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1 Name Period Day Date Assignment (Due the next class meeting) Wednesday Thursday Friday Monday Tuesday Wednesday Thursday Friday Monday Tuesday Wednesday Thursday Friday Monday Tuesday Wednesday Thursday Friday 3/21/18 (A) 3/22/18 (B) 3/23/18 (A) 4/09/18 (B) 4/10/18 (A) 4/11/18 (B) 4/12/18 (A) 4/13/18 (B) 4/16/18 (A) 4/17/18 (B) 4/18/18 (A) 4/19/18 (B) 4/20/18 (A) 4/23/18 (B) 4/24/18 (A) 4/25/18 (B) 4/26/18 (A) 4/27/18 (B) 10.1 Worksheet Graphing Quadratics in Vertex Form 10.2 Worksheet Completing the Square 10.3 Worksheet Solving by Factoring 10.4 Worksheet Graphing Quadratics in Intercept Form 10.5 Worksheet Solving by Square Rooting 10.6 Worksheet The Quadratic Formula 10.7 Worksheet Systems and Word Problems Unit 10 Practice Test Unit 10 Test NOTE: You should be prepared for daily quizzes. Every student is expected to do every assignment for the entire unit. Students with 100% HW completion at the end of the semester will be rewarded with a 2% grade increase. Students with no late or missing HW will get a free pizza lunch. HW reminders: If you cannot solve a problem, get help before the assignment is due. Extra Help? Visit or Do you need a worksheet or a copy of the teacher notes? Go to 1

2 10.1 Notes: Graphing Quadratics in Vertex Form Warm-Up: Graph each function. 1) y = 1 (x 1) + 2 2) f(x) = 2 x Quadratic Functions: 2

3 The Parent Function of the Quadratic: y = x 2 x y = x Find the coordinates of the vertex. What are the coordinates for the x-intercept(s)? What is the y-intercept? Find the minimum (height) of the function. Is there a maximum (height) of the function? Explain. Find the domain and range of this function. Vertex Form of a quadratic function: y = a(x h) 2 + k Exploration: Graph the following functions: y = 2(x + 1) 3 y = 2 x y = 2(x + 1) 2 3 How are they the same? How are they different? 3

4 Example 1: Sketch each quadratic function. Show the vertex and y-intercept. a) y = (x 1) 2 b) f(x) = x 2 3 c) g(x) = (x + 2) ) Sketch the graph of the quadratic function and find the requested information: y = x Vertex: Domain: y-intercept: Does the function have a max or min? Range: Transformation from y = x 2 3) You try! Sketch the graph of the quadratic function and find the requested information: y = (x + 2) 2 3 Vertex: Domain: y-intercept: Does the function have a max or min? Range: Transformation from y = x 2 4

5 Vertical Stretch/Compress for a Quadratic Function: Reflections in the x-axis: Examples: For each quadratic function, sketch the graph, and then find the requested information. 4) y = 2(x 3) 2 5 Vertex: y-intercept: Domain: opens up or down? Range: Transformation from y = x 2 5) y = 1 3 x2 + 2 Vertex: y-intercept: Domain: opens up or down? Range: Transformation from y = x 2 6) y = 3(x + 2) 2 Vertex: y-intercept: Domain: opens up or down? Range: Transformation from y = x 2 5

6 Mosquitos (in millions) Algebra 1 Unit 10 Solving and Graph Quadratics ) y = 1 (x 4 1)2 6 Vertex: y-intercept: Domain: Range: Transformation from y = x 2 8) Describe how g(x) = x 2 changes after the transformation g(x 4) is applied. For #9 11: The number of mosquitoes is Anchorage, Alaska (in millions of mosquitoes) is a function of rainfall (in cm) is modeled by m(x) = (x 3) 2 + 5, as shown in the graph below. 9) How many cm of rainfall would result in 4 million mosquitos? 10) What is the maximum number of mosquitos? 11) How many cm of rainfall would result in the maximum number of mosquitos? Rainfall (in cm) For #12 13: Consider the function h(x) = x ) If h(x) is reflected in the x-axis and then translated up 2 units, what would be its new equation? 13) If h(x) translated up 2 units and then is reflected in the x-axis, what would be its new equation? Compare your answer with #14. What do you notice? 6

7 14) Which of the following statements is true for f(x) = 3x 2 2? Choose all that apply. A) The y-intercept is at (0,-2). B) The vertex is at (3, -2). C) The function opens upward. D) The range of the function is all real numbers. E) The domain of the function is all real numbers. F) The range of the function is y : Completing the Square Warm up: 1. Multiply: (x 3) 2 2. Simplify: (x + 2) 2 3. Factor: x x Factor: 4x 2 12x

8 Trinomials that are Perfect Squares when factored: Examples: Find the missing value that would make the trinomial a perfect square. Then factor each trinomial. 1) x 2 + 6x + 2) x 2 10x + 3) x 2 + 8x + (x ) 2 (x ) 2 (x ) 2 Completing the Square is a process that allows us to rewrite a quadratic equation from standard form y = ax 2 + bx + c into vertex form, which is also known as form: y = a(x h) 2 + k. This will allow us to easily find the. Steps for Completing the Square: For #4 5: Complete the square to rewrite the equation in vertex form, and then identify the vertex. 4) y = x 2 + 4x

9 You try! 5) y = x 2 6x 2 If a quadratic function is in standard form (y = ax 2 + bx + c), then the y-intercept is 6) Write the function y = x 2 18x + 4 in vertex form, and then draw a sketch of the graph that includes the vertex and y-intercept. the value of c. You try! 7) Write the function y = x 2 + 8x + 5 in vertex form, and then draw a sketch of the graph that includes the vertex and y-intercept. Example 8: Write the function y = 2x x + 6 in vertex form, and then draw a sketch of the graph that includes the vertex and y-intercept. 9

10 You try! 9) Write the function y = 3x 2 6x 2 in vertex form, and then draw a sketch of the graph that includes the vertex and y-intercept. Example 10: Write the function y = x x + 2 in vertex form, and then draw a sketch of the graph that includes the vertex and y-intercept. 11) Which of the following functions has a vertex at (4, -3)? Show work to justify your choice. a) y = x 2 + 4x + 3 b) y = x 2 + 8x + 13 c) y = x 2 + 8x 19 Example 12: Find the domain and range for the function y = 2x x

11 13) Find the coordinates of the max or min of the function y = x x Notes: Solving Quadratics By Factoring Warm-Up: Factor each expression. 1) 6x 4 9x 3 2) x 2 3x

12 Exploration: Given that ab = 0. What must be true about a and/or b? Given that (x 2)(x + 5) = 0. What would x have to equal in order for this equation to be true? Zero-Product Property Let a and b be real numbers. If ab = 0, then a = 0 and/or b = 0. Quadratic Equations: For #1 4: Solve each equation for x. 1) x(x 6) = 0 2) 2.5x(x + 1) = 0 3) 3(x 2)(x + 2) = 0 4) 4x(2x 3)(x 100) = 0 Steps for Solving Quadratic Equations by factoring: 1) Get a zero on one side of the equation. 2) Factor. 3) Set each factor equal to zero and solve. The solutions are also called roots, x-intercepts, solutions, or zeros. 12

13 Example 5: Given the quadratic equation: x 2 + 5x + 6 = 0 a) Use factoring to solve for x. b) Consider the graph of y = x 2 + 5x + 6, as shown. What do you notice? (Hint: compare your answer from part a) to the graph below. For #6 9: Solve the following equations. Sketch a graph that includes the x-intercepts ( roots ). 6) x 2 + 7x 18 = 0 7) x 2 + 8x + 16 = 0 8) 2x 2 10x = 0 9) 2x 2 8x =

14 For #10 13: Solve the following equations. 10) x 2 30 = x 11) a 2 + 6a + 15 = 1 12) 3x x 55 = 0 13) 3b 3 b 2 = 10b For #14 19: Solve the following equations. 14) 4x = 0 15) 10 = 9a ) y 2 5y + 3 = 3y 2 17) 4x 2 2x =

15 2x + 16 Algebra 1 Unit 10 Solving and Graph Quadratics ) x 3 x x = 0 19) 3r r = 21r 2 20) Find the x-intercept(s): h 2 + 6h + 9 = y 21) Find the x-intercept(s): 4x x = f(x) 25 A) (3, 0) A) 5 2, 5 2 B) (-3, 0) B) 5 2 C) (3, 0) and (-3, 0) C) 5 and 2 D) (3, 0) and (6, 0) D) -5 22) A garden measuring 12 ft by 16 ft has a walkway installed all around it. Including the walkway, the total area of the region is 396 ft 2. What is the width x of the walkway? 2x

16 23) Given (x + 4) is a factor of 2x x + 2m, determine the value of m. 24) Solve the equation for x: 6x + Cx = : Graphing Quadratics in Intercept Form 16

17 Exploration: Consider the quadratic function y = x 2 6x + 8. What is the y-intercept of this quadratic? Does this function open up or down? How do you know? Solve for x: 0 = x 2 6x + 8. Draw a sketch of this function by using the x- and y-intercepts. Where do you think the vertex would be for this quadratic function? Intercept Form of a Quadratic Function: y = a(x p)(x q) Standard Form of a Quadratic Function: y = ax 2 + bx + c When a quadratic is written in standard form, the y-intercept is the vertex is at 17

18 Reflect: How could you change a quadratic function from standard form to intercept form? Example #1: Sketch the graph of y = (x + 3)(x 1). Include the x-intercepts, the y-intercept, and the vertex. 2) You Try! Sketch the graph of y = 2x(x 6). Include the x-intercepts, the y-intercept, and the vertex. Terms for x-intercepts: 3) Given the parent function y = x 2, what are the zeros of the function after the graph is reflected, translated up 4 units and left 3 units? 18

19 Example 4: Given the quadratic function y = 2x 2 8x 10. Sketch a graph. Include the intercepts and vertex. 5) You try! Given the quadratic function y = x 2 + 8x Sketch a graph. Include the intercepts and vertex. For #6 7: The storage building shown can be modeled by the graph of the function y = x x 44 where x is the horizontal width of the building (in feet) and y is the height (in feet). 6) What is the width of the building at the base? 7) What is the maximum height of the building? 19

20 8) Given the quadratic function f(x) = 5x x. Sketch a graph. Include the intercepts and vertex. 9) Given the quadratic function y + 9 = x 2 + 6x. Sketch a graph. Include the intercepts and vertex. What do you notice about this graph? 10) Given the quadratic function g(x) = x 2 + 4, sketch a graph. Include the intercepts and vertex. What do you notice about the x-intercepts for this graph? Why does this happen? 20

21 11) Sarah is a cliff diver standing on the top of a cliff with a height of 50 feet. When she dives off, she reaches a height of 30 feet in 2 seconds. Determine the quadratic function, f(t), that would model Sarah s height t seconds after she jumps off the cliff. A. f(t) = 5(t 0) B. f(t) = 5(t 0) C. f(t) = 5(t 0) D. f(t) = (t 0) ) Given the quadratic function y x = 3x 2, sketch a graph. Include the intercepts. For #13 16: A ball is thrown, and the height h (in meters) as a function of time t (in seconds) is given by h(t) = 2t 2 + 8t ) At what time does the ball hit the ground? 14) What is the maximum height of the ball? 15) At how many seconds after the ball was thrown did it have a height of 16 meters? 16) What is the height of the ball when t = 4 seconds? 21

22 10.5 Notes: Solving Quadratics by Square Rooting Warm Up: 1) When a number is squared, the result is 25. What could the original have as its value? (Hint: there are two answers.) 2) Solve for x: 2 x = 13 Solving Quadratics by Square Rooting *Use this strategy when a function is in vertex form, or if there is not a b term. Step 1: the variable or ( ) 2 by using inverse operations. Step 2: Square root both. PLUS OR MINUS! Simplify radical answers. Note: When a variable 2 or ( ) 2 is isolated, it cannot equal a number. (If it does, then there is no solution.) We can have solution, solution, or solutions. 22

23 Examples: Solve each equation for the variable by square rooting. 1) z 2 5 = 4 2) r = 4 3) 4x = 3 You try! Solve each equation for the variable by square rooting. 4) 3x 2 = 27 5) t = 17 6) 4p 2 4 = 0 7) Solve for x: 5(x + 1) 2 = 80 8) Solve for a: 4(a 3) 2 8 =

24 Example 9: Pick one of the following problems to find the solutions. The problems go in order from easiest to more challenging from left to right. a) 2x 2 7 = 9 b) 3(m 4) 2 = 12 c) 4(a 3) 2 40 = 20 Example 10: Find the zeros (x-intercepts) of the function f(x) = 3x 2 9, if possible. If needed, write your answer as a simplified radical. Then draw a sketch of the quadratic function. Include the roots (x-intercepts), y-intercept, and vertex. Example 11: Find the roots (x-intercepts) of the function f(x) = 2(x 3) 2 + 4, if possible. If needed, write your answer as a simplified radical. Then draw a sketch of the quadratic function. Include the vertex and intercepts. Solving Quadratic Equations by Completing the Square When solving quadratic equations by completing the square, we want to keep the equation. So whatever is added (or subtracted) to one side of an equation, we must add or subtract) the to the other side of the equation. Once we have a perfect square on one side of the equation, we can then solve by square rooting. 24

25 Example 12: Solve for x by completing the square: x 2 8x + 5 = 0 Example 13: Solve for x by completing the square: 2x 2 8x = 12 Example 14: Find the x-intercepts of y = x 2 2x + 3 by factoring. Now find the x-intercepts of y = x 2 2x + 3 by completing the square. 25

26 Example 15: Maple sells and manufactures sunglasses. Let P represent the profit Maple makes in one year when selling sunglasses at a price of d dollars each. The profit can be modeled by the equation P = 4(d 55) a) What is the maximum profit that Maple can make in one year? b) At what selling price d will Maple make the maximum profit? c) If Maple makes the sunglasses too expensive, she might have less buyers and could actually lose money. What is the highest price that Maple could sell the sunglasses at before losing money? Example 16: Consider the function f(x) = 3x a) What is the vertex for this function? b) Will this function open up or down? c) Draw a sketch of this function. What do you notice about the x-intercepts? d) Solve f(x) for the zeros (x-intercepts.) Does your solution support your conclusion from part c? 26

27 10.6 Notes: The Quadratic Formula Solve ax 2 + bx + c = 0 for x by completing the square: 27

28 The Quadratic Formula: Example 1: Consider the function y = x 2 + 7x 3. A) Solve 0 = x 2 + 7x 3 for x by using the quadratic equation. If needed, write your answers as simplified radicals. B) Write the solutions for x as decimals rounded to the nearest tenth. C) The graph of the function y = x 2 + 7x 3 is shown to the right. What did you find when you solved this equation for x? Example 2: Consider the function 3x 2 x = f(x) + 4. A) Solve 3x 2 x = 4 for x by using the quadratic equation. If needed, write your answers as simplified answers. B) The graph of the function 3x 2 x = f(x) + 4 is shown to the right. What did you find when you solved this equation for x? 28

29 Example 3: Solve for x: x = 7x. If needed, write your answers as simplified radicals. Example 4: Consider the function y = 2x 2 2x + 4 A) Solve 0 = 2x 2 2x + 4 for x by using the quadratic equation. If needed, write your answers as simplified answers. B) The graph of the function y = 2x 2 2x + 4 is shown to the right. What did you find when you solved this equation for x? You try! For #5 7, solve each equation for x by using the quadratic formula. If needed, write your answers as simplified radicals. 5) 4x 2 x + 20 = 0 6) 6x 2 + 7x = 3 7) x 2 4x =

30 8.) Which of the following is the solution set of 2x x = 18? A. x = 7 ± 85 2 C. x = 7 ± 85 2 B. x = 7 ± 13 2 D. x = 14 ± ) Consider the function x 2 10x = y 25. a) Find the x-intercept(s). b) Find the vertex. c) Find the y-intercept. d) Sketch the function. Include the vertex and intercepts. e) Draw a vertical line through the vertex. What is the equation of this line? Axis of Symmetry: 30

31 Example 10: A football is kicked in the air, and its path can be modeled by the equation f(x) = 16(x 5) , where x is the horizontal distance (in feet) and f(x) is the height. A. What is the maximum height of the football? B. What is the equation of the axis of symmetry for this function? C. At what horizontal distance will the football hit the ground? Write your answer as a decimal, rounded to the nearest tenth Systems and Word Problems with Quadratic Functions Warm Up: Solve the system for x using any method. { f(x) = 3 g(x) = x

32 Solutions of a quadratic and linear system: 2 solutions 1 solution no solution f(x) = 2 Example 2: Find the solution for x for the system: { g(x) = (x + 3) 2 1 A) Solve by graphing. B) Solve by using substitution. 32

33 For #2 4: Match the equations with the graph that best models the solution A) 3 = (x + 2) 2 4 B) (x 2) = 3 C) 2x = 3 f(x) = 3x 5) Which graph models the solutions to the system of equations? { g(x) = 1 2 x2 1 A) B) C) 6) You try! Find the solution for x for the system: { f(x) = 1 4 (x + 2)2 + 4 g(x) = 1 2 x

34 Example 7: Set up as a system and find the solutions for x: (x 2) 2 3 = 2. A) Solve by graphing. B) Solve by square rooting. You try! 8) Set up as a system and find the solutions for x: (x + 1)2 2 = 6 A) Solve by graphing. B) Solve by square rooting. For #12 14: The height (h), in feet, of Jon jumping off a rock into a lake can be modeled by the equation h(t) = 16t 2 + 4t + 24, where t represents the time in seconds after Jon has jumped off the rock. 12) What is Jon s height after 1 second? 13) After how many seconds does Jon enter the water? Round to the nearest hundredth. 14) What is the height of the rock? 34

35 For #15 19: A rocket was shot up into the air. The graph shows the height of its flight t seconds after it was shot. 15) At about what height was the rocket after 6 seconds? 16) What is the maximum height reached by the rocket? 17) At what time was the height of the rocket 30 yards? 18) At what time(s) was the rocket on the ground? 19) How long did it take the rocket to come back down to the ground? 35

36 Unit 7 Study Guide Vertex Form Graphing Quadratics Form What it tells us Read about it in your notes! y = a(x h) 2 + k Intercept Form y = a(x p)(x q) Standard Form y = ax 2 + bx + c Vertex at (h, k) Opens up if a is positive Opens down if a is negative Vertical stretch if a > 1 Vertical compression of 0 < a < 1 Find the y-intercept by substituting a 0 for x and solving for y. x-intercepts at (p, 0)and (q, 0) The x-coordinate of the vertex is exactly mid-way between the x-intercepts. Substitute this value of x into the function to find the y-value of the vertex. The y-intercept is at c. Complete the square to put into vertex form, or use b to find 2a the x-value of the vertex. Substitute this value of x into the function to find the y-value of the vertex. Factor the function, set equal to 0, and solve to find the zeros (xintercepts.) Solving Quadratics Section 10.1 Section 10.2 and 10.4 Section 10.3 and 10.4 Technique Hints and Steps Read about it in your notes! Solving by Factoring ax 2 + bx + c = 0 Solving by Square Rooting 0 = a(x h) 2 + k 0 = ax 2 + c Solving by the Quadratic Formula ax 2 + bx + c = 0 Get a 0 on one side of the equation. Factor completely. Set each factor = 0 and solve Isolate variable 2 or (variable h) 2 Square root each side (±). Isolate the variable Get a 0 on one side of the equation Use x = b± b2 4ac 2a Section 10.2 Section 10.5 Section