no real number whose square is negative
There is no real solution

52. Using Square Roots to Solve Quadratic Equations
If necessary, use inverse operations to isolate the squared part of a quadratic equation before taking the square root of both sides
𝑥 2 +5=5
𝑥 2 =0
𝑥=± 0 =0
The solution is 0

53. Using Square Roots to Solve Quadratic Equations
4𝑥 2 −25=0
4𝑥 2 =25
𝑥 2 = 25 4
𝑥=±
𝑥=± 5 2
The solutions are 𝑎𝑛𝑑 − 5 2

54. Additional Practice Workbook Page 457 Finish Project
Standards HRW DUE 2/20

55. Completing the Square
Lesson 8.8

56. Completing the Square
When a trinomial is a perfect square, there is a relationship between the coefficient of the x-term and the constant term.
An expression in the form 𝑥 2 +𝑏𝑥 is not a perfect square. However, you can use the relationship shown above to add a term to 𝑥 2 +𝑏𝑥 to form a trinomial that is a perfect square. This is called completing the square.

57. Completing the Square
𝑥 2 +10𝑥+
𝑥 2 +10𝑥
= 5 2 =25
𝑥 2 +10𝑥+25

58. Solving a Quadratic Equation by Completing the Square

59. Solving 𝑥 2 +𝑏𝑥=𝑐 by Completing the Square
𝑥 2 +14𝑥=15
= 7 2 =49
𝑥 2 +14𝑥+49=15+49
𝑥+7 2 =64
𝑥+7=±8
𝑥+7=8 or 𝑥+7=−8
𝑥=−1 𝑜𝑟 𝑥=−15

60. Solving 𝑎𝑥 2 +𝑏𝑥=𝑐 by Completing the Square

61. Additional Practice
Guided Practice p. 579 #’s 2-32 even
Test Thursday

62. The Quadratic Formula and the Discriminant
Lesson 8.9

63. Using the Quadratic Formula

64. Using the Quadratic Formula

65. Using the Quadratic Formula to Estimate Solutions

66. Using the Discriminant
If the quadratic equation is in standard form, the discriminant of a quadratic equation is 𝑏 2 −4𝑎𝑐, the part of the equation under the radical sign

67. Using the Discriminant

68. Using the Discriminant

69. Solving Using Different Methods
Factoring
Completing the Square
Using the Quadratic Formula

70. Additional Practice
Workbook page 477

71. Nonlinear Systems
Lesson 8.10

72. Solving a Nonlinear System by Graphing
A nonlinear system of equations is a system in which at least one of the equations is nonlinear

73. Solving a Nonlinear System by Graphing

74. Solving a Nonlinear System by Substitution

75. Solving a Nonlinear System by Elimination

76. Additional Practice
Workbook page 485
Quiz Tuesday 2/20 on 8.5 – 8.10