Name: Teacher: Per: Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7 Unit 8 Unit 9 Unit 10 Unit 9a [Quadratic Functions] Unit 9 Quadratics 1
To be a Successful Algebra class, TIGERs will show #TENACITY during our practice, have I attempt all practice I attempt all homework I never give up when I don t understand #INTEGRITY as we help others with their work, maintain a #GO-FOR-IT attitude, continually I always check my answers I correct my work, I never just copy answers I explain answers, I never just give them I write down all notes, even if I m confused I remain positive about my goals I treat each day as a chance to reset #ENCOURAGE each other to succeed as a team, and always #REACH-OUT and ask for help when we need it! I offer help when I understand the material I push my teammates to reach their goals I never let my teammates give up I ask my questions during homework check I ask my teammates for help during practice I attend enrichment/tutorials when I need to Unit 9 Quadratics 2
Unit Calendar MONDAY TUESDAY WEDNESDAY THURSDAY FRIDAY March 16 17 18 19 20 Domain and Range for Discrete and Continuous Functions Introduce Quadratic Graph and Transformations Identify Key Features from the Table Identify Key Features from the Graph QUIZ Identify Key Features from the Calculator 23 24 25 26 27 Mixed Practice Applications Review TEST A DLA 30 31 March 1 2 3 Solve by Factoring Solve by Factoring Solve by Quadratic Formula English I EOC Holiday 6 7 8 9 10 Solve all 4 Ways Solve Practice Classify Functions Review QUIZ TEST B Essential Questions What are the similarities and differences between a linear and quadratic function? What do zeroes, solutions, roots, and x-intercepts have in common? How do they differ? Unit 9 Quadratics 3
Critical Vocabulary Quadratic Parabola Roots Zeroes x-intercepts Solutions Vertex Axis of Symmetry Unit 9 Quadratics 4
Domain and Discrete and Continuous Discrete A relation/function/situation that is represented by. Continuous A relation/function/situation that is represented by a. Domain For a discrete relationship, it is a list of the. Range For a discrete relationship, it is a list of the. For a continuous relationship, it is written as an inequality from Unit 2: Unit 9 Quadratics 5
Reminders: Discrete {(2, 3), (4, 6), (3, 1), (6, 6), (2, 3)} Find the range of this function for the given domain. f ( x) 5x 2 1 domain: range: domain: { -4, 1, 5 } range: x y -3 5-2 5-1 5 0 5 1 5 2 5 domain: range: domain: range: Unit 9 Quadratics 6
Examples: Continuous Graphs that continue onward in at least one direction: It is possible for either the domain or the range to include every possible number, which we write as All Real Numbers otherwise, it is written as an inequality. for Graphs that have the domain and/or range bounded on both sides: When bounded on both sides, we use a special double inequality to demonstrate the two endpoints that the x- or y-values fall between. Unit 9 Quadratics 7
Unit 9 Quadratics 8
Unit 9 Quadratics 9
Quadratic Function: The Parabola and Transformations Parent Functions: The most BASIC form of a graph Linear y = x Quadratic y = x 2 x -3-2 -1 0 1 2 3 f(x) x -3-2 -1 0 1 2 3 f(x) y = mx + b y = ax 2 + c Changes how or the line is. - intercept Changes how or the parabola is. - intercept Unit 9 Quadratics 10
Examples: What happens when we change c in ax 2 + c? y = x f(x) = x + 1 y = x + 3 f(x) = x 5 y = x 2 f(x) = x 2 + 1 y = x 2 + 3 f(x) = x 2 5 We notice that the c is the -. What happens when we change a in ax 2 + c? f(x) = - x y = 2x f(x) = 1 2 x y = - 4x f(x) = - x 2 y = 2x 2 f(x) = 1 2 x2 y = - 4x 2 We notice that when a is the graph opens. We notice that when a is the graph opens. Ignoring the positive or negative sign We notice that when a is than 1, the parabola is. We notice that when a is than 1, the parabola is. Unit 9 Quadratics 11
Practice: Equation Width Translation Reflection y = 1 8 x2 + 5 y = 4x 2 y = x 2 27 y = 3x 2 8 y = 3 2 x2 6 y = 2x 2 + 4 y = 1 4 x2 Narrower / Wider / No Change Shift up / Shift Down / No Change Reflection / No Change Narrower / Wider / No Change Shift up / Shift Down / No Change Reflection / No Change Narrower / Wider / No Change Shift up / Shift Down / No Change Reflection / No Change Narrower / Wider / No Change Shift up / Shift Down / No Change Reflection / No Change Narrower / Wider / No Change Shift up / Shift Down / No Change Reflection / No Change Narrower / Wider / No Change Shift up / Shift Down / No Change Reflection / No Change Narrower / Wider / No Change Shift up / Shift Down / No Change Reflection / No Change Put the 7 equations above in order from Narrowest to Widest:,,,,,,, If you translate the equation y = 2x 2 + 4 up 2 units, what is the new equation?. If you translate the equation y = x 2 27 down 3 units, what is the new equation?. If you reflect the equation y = 3 2 x2 6, what is the new equation?. Write an equation that is wider than y = 1 4 x2. Unit 9 Quadratics 12
Unit 9 Quadratics 13
Quadratic Functions: Key Features from the Graph Quadratic y = x 2 y = ax 2 + c Changes how or the parabola is. - intercept Examples: f(x) = -x 2 + 4 Roots: Transformations: y = 3x 2 3 Zeroes: Transformations: y = 4x 2 + 1 x-int: Transformations: f(x) = 1 2 x2 Solutions: Transformations: Roots: y-int: Unit 9 Quadratics 14
Practice: Roots: y-int: Zeroes: y-int: y = -x 2 + 1 f(x) = 4x 2 4 y = -5x 2 + 5 f(x) = x 2 + 4 Roots: Transformations: Zeroes: Transformations: x-int: Transformations: Solutions: Transformations: Unit 9 Quadratics 15
Quadratic Functions: Key Features from the Table x y -2 5-1 0 0-3 1-4 2-3 3 0 4 5 Roots: 0 X S Examples: f(x) = -x 2 + 6x 5 y = x 2 10x + 25 x f(x) 0-5 1 0 2 3 3 4 4 3 5 0 6-5 x y 2 9 3 4 4 1 5 0 6 1 7 4 8 9 Roots: x-intercept(s): g(x) = x 2 + 2x 3 h(x) = x 2 4x x g(x) -4 5-3 0-2 -3-1 -4 0-3 1 0 2 5 x -2-1 0 1 2 3 4 h(x) Solutions when g(x) = 0: Solutions when g(x) = 5: Solutions when h(x) = 0: Solutions when h(x) = -3: Unit 9 Quadratics 16
Practice: f(x) = x 2 8x + 7 y = -x 2 6x 9 x f(x) 1 0 2-5 3-8 4-9 5-8 6-5 7 0 x y -6-9 -5-4 -4-1 -3 0-2 -1-1 -4 0-9 Roots: Zeroes: Solutions when f(x) = -8 Solutions when y = -4 g(x) = -x 2 2x h(x) = x 2 + 6x + 8 x g(x) -5-15 -4-8 -3-3 -2 0-1 1 0 0 1-3 x -5-4 -3-2 -1 0 1 h(x) x-intercepts: Solutions when h(x) = 0 FACTOR!!! (NO, you can t forget this) n 2 + 11n + 10 2x 2 + 11x + 14 2 6x 19x 10 x 2 + 14x + 48 2 2 5x 17x 6 4x 16x 15 Unit 9 Quadratics 17
Quadratic Functions: Key Features from the Calculator Quadratics on the Graphing Calculator: Examples: Calculator On (You can press ON to return to the HOME SCREEN at any time) Press NEW DOCUMENT, select NO TO SAVE CHANGES, select GRAPH Type in equation f1(x) =, ENTER Press CTRL T to bring up table (* press CTRL T again to remove the table if needed) Find the Vertex and ROXS (roots, zeroes, x-intercepts, solutions) in the table VERTEX: find where table becomes symmetrical ROXS: find where y=0 or changes sign f(x) = x 2 + 8x + 15 y = -x 2 + 4 x f(x) x y Roots: Zeroes: g(x) = x 2-16 h(x) = -x 2 4x + 5 x g(x) Solutions: x-intercepts: Unit 9 Quadratics 18
Practice: f(x) = x 2 16x + 63 y = -x 2 + 9 x f(x) x y Roots: Zeroes: g(x) = -x 2 + 2x - 3 h(x) = x 2 + 14x + 48 Solutions: Solutions: FACTOR!!! (NO, you can t forget this) n 2 + 4n 12 a 2 13a 30 7x 2-5x 2 2x 2 6x + 4 2 5x 7x 6 3x 2 + 15x + 18 Unit 9 Quadratics 19
Quadratic Functions: Key Features Scavenger Hunt Graph the equation f(x) = x 2 + 4x 12 Roots: Axis of Symmetry: Examples: Use the table to answer the questions. Given the following function, Given the following function, x -3 12-2 5-1 0 0-3 1-4 2-3 3 0 y f(x) = x 2 13x 40 What are the roots? y = 3x 2 + 12x + 15 What is the vertex? Axis of Symmetry: Zeroes: Which of the following is the vertex of the graph of the equation y = x 2 4x? What are the x-intercepts of the graph of the equation y = x 2 + 5x 4? A. (-3,3) C. (-2,4) B. (-4,0) D. (0,0) A ( 5. 7, 0), (. 702, 0) C (. 702, 0), (5. 7, 0) B ( 5. 7, 0), (. 702, 0) D (. 7, 0), (5, 0) Unit 9 Quadratics 20
Unit 9 Quadratics 21
Quadratic Functions: Application Problems What maximum height did the rocket reach? How many seconds was the rocket in the air? From what height was the rocket launched? For what interval of time was the rocket above 35 meters? Between and For how long was the rocket above 35 meters? What is the y-intercept (, )? What does this represent? What is the x-intercept (, )? What does this represent? What is the vertex (, )? What does this represent? What time did the rocket reach its maximum height? What is the Domain? What is the Range? Practice: The graph represents the relationship between the height of a ball and the distance it traveled after being thrown. Are the following statements True or False? The ball reaches a maximum height of 14 feet. The ball reaches its maximum height after traveling 14 feet. The ball was thrown from a height of 6 feet. It took longer than 30 seconds for the ball to hit the ground The axis of symmetry for this graph is y = 14. Unit 9 Quadratics 22
The graph below shows the height of a baseball from the time it is thrown from the top of a building to the time it hits the ground. About what height is the building? How long did it take for the ball to hit the ground? Between what times is the baseball 80 meters or more above the ground? How much time passes while the baseball is 80 meters or more above the ground? What was the maximum height of the ball? A farmer wants to create a rectangular fence. He has 120 feet of fencing and plans to use his barn as one of the sides of the rectangle. Here is a graph of the length of one side and the area. What is the maximum area he can achieve? How long is the side when he achieves this area? If he makes the side length 20 feet, what would be his area? If he wanted an area of 800 square feet, what two side lengths could he have used? What is the Domain? What is the Range? The graph shows h, the height in meters of a model rocket, versus t, the time in seconds after the rocket is launched. At what time does the maximum height occur? What is the maximum height of the rocket? About how long did it take for the rocket to land? What is the Domain? What is the Range? Unit 9 Quadratics 23
Unit 9 Quadratics 24
Unit 9 Quadratics 25