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 QuadraticsObjective: 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 Quadraticsparabola – 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 Quadraticsaxis of symmetry – the line passing through the vertex about which the parabola is symmetric (the same on both sides).
5 Properties of QuadraticsFind 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 + c1) 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 + c2) 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 + c3) 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?
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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
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29 Factoring Quadratic ExpressionsObjective: 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 Factorgreatest 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
31
32 Finding Binomial Factors𝒙 𝟐 + 14x + 40
33 Finding Binomial Factors𝒙 𝟐 + 12x + 32
34 Finding Binomial Factors𝒙 𝟐 − 11x + 24
35 Finding Binomial Factors𝒙 𝟐 − 17x + 72
36 Finding Binomial Factors𝒙 𝟐 − 14x − 32
37 Finding Binomial Factors𝒙 𝟐 + 3x − 28
38 Finding Binomial Factors𝟐𝒙 𝟐 + 11x + 12
39 Finding Binomial Factors𝟔𝒙 𝟐 − 31x + 35
40 Finding Binomial Factors𝟏𝟐𝒙 𝟐 + 32x − 35
41 Finding Binomial Factors𝟑𝒙 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 AssignmentIn 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 RootsObjective: 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 RootsQuadratic equations in the form 𝒂 𝒙 𝟐 =𝒄 can be solved by finding square roots. 𝟑𝒙 𝟐 = 243
57 Solve Using Square Roots𝟓𝒙 𝟐 − 200 = 0
58 Solve Using Square Roots*𝟒𝒙 𝟐 − 25 = 0
59 Honors Math 2 AssignmentIn 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
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64 Unit 4, Lesson 5: Complex Numbers Objective: To define imaginary and complex numbers and to perform operations on complex numbers
65 Introducing Imaginary NumbersFind the solutions to the following equation:
66 Introducing Imaginary NumbersNow 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, 41-46In the Algebra 2 Textbook, Pgs #s 1-17 odd, odd, 41-46
74 End of Day 5
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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 squareSolve 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 squarex2 + 4x = 21 x2 – 8x – 33 = 0 4x2 + 4x = 3
82 Solving Quadratic Equations: Quadratic FormulaObjective: 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 Formula𝟐𝒙 𝟐 + 5x − 8 = 0
84 Solve using Quadratic Formula𝟑𝒙 𝟐 + 23x + 40 = 0
85 Solve using Quadratic Formula𝟗𝒙 𝟐 +𝟔𝐱−𝟏=𝟎
86 Solve using Quadratic Formula*𝟒𝒙 𝟐 −𝟖𝒙=− 𝟏𝟎
87 Solve using Quadratic Formula𝟐𝟓𝒙 𝟐 −𝟑𝟎𝒙+𝟏𝟐=𝟎
88 Solve using Quadratic Formula𝟑𝒙 𝟐 −𝟐𝒙+𝟒=𝟎
89 Solve using Quadratic Formula𝟐𝒙 𝟐 = -6x – 7
90 Honors Math 2 AssignmentIn 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: GraphingObjective: 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 GraphingSolution
95 Solve Quadratic Equations by GraphingStep 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 Equation𝒚= 𝒙 𝟐 + x − 𝟐 𝒚=−𝒙+𝟑
106 Solve a System with a Quadratic Equation𝒚=𝟐𝒙 𝟐 + x 𝒚= 𝟒 𝟑 𝒙+𝟒
107 Solve a System with a Quadratic Equation𝒚= 𝒙 𝟐 + 𝟒𝒙 + 𝟕 𝒚=−𝟐𝒙
108 Solve a System with a Quadratic Equation𝒚= 𝒙 𝟐 −𝟔x + 𝟏𝟎 𝒚=𝟏
109 Solve a System with Quadratic Equations𝒚= 𝒙 𝟐 − 𝟔𝒙+𝟓 𝒚=−𝟐𝒙 𝟐 +𝟓𝒙
110 Solve a System with Quadratic Equations𝒚= 𝒙 𝟐 + 𝟕𝒙 𝐲= 𝟏 𝟒 𝒙 𝟐 −𝟓𝒙−𝟗 𝒚= 𝒙 𝟐 − 𝟔𝒙+𝟓 𝒚=−𝟐𝒙 𝟐 +𝟓𝒙
111 Honors Math 2 AssignmentIn the Algebra 2 textbook, pp #20-31, 35, 54-56 Solve each quadratic equation or system by graphing.
112 Modeling Data with Quadratic EquationsObjective: 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 Model1) 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 Model3) 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 Model4) 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 AssignmentIn 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, 70-72In 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