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3 Daily Plan Unit 3 FCC3 Day Class Lesson Homework Day 1: Thurs 10/ Day : Fri 10/3 Day 3: Mon 10/6 Day 4: Tues 10/7 Day 5: Wed 10/8 Day 6: Thurs 10/9 Day 7: Fri 10/10 Day 8: Mon 10/13 Day 9: Tues 10/14 Wed 10/15 Day 10: Thurs 10/16 Day 11: Fri 10/17 Mon 10/18 Graphing Quadratics, vertex and standard forms, vertex axis of sym, solutions (p 1-) Solve quad by factoring and taking sq root (p. 4-5) More solve by factoring practice (p 7. Evens) Quad applications using the calculator (p8 ex 1-6) QUIZ #1 (covers days 1-4) Imag and Complex Numbers (p1) Solving Quad by comp the sq (p evens) More comp the sq practice in groups (p17 evens) P all P all P 7. odds p all, p10-11 all p all P odds (answers on p16) P 17 odds Solve using the quad formula (p 18) p all Review for quiz (p 0-1) QUIZ # (covers days 5-7) PSAT Students in class review (p -4) Solving quad applications (no calc) p all EARLY RELEASE P6-7 evens in groups PLAN Students in class begin review p. 8-31(evens) If you miss class due to the PSAT, do p -4 for HW P6-7 odds Students who miss class for the PLAN do the evens on the review for HW p.8-31 evens Day 1: Tues 10/19 Finish review p8-31 odds Study for TEST Day 13: Wed 10/0 Unit 3 TEST

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5 Unit 3 Day 1 Notes Quadratics Quadratic Equations: ALL GRAPH TO BE PARABOLAS General Form: f(x) = ax + bx + c where a 0. allows us to solve ^ ^ ^ easily for the roots The vertex is (, f( ) ) and the axis of symmetry is a vertical line x= Vertex Form f(x) = a(x h) + k allows us to graph easier The vertex is (h,k) and the axis of symmetry is a vertical line x=h Real life applications: All upward projectiles like baseballs, rockets, arrows follow a path that is shaped like an upside down parabola. Degree (greatest exponent) = # of roots, the equation will have at most that # of solutions. Quadratics have two solutions The SOLUTION of a quadratic equation can be found by graphing, factoring, completing the square or taking square roots of both sides. Today we will focus on graphing to find solutions. When you graph the solutions are the x-intercepts of the parabola Let s Graph the Following: Remember your transformations After plotting the vertex, remember that the NORMAL parabola goes over 1 up 1 then over up 4 make sure you always show 5 points Parent Function y = x y = (x ) + 1 y = (x + 3) 4 y= (x + 1) + 3 y = (x 3) 5 y = (x) 1

6 Put the following in General Form ax + bx + c. Name the vertex and axis of symmetry 1. f(x) = (x 3) + 4. f(x) = (x + 1) 3 3. f(x) = (x 4) 3 Solutions of quadratics are the x-intercepts you can find them by graphing Name the vertex of the graph Name the axis of symmetry What are the x-intercepts? Let s try to write the equation Parent Function y = x y = (x 3) 1 y = -(x + ) + 4 x- intercepts: y = 3(x + 4) + 3 y = (x 1) y = (x 5) x- intercepts:

7 FCC3 Day 1 Homework Identify the quadratic term, the linear term, and the constant term for each function. 1. f(x) = x + 14x f(x) = -3x + x - 4 Graph each function. Name the vertex, x intercepts and the axis of symmetry. 3. f(x) = (x 5) 4. f(x) = (x + 4) 6 5. f(x) = (x -3) 4 6. f(x) = -(x + 5) f(x) = x + x 8. f(x) = x 8x + 6 Draw the graph of each equation: 9) f(x) = (x 3) + 10) f(x) = x + x + 6 3

8 Unit 3 Day Notes Quadratics Remember that you can solve a QUADRATIC by graphing, factoring, taking the square root or completing the square. Today we focus on factoring and taking the square root.. Factoring- express a polynomial as the product of its prime factors. FACTORING REVIEW AND PRACTICE Method #1 --G.C.F. 1) 4a b 18ab ) -15x 5x 3) 3s t + 4st + st Method # --Difference of squares: Can be thought of as a trinomial ax + 0x c 4) x 49 5) 9x 5y 6) b 64 7) 5x 1 8) a + b 9) 4z 6-81 REMEMBER FOIL?? (x + ) (x + 3) = x + 5x + 6 (3x + 4) (x 1) = 3x + x 4 Try these: (x + 1)(x + 5)= (x 1) (x + 6) = Method #3 --Factoring Trinomials: 10) x 3x 10 11) x + 3x ) x + 8x ) x 5x 84 Method #4 Trial and Error 14) 3z + 16z 35 15) 16x 4x ) 4m 1m ) 6n 11n 18) m 1m + 18m 19) 9x 4x ) 5y 0y + 1 1) x + 3x 54x Method #5 Grouping ) b 3b + 4b 1 3) x + x x 4

9 Solve by Factoring: Make sure to 1) set the equation equal to 0 ) Factor completely 3) Set each factor equal to 0 and solve. 1) x x 15 = 0 ) z 5z = 0 3) x + 6x = -9 4) q + 11q = 1 5) x 8x + 16 = 0 6) x = 4x 7) x + 9x = -14 8) 3x + 10x = 8 9) 9y = 49 10) 18r + 6r = 4r Solving by taking the Square Root: ONLY WORKS WHEN THE LINEAR TERM IS MISSING 11): x 16 = 0 1) x = 4 13) x = 18 14) 3x 48 = 0 15) 6y = ) 5x = 15 5

10 FCC3 Day Homework Solve each equation by factoring. Remember to set equal to zero. You can solve some of the following by isolating the squared term, then square rooting both sides. 1. x 4x 1 = 0. x 16x + 64 = 0 3. x + 5 = 10x 4. 9z = 10z 5. 7x 4x = 0 6. x = x w 35w + 60 = x + 4x + 45 = m + 19m + 6 = x + 6 = 11x x = x 3 8x = 15x 13. 6x 3 = 5x + 6 x = 64x 6

11 FCC3 Day 3 Solving Quadratic Equations Solve using any method. 1. x = x + 6x = x = x + 8 = x 3x + 0 = x + 8x + 4 = x + 8 = x + 15 = n 6n 45 = x + 8 = x 1 = y + y 4 = x 3x = 0 7. b - 1b = b 45. 3x 8x = 0 8. x = 8x x = x + 6x 1 = x = 9 1. x + 1 = x 1 = y - 7y = x -7x + 10 = x = 6x 7. 3x +4x 1 = x +17x + 5 = x - 7x = y = -11y x = 1x x = 3-7x 7

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13 Unit 3 Day 4 Homework FCC3 Quadratics 1. Finish any of the above examples that we did not cover in class.. T rajectory of a Ball: The height y (in feet) of a ball thrown by a child is: 1 y #" x x 4 1 Where x is the horizontal distance (in feet) from where the ball is thrown. Using a calculator to graph the path of a ball answer the following: a) "#$%&'%$&($)%*$+,--$#%*.$&)$-*,/*($)%*$0%&-1($%,.13$4Hint: Find y when x = 0) b) How high is the ball when it is at its maximum height? c) How far from the child does the ball strike the ground? 3. Business: The profit P (in hundreds of dollars) that a company makes depends on the amount x (in hundreds of dollars) the company spends on advertising according to the model: P = -0.5x + 0x + 30 What expense for advertising results in the maximum profit? 4. Business: A textile manufacturer has a daily production costs of : C = 0.45x 5 110x + 10,000 where C is the total cost (in dollars) and x is the number of units produced. How many units should be produced each day to yield a minimum cost? 9

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17 Unit 3 day 5 Complex Numbers Worksheet Simplify (7 6i) + (9 + 11i) ( 8i) + 3(5 + 7i) 1. 4(7 i) - 5( 6i) 13. (3 4i) (6 4i)(6 + 4i) ( + 3i) + 6(8 5i) 18. (4 + 3i)( 5i)(4 3i) Find the values of x and y for which each equation is true x 5yi = 15 0i 0. 5x + 7yi = 6 i Solve each equation by taking the square root. 1. n + 5 = 0. (m-) + 10 = (y-3) + 4 = r + 64 = 0 13

18 Name Date Period FCC3 - - Day 6 QUADRATIC EQUATIONS - - COMPLETING THE SQUARE - - Notes Sheet I. Solve each equation by taking the square root of both sides II. What should be added to form a perfect square trinomial? III. DIRECTIONS FOR COMPLETING THE SQUARE. 1. The coefficient of the squared term must be one. If not, divide each side of the equation by the coefficient.. Add the opposite of the constant to both sides of the equation. 3. Take half of the coefficient of the linear term, square it, and add it to both sides of the equation. 4. Factor the perfect square trinomial on the left combine the numbers on the right. 5. Take the square root of both sides of the equation. (REMEMBER THE ) 14

19 6. Solve for the variable. Solve each equation by completing the square

20 Unit 3 Day 6:Graphing Quadratics Write each equation in the form f(x) = (x h) + k. Then name the vertex and the axis of symmetry for the graph of each function. 1) f(x) = x 10x + 5 ) f(x) = x + 1x ) f(x) = x + 4) f(x) = x 6x 5) f(x) = x 3x 1 6) f(x) = x x 1 Draw the graph of each equation: 7) f(x) = (x 3) + 8) f(x) = (x + 5) 1 9) f(x) = x + x ) f(x) = x 4x

21 FCC3 Unit 3 Day 7 More Completing the Square Practice SHOW EVERY STEP FOR COMPLETING THE SQUARE 1.) x 6x + 5 = 0.) y - 8y -9 = 0 3.) n + 4n + 9 = 0 4.) x 8x + 1 = 0 5.) a + 10a = 6.) y 6y = ) x x = -5 8.) w + 1w 4 = 0 9.) y 18y = 9 10.) x 10x + 9 = 0 "#

22 x # Unit 3 Day 8 The Quadratic Formula b " b 4ac 1. Quadratic Formula: a. Solving using the quadratic formula: a. Set the equation equal to 0 b. Identify a, b, and c c. Plug values into formula and simplify 3. Examples: Solve each using the quadratic formula. a. 3x $ 8x # 35 b. 1x 5 # x $ 13 c. x # x 5 d. x $ x # x x $ 4 e. x $ 64 # 16x f. x # x $ 3 g. 3p 5p + 9 = 0 h. x 8x + 16 = 0 i. x + x = 5 j. y 16 = 0 18

23 FCC3 Homework Day 8 Name Solve each equation using the Quadratic Formula. 1. x 8x + 4 = 0 6. x + x 3 = 0. x x + 1 = 0 7. x + 5x + 6 = 0 3. x 10x + 30 = 0 8. x + 5x + 7 = 0 4. x + x + 4 = x 4x = 5. 4x 1x + 9 = x + 0x = 5 19

24 FCC3 -Unit 3 Review Day 5 8 Name 1. What value would be added to both sides to complete the square for the equation x # 10x " 3 0? A. -5 B. -5 C. 5 D Simplify ""#$"%" &$ A. 6 " 8i B. 8i C. "'" %% D. %"%. 3. Simplify %" &$% # &$ A. -5 B. ""( C. 13 D. )&")% 3. Solve by completing the square. 4. x + 4x 7= y 8y = x x + 1 = 0 6. Solve using the Quadratic Formula. Show all work and simplify 7. x " x = -6x

25 9. 6x 5x + 4 = 0 9. Simplify these expressions completely 10.) ( - 3i ) - ( -6 i ) ) ( 4 9i ) + 3( - + 5i ) ) ( - + i )( 4 3i ) ) 3i + i 5i + 1i ) 6 ( i ) - 3( -5 + i) 14. BONUS: what does i 1 i 3 equal?? 1

26 F C C3 Review for PSA T D A Y Fill in the blanks, and graph (use vertex & 4 more points for each). 1. y # ( x " 1) 3. y # ( x 1) " 8 Vertex: Axis of sym X intercepts: Vertex: Axis of sym X-intercepts: Multiple Choice: 3. What is the direction of opening of the parabola "" # "" # $" " %? A. up B. down C. right D. left What is the equation of the graph of # # &" # shifted two units to the right? A. # # &" # #" B. # # &" #" # C. # # &" " #" # D. # # &" # # Write y # ( x 5) 1 in standard form. A. # # " # #' B. # # " # " #& C. # # " # " ()" " #& D. y # x 10x 6 5.

27 6. What is the equation of the parabola shown? A. y " ( x 1) 1 B. " " $" C. " " " " ( x $ 1) D. y A ball is thrown upward vertically with an initial speed of 48 feet per second. The equation h = 3t 16t gives the height of the ball where t is the number of seconds after the ball is released. "#$%&"'$()$'&*$+",,-)$."/(.0.$&*(1&' 7a. b) How long is the ball in the air? 7b.. Solve each equation by factoring. 8. x 3$4/$3$56$7$8 9. 4x + 0x = d 3$69:$7$ x + 9x + 3 = 0 Solve each equation by completing the square. 1. x + 4 = 8x 13. x 3 5x = x 3$;6/$= x 3;6/$7$;< 3

28 Solve each equation by using the quadratic formula. 16. x + x = x 1x + 5 = x 5x + 3 = x 3x + = 0. Given a = 5 - i and b= i 0. Find a+ b 1. Find a b. Find the product of a and b 3. Find a 3b 4. Find a b 5. Find 3i + i 4 4

29 Day Q U A DR A T I C E Q U A T I O N APPL I C A T I O NS - - Class Examples 1. Find two consecutive positive integers whose product is 56.. The sum of the squares of two consecutive negative odd integers is 130. Find the integers. 3. If the length of a rectangle is 3 ft greater than the width and it has an area of dimensions. 8 ft, find the 4. The sum of the squares of 3 consecutive integers is 77. Find the integers. 5. A farmer makes a rectangular enclosure using a stone wall for one side and 13 ft of fencing for the other three sides. Find the dimensions of the enclosure if the area is 15 ft. 6. Find the dimensions of a rectangle with an area of 4 ft if its length is ft greater than twice its width. 7. A rectangular plate has a rectangular piece cut from one corner as shown in the figure. The area after the rectangular piece is cut out is 40 square units. Use the information given in the figure to find the length and width of the original rectangular plate. x 4 3 x x x 5

30 F C C3 -- Day 10: Q U A DR A T I C E Q U A T I O NS - - W O RD PR O B L E MS 1. Find three consecutive odd integers such that the product of the first and the third is 5 more than 8 times the second integer.. Find two numbers that have a sum of 9 and a product of The sail on a boat is shaped like a triangle and has an area of 68 ft. If the height of the sail is one foot longer than twice the base, find the height of the sail. 4. One side of a rectangular garden is yd less than the other side. If the area of the garden is 63yd, find the dimensions of the garden. 5. A farmer makes a rectangular enclosure. He uses his barn for one side and 00 ft of fencing for the other three sides. Find the dimensions of the enclosure if its area is 5000 ft. 6. A cement walk of uniform width sur rounds a rectangular swimming pool that is 0 ft wide and 40 ft long. Find the width of the walk if its area is 396 ft. 7. Find three consecutive odd integers such that the product of the smallest and the largest is 6 more that three times the second. 8. A rectangular field is to be fenced on three sides with the fourth side bounded by the river. If the area of the field is 300m and the total fencing used is 50 m, what is the length of the side parallel to the river? 9. F ind two consecutive integers whose product is The length of a rectangular pool is 4 yd longer than its width. The area of the pool is 60yd. What are the dimensions of the pool? 6

31 11. The legs of a right triangle have length of 4 ft and 5 ft. When the legs are decreased in length by equal amounts, the area of the resulting triangle is Find the lengths of the legs of the new triangle. 4 ft less than that of the original triangle. 1. The base of a triangle is 4 ft greater that its altitude. The area of the triangle is the length of the base. 96 ft. Find 13. When a border of uniform width is added to a rectangular lot with dimensions of 30yd x 0 yd, the total area is double that of the original lot. Find the width of the border. 14. Find the negative integer whose square is 10 more than three times the integer. 15. A rectangular yard is 40 ft x 60 ft. G rass is cut in a uniform strip along the edges until of the 3 grass is cut. What is the width of the strip? 16. A rectangular field is to be fenced in on four sides, but one side has a gap of 5 ft for a gate. If the area of the field is 600 ft and the total amount of fencing used is 95 ft, what is the length of the side with the gate? 17. A rectangular photograph is mounted inside a rectangular poster. The area of the photograph is 600in. T here are 3 in borders between the top of the photograph and the top of the poster and between the bottom of the photograph and the bottom of the poster. There are in borders between the sides of the photograph and the sides of the poster. If the perimeter of the poster is 10 in, what are the dimensions of the photograph? 7

32 FCC3 Name Unit 3 REVIEW G raph each of the following. Use your transformations.(not your calculator) 1. y = (x + 1) 3. y = (x 3) y = (x ) + 3 vertex: vertex: vertex: y y y x x x Find the vertex for each of the following. 4. f(x) = x + 4x f(x) = -3x + 6x 5 6. f(x) = x + 8x vertex: vertex: vertex: For the multiple choice questions, write the letter for the correct answer in the blank at the right. 7. What is the equation of the axis of symmetry of y = (x 4) + 1? A. x = -1 B. x = 4 C. x = 1 D. x = What is the direction of opening of the parabola y = -x + 7x 8? A. up B. right C. down D. left Write an equation of the parabola obtained by shifting the graph of y = -x exactly two units to the left? A. y # ( x " ) B. y # x " C. y # ( x ) D. y # ( x " ) 9. 8

33 10. What is the equation of the parabola shown below? 10. y A. y # ( x " 1) 1 B. y # x " 1 C. y # x 1 D. y # ( x 1) 1 x 11. If a parabola has a vertex at (3, -1), the axis of symmetry will be: A. y = -1 B. y = 3 C. x = -1 D. x = For #1 & 13, find the value of c that makes each a perfect square. 1. x 40x + c 1. A. 0 B. 100 C. 400 D x + 6x + c 13. A.3 B. 9 C. 0 D Simplify the following (8 5i) ( 3i). 14. A. 10 8i B. 6 8i C. 1 34i D. 6 i 15. Simplify the following ( 3i)( + 3i). 15. A. -5 B. 4 9i C. 13 D. 13 1i 16. Solve the following equation using any method: x + 9 = Solve the following equation by factoring: x 9 =

34 18. Solve the following equation by graphing: x + 4x + 3 = 0 y x x- intercepts: Solve each equation by factoring x 3 1x = x 8x = x + 13x + 3 = Solve x + 6x 8 = 0 by completing the square.. Solve each equation using any method. For the multiple choice questions, write the letter for the correct answer in the blank at the right. 3. 4x 5x = 0 3. A. 5 i 8 7 B C. 5 i 8 57 D x = 9x x = 4x

35 Solve using the Quadratic Formula. Show all work and simplify 6. 7x 6x + 1 = x 5x + 3 = 0 7. A B. 5 i 97 1 C. 5 i 47 1 D A ball is thrown upward vertically with an initial speed of 7 feet per second. The equation h = 7t 16t gives the height of the ball in t in seconds. Use the calculator to find the maximum height reached by the ball Write an equation, then solve using the calculator: Find two numbers whose sum is 16 and whose product is a maximum. Equation Write an equation and solve using the calculator: Gary plans to put a fence around his garden. He has 36 meters of fencing. His garage is used for one side of the garden. What would the dimensions be for the maximum area? Equation

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