1. Quadratic Functions and Factoring
Unit 4
Quadratic Functions and Factoring

2. Unit Essential Question:
What are the different ways to graph a quadratic function and to solve quadratic equations?

3. Graphing Quadratic Functions in standard Form
Lesson 4.1
Graphing Quadratic Functions in standard Form

4. Lesson Essential Question
How do we graph a quadratic function in standard form?

5. Standard Form of a Quadratic Function:
𝑦=𝑎 𝑥 2 +𝑏𝑥+𝑐
The graph of a quadratic function is a parabola.

6. Parent Function for Quadratic Functions
𝑦= 𝑥 2

7. To graph:
Find the vertex. The x-value of the vertex = − 𝑏 2𝑎 , then use this x-value to find the y-value.
Find the y-intercept. Substitute 0 in for x, and solve for y.
Use the axis of symmetry, 𝑥=− 𝑏 2𝑎 , to reflect the y-intercept to find a third point.

8. Pay attention to the value of a.
If a > 0, then the graph opens up.
If a < 0, then the graph opens down.

9. Example 1: Graph 𝑦=𝑎 𝑥 2 +𝑏𝑥+𝑐

10. What if the vertex is on the y-axis?
Then find another point to use instead of the y-intercept. Substitute another number in for x, and solve for y. Then reflect this point across the axis of symmetry.

11. Example 2: Graph 𝑦=𝑎 𝑥 2 +𝑐
Graph 𝑦=− 1 2 𝑥 2 +3.

12. On Your Own 𝑦= 𝑥 2 −2𝑥−1 𝑦=2 𝑥 2 +6𝑥+3 𝑓 𝑥 =− 1 3 𝑥 2 −5𝑥+2
𝑓 𝑥 =− 1 3 𝑥 2 −5𝑥+2
Do not Graph
Find the Vertex
Y-intercept
3rd Point

13. 1) ) )

14. Minimum/Maximum Values
A parabola can have either a minimum or maximum value depending upon the value of the x² term. If a is positive, then the function will have a minimum. If a is negative, then it will have a maximum.
The minimum or maximum value of a given quadratic function will be equal to the y-value in the vertex!

15. Example 3: Min or Max
Tell whether the function 𝑦=3 𝑥 2 −18𝑥+20 has a minimum value or a maximum value. Then find the minimum or maximum value.
Because a > 0, the function has a minimum value. To find it, calculate the coordinates of the vertex.
The minimum value is 𝑦=−7.

16. On Your Own
Find the minimum or maximum value of 𝑦=4 𝑥 2 +16𝑥−3.

17. Example 4:
Amber is selling sticky buns at the Bloomsburg Fair to raise money for the softball team. She charges $5 per sticky bun, and sells about 800 a day. She found that for every time she raises the price by $0.25, she sells 20 less sticky buns. Write a quadratic function and find out how she can maximize her revenue.

18. Example 5:
Papa John’s in Bloomsburg sells about 500 pizzas in a day when they charge $12. For every time they decrease the price by $0.50, they sell 60 more pizzas. Find the price per pizza and number of pizzas sold each day that will maximize their profit.

19. Example 6:
The height h (in feet) of a toy rocket, t seconds after it is launched is given by ℎ=−16 𝑡 𝑡. What is the maximum height the rocket reaches and how long did it take to get there?

20. 4.1 Homework:
Pages 240 – 242 #’s 11 – 39 odds, 55 – 59 (59 part a only, then find how they can maximize the profit), 60 (part a only, then find the maximum height of the ball on Earth and on the Moon), 61

21. Bell Work Graph the quadratic function 𝑦=3 𝑥 2 −6𝑥+4 𝑔 𝑥 =−2 𝑥 2 −5
𝑦=− 3 4 𝑥 2 −4𝑥−1

22. 4.1 Quiz Monday
Graphing Quadratic Functions Minimum/Maximum Values Revenue

23. Review Graph the quadratic function: 𝑦= 𝑥 2 −6𝑥+5 𝑓 𝑥 =−2 𝑥 2 +8
𝑔 𝑥 =−(𝑥−1)(𝑥−5)
𝑦= 3𝑥 2 −9

24. Review
5) An electronics store sells about 100 digital cameras per week at a price of $450 each. For each $20 decrease in price, they sell about 6 more cameras per week. How can the electronics store maximize their weekly revenue?

25. Review
6) The equation 𝑦=− 1 4 𝑥(𝑥−30) describes that pattern of a baseball being thrown where x is the horizontal distance in feet and y is the vertical distance in feet. What was the maximum height of the ball?

26. Review
7) A football is thrown straight upwards with an initial speed of 64ft/sec. The height (h) in feet above the ground after time (t) is given by h=−8 𝑡 2 +32𝑡. a) At what time will the football be at its maximum height? b) What is the maximum height it reaches?

27. Graphing Quadratic functions in Vertex form and intercept form
Lesson 4.2
Graphing Quadratic functions in Vertex form and intercept form

28. Lesson Essential Question:
What is vertex form and intercept form for a quadratic function and how can we use it?

29. Vertex Form
𝑦=𝑎(𝑥−ℎ ) 2 +𝑘
Where:
The vertex is (h,k).
The axis of symmetry is x = h.
The graph opens up if a > 0 and down if a < 0.

30. Example 1: Vertex Form
Graph 𝑦=3 𝑥−7 2 −1.

31. Example 2: Vertex Form
Graph 𝑦=−4 𝑥−

32. On Your Own Graph the function. 𝑦= 𝑥+2 2 −3 𝑦=− 𝑥−1 2 +5
𝑦= 𝑥+2 2 −3
𝑦=− 𝑥−
𝑓 𝑥 = 𝑥−3 2 −4 1

33. Intercept Form 𝑦=𝑎(𝑥−𝑝)(𝑥−𝑞) Where: p and q are the x-intercepts.
The axis of symmetry is halfway between p and q.
The graph opens up if a > 0 and down if a < 0.

34. Example 2: Intercept Form
Graph 𝑦= 𝑥+1 𝑥−3 .

35. Example 2: Intercept Form
Graph 𝑦=3 𝑥+2 𝑥+6 .

36. On Your Own Graph the function. 𝑦= 𝑥−3 𝑥−7 𝑓 𝑥 =2 𝑥−4 𝑥+1
𝑦= 𝑥−3 𝑥−7
𝑓 𝑥 =2 𝑥−4 𝑥+1
𝑦=−(𝑥+1)(𝑥−5)

37. Example 3: Intercept to Standard
Write 𝑦=4 𝑥+1 𝑥−6 in standard form.
𝑦=3 𝑥 2 +6𝑥−72

38. Example 4: Vertex to Standard
Write 𝑓 𝑥 =12 𝑥− in standard form.
𝑓 𝑥 =− 1 2 𝑥 2 −8𝑥+3

39. On Your Own
𝑦=3(𝑥+4)(𝑥+3)
𝑦=− 𝑥

40. Example 5: Finding a Function
Write an equation for a quadratic function based upon the given information:
Vertex (-3,4) and passes through the point (2,-71)
x-intercepts at 5 and -1 and passes through the point (-3,4)
x – intercept at -4 and a maximum value at (2,180).

41. Example 6: Word Problems
The path of a frogs jump can modeled as a quadratic function. The length of the jump is 8 feet and the maximum height during the jump is 3 feet. Write an equation in standard form to model the jump.
The flight of a projectile is modeled as ℎ=−16 𝑡 2 +96𝑡 where h is the height in yards and t is the time in seconds. How long will it take for the object to hit the ground?

42. Homework:
Page # 5, 8, 11, 13, 16, 19, (odd) 51, 53, 55 (a and b), 56

43. Bell Work: Sketch the graph of each quadratic function: 𝑦=− 𝑥 2 +3𝑥+4
𝑦= 1 2 (𝑥−2 ) 2
𝑦=−2𝑥(𝑥+2)

44. Bell Work:
Jessica is performing the long jump for the track team. She is able to jump a total length of 18 feet and reach a maximum height of 3 feet. Write a quadratic function to model Jessica’s jump.
A suspension bridge is shown on the board. The two main supports have a height of 90 feet and the entire bridge has a length of 500 feet. If the lowest point of the suspension cable is 45 feet above the bridge, write a quadratic function to model the suspension cable.

45. Quiz Tomorrow! Quadratic Functions: Standard Form Vertex Form
Intercept Form
Minimum/Maximum
Word Problems
You need to have the forms memorized!!!

46. On Your Own:
Graph the following quadratic functions using the given form:
𝑦=−3 𝑥 2 +6𝑥−2
𝑦=2(𝑥+4 ) 2 −3
𝑦=−(𝑥+3)(𝑥−7)

47. On Your Own: Sketch the graphs of the following functions:
𝑦=− 1 2 𝑥 2 +4𝑥+2
𝑦= 1 3 (𝑥+2 ) 2 −6
𝑦= 1 2 (𝑥−4)(𝑥+4)

48. On Your Own:
Find the equation for a quadratic function if it has a vertex of (2,-4) and passes through the point (4,12).
Find the equation of the quadratic function if is has x- intercepts at -2 and 4, and also passes through the point P(6,-32).

49. Example 9
The path that a football travels on the opening kickoff is given as 𝑦=−0.026𝑥(𝑥−46) where x is the horizontal distance the ball traveled in yards, and y is height in yards.
How far was the football kicked?
What was the maximum height of the football?

50. On Your Own:
10) The path of a frog jumping can be modeled by the equation 𝑦=−0.025𝑥(𝑥−6) where x is the horizontal distance in feet and y is the vertical distance in feet from its starting point.
How far can the frog jump?
How high can the frog jump?

51. Example 11:
Megan sells lemonade to save money for a trip to the Bahamas. She sells lemonade for $1 a cup, and sells about 100 cups a week. When she raises the price by $0.10, she sells 4 less cups each week. Write a quadratic function to model Megan’s income. How can Megan maximize her revenue?

52. On Your Own: Sketch the graphs of the following functions:
𝑔(𝑥)=2 𝑥 2 +4𝑥+2
𝑦=− 1 2 (𝑥+3 ) 2 +5
𝑓(𝑥)=2 (𝑥−3)(𝑥+3)
𝑦=−3 𝑥 2 +8

53. On Your Own:
A quadratic function has an x-intercept at -2 and a vertex of (2,64). Write the equation for this function in standard form.
A quadratic function has x-intercepts at -3 and -9, and it passes through the point (2,165). Write the equation for this function in intercept form.
A quadratic function has a vertex V(4,8) and passes through the point (8,4). Write the equation for this function in vertex form.

54. Examples:
Abby jumps from point A to point B 12 feet away. Her maximum height during the jump was 6 feet. Write the equation for a quadratic function that describes her jump.
The equation for how Abby the prairie dog jumps is given as 𝑦= −0.55𝑥 𝑥−30 where x is the horizontal distance traveled in inches, and y is the vertical distance traveled in inches. Find the maximum height that Abby the prairie dog can jump, and how far she can jump.

55. On Your Own:
Abby was walking underneath an arch, and found that the equation that models the arch is 𝑦=−30 𝑥 2 −1500𝑥 where x is the horizontal distance in feet and y is the vertical distance in feet. What is the height of the arch at its apex?
Why would the height to the top of the arch be same from ¼ of the way across the arch as ¾? Prove that it is the same.
How wide is the arch?

56. Solving Quadratic equations by factoring when a=1.
Lesson 4.3
Solving Quadratic equations by factoring when a=1.

57. Lesson Essential Question
How do we factor trinomials and binomials, and how will this help us solve quadratic equations?

58. To factor 𝑥 2 +𝑏𝑥+𝑐, find the integers m and n such that:
Factoring: 𝒙 𝟐 +𝒃𝒙+𝒄
To factor 𝑥 2 +𝑏𝑥+𝑐, find the integers m and n such that:
𝑥 2 +𝑏𝑥+𝑐= 𝑥+𝑚 𝑥+𝑛 = 𝑥 2 + 𝑚+𝑛 𝑥+𝑚𝑛
What factors of c add up to b?

59. Example 1: Factoring 𝑥 2 +𝑏𝑥+𝑐
𝑥 2 +20𝑥+64
𝑥 2 −12𝑥+27
𝑥 2 −2𝑥−120

60. On Your Own: Factor the expression. 𝑥 2 −9𝑥+20 𝑥 2 −𝑥 −12 𝑥 2 −3𝑥−18
𝑟 2 +2𝑟−63

61. Special Factoring Patterns
Difference of Two Squares
𝑎 2 − 𝑏 2 =(𝑎+𝑏)(𝑎−𝑏)
Example: 𝑥 2 −4=(𝑥+2)(𝑥−2)

62. Special Factoring Patterns
Perfect Square Trinomials
𝑎 2 +2𝑎𝑏+ 𝑏 2 = 𝑎+𝑏 𝑎 2 −2𝑎𝑏+ 𝑏 2 = 𝑎−𝑏 2
Examples:
𝑥 2 +6𝑥+9= 𝑥+3 2
𝑥 2 −4𝑥+4= 𝑥−2 2

63. Example 2: Factor with Special Patterns
Factor the expression.
𝑥 2 −49
𝑑 2 +12𝑑+36
𝑥 2 −121
(x+7)(x-7)
𝑑+6 2
(𝑥+11)(𝑥−11)

64. Solving Quadratic Equations:
Three Easy Steps:
Make sure the equation is set equal to zero.
Factor the equation as much as possible.
Set each factor equal to zero and solve.
Your answers to each equation are the roots of the equation.

65. Finding Zeros of Quadratic Functions
A regular quadratic function 𝑦=𝑎 𝑥 2 +𝑏𝑥+𝑐 has zeros if it crosses the x-axis. The zeros are x-intercepts.
If the function has only one zero, then that x-intercept is the vertex!!!
What if the function is unfactorable? What does that tell us about the function?

66. Example 3: Finding zeros
Find the zeroes of the function.
y= 𝑥 2 −𝑥−42
𝑦= 𝑧 2 −11𝑧+28
𝑥=−6 and 𝑥=7; The zeroes of the function are -6 and 7.
𝑧=7 and 𝑧=4; The zeroes of the function are 7 and 4.

67. On Your Own: Find the zeroes of the function. 𝑦= 𝑥 2 +5𝑥−14
𝑦= 𝑥 2 −7𝑥−30
𝑓 𝑥 = 𝑥 2 −10𝑥+25
-7, 2
-3, 10 5

68. Little More Challenging:
Solve each quadratic equation by factoring:
𝑥 2 −90=−9
3 𝑥 2 −𝑥−30=2 𝑥 2 +12
𝑥 2 +9𝑥−10=−10
𝑥 2 −10𝑥−12=2𝑥−48

69. Example 4: Application Problem
A city’s skate park is a rectangle 100 feet long by 50 feet wide. The city wants to triple the area of the skate park by adding the same distance x to the length and the width. Write and solve an equation to find the value of x. What are the new dimensions of the skate park?
= 50+𝑥 100+𝑥 ;−200,50;
100 ft by 150 ft

70. Homework:
Page #1-39 Odd, 43 – 63 odd, 66 – 67, and 69 – 71

71. Bell Work:
Page 258 # 72

72. Solving Quadratic equations by factoring when a≠1.
Lesson 4.4
Solving Quadratic equations by factoring when a≠1.

73. Lesson Essential Question:
How do we factor trinomials differently when the a value does not equation 1?

74. Example 1: Master Product Method
Factor.
2 𝑥 2 +5𝑥−12

75. Example 2: Master Product Method
Factor.
6 𝑥 2 +17𝑥+5

76. On Your Own: Factor. 14 𝑥 2 −53𝑥+14 9 𝑥 2 +48𝑥+64 6𝑥 2 +11𝑥+3
32𝑥 2 −52𝑥+15
−20𝑥

77. On your own: Factor each expression: 6𝑥 2 +11𝑥+3 32𝑥 2 −52𝑥+15
−20𝑥

78. Example 3: Solving Quadratic Equations
Solve:
3 𝑥 2 −10𝑥−25=0
2 𝑥 2 +23𝑥=12

79. On Your Own:
Solve:
4𝑥 2 +8𝑥−96=0
25𝑥 2 −49=0
30𝑥 2 +50𝑥=0

80. On Your Own: Find the zeros for each function: 𝑓 𝑥 =4 𝑥 2 −33𝑥+50
𝑔 𝑥 =25 𝑥 2 −4
ℎ 𝑥 =−12 𝑥 2 +9𝑥
𝑘 𝑥 =16 𝑥 2 +40𝑥+25

81. On Your Own: Solve each equation: 5 𝑥 2 −3𝑥=8 18 𝑥 2 −12𝑥+10=2𝑥(4𝑥+4)
48 𝑥 2 +10𝑥=−2𝑥+90

82. Examples:
A 10x12 inch picture is to be framed. The thickness of the frame on all sides will be the same. If the area of just the frame is 75 square inches, find the thickness of the frame.
A box with an open top is to be constructed by cutting 3 inch squares from the corners of a rectangular piece of cardboard whose length is twice its width. What size piece of cardboard would produce a box with a volume of 60 cubic inches?

83. Homework:
Pages 263 – 265 #’s 33 – 61 odds and 62– 67

84. Bell Work: Find the zeros for each function: 𝑓 𝑥 = 12𝑥 2 +9𝑥−30
𝑔 𝑥 = 40𝑥 2 −37𝑥+4
ℎ 𝑥 = 18𝑥 2 −32
𝑘 𝑥 = −24𝑥 2 −18𝑥

85. 4.3-4.4 Quiz Tomorrow Factoring when a does and does not equal 1
Finding zeros
Word Problems

86. Examples: Find the zeros for each function: 𝑓 𝑥 = 𝑥 2 −14𝑥−120
𝑔 𝑥 = 𝑥 2 −32𝑥+87
ℎ 𝑥 = 20𝑥 2 +9𝑥−18
𝑘 𝑥 = 27𝑥 2 −75

87. Examples:
Write a quadratic equation in the form 𝑥 2 +𝑏𝑥+𝑐=0 that has zeros of -8 and 11. For what integer(s) for b can the expression 𝑥 2 +𝑏𝑥+12 be factored?

88. Examples:
The length of a rectangle is (3𝑥−2) and the width is (𝑥+4). If the area of a rectangle is 209 square feet, write and solve a quadratic equation to find the value(s) of x.

89. Examples:
A farmer has a fenced in field that measures 40x50 yards. He decides to increase the length and width by a certain amount (x). If the area was increased by 1575 square yards, find the new dimensions of the fenced in field.
A picture has a length that is twice its width. If you put a frame around the picture with a uniform width of four inches, and the total area of the picture and the frame is 640 square inches, find the dimensions of the picture.

90. On Your Own: Solve each quadratic equation: 4𝑥−20= 𝑥 2 −5𝑥
12 𝑥 2 −3𝑥+5=−2 𝑥 2 −60𝑥+32
45 𝑥 2 −4𝑥=245−4𝑥

91. On Your Own:
The area of a rectangle is 360 square miles. If the length of the rectangle is six less than twice the width, find the dimensions of the rectangle.
The area of a triangle is 360 square kilometers. If the height is two more than half the base, find the dimensions of the triangle.
A rectangular piece of tin has a length that three times its width. Four five inch squares are going to be cut out of the corners of the rectangle so that the sides can be folded to form a box with an open top. If the volume of the box is 16,500 cubic inches, find the dimensions of the original rectangular piece.

92. Bell Work:
A suspension bridge has supports that rise up 90 feet from the bridge, and there is 400 feet between each support. The lowest point of the suspension cable connecting the two supports is 10 feet above the road. a) Write an equation in vertex form for the parabolic shape of the suspension cable.

93. Unit 4 Test (Part 1) Tomorrow
Graphing Quadratics in various forms
Solving Quadratics
Revenue
Minimum/Maximum

94. Bonus for Unit 4 Test 1:
For what values of c can the following expression be factored if we know that c is positive?
𝑥 2 +11𝑥+𝑐
You must have all values to receive credit!