Changing from Standard to Vertex Form Date: Per:

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1 Math 2 Unit 11 Worksheet 1 Name: Changing from Standard to Vertex Form Date: Per: [1-9] Find the value of cc in the expression that completes the square, where cc =. Then write in factored form. 1. xx xx + 2. xx 2 + 8xx + 3. xx 2 6xx + 4. xx 2 + 2xx + 5. xx 2 14xx + 6. xx 2 + 4xx + 5. xx 2 + 6xx + 8. xx 2 + 5xx + 9. xx 2 + 7xx + [10-21] a) Write the following functions in vertex form by completing the square. b) Find the vertex. 10. yy = xx 2 + 6xx yy = xx 2 10xx 26 a) Vertex Form: a) Vertex Form: b) Vertex:, b) Vertex:, 12. yy = xx 2 18xx yy = xx 2 + 4xx + 12 a) Vertex Form: a) Vertex Form: b) Vertex:, b) Vertex:, Math 2 Unit 11 Worksheet 1

2 14. yy = xx xx yy = xx 2 12xx 8 a) Vertex Form: a) Vertex Form: b) Vertex:, b) Vertex:, 16. yy = 2xx 2 + 8xx yy = 2xx xx + 31 a) Vertex Form: a) Vertex Form: b) Vertex:, b) Vertex:, 18. yy = xx xx yy = 3xx 2 18xx + 16 a) Vertex Form: a) Vertex Form: b) Vertex:, b) Vertex:, 20. yy = xx xx yy = xx 2 2xx 4 a) Vertex Form: a) Vertex Form: b) Vertex:, b) Vertex:, Math 2 Unit 11 Worksheet 1

3 Math 2 Unit 11 Worksheet 2 Name: Solving by Completing the Square Date: Per: [1-12] Solve for xx by completing the square and square rooting. If no real value of xx makes the equation true, write none. 1. xx 2 + 8xx 12 = 0 2. xx xx + 14 = 0 3. xx 2 18xx = xx 2 + 6xx + 12 = 0 5. xx 2 10xx = xx 2 20xx 46 = 2 5. xx xx + 47 = 5 8. xx 2 + 8xx + 24 = 0 9. xx 2 16xx + 20 = xx xx + 27 = xx 2 8xx + 20 = xx 2 + 2xx 20 = 3 [13-17] Write the following equations in vertex form and find the vertex, xx-intercept(s), yy-intercept, and graph for each quadratic function. 13. yy = xx 2 + 2xx 15 Vertex Form: Vertex: xx-intercept(s): yy-intercept: Math 2 Unit 11 Worksheet 2

4 14. yy = xx 2 12xx + 32 Vertex Form: Vertex: xx-intercept(s): yy-intercept: 15. yy = xx 2 2xx + 5 Vertex Form: Vertex: xx-intercept(s): yy-intercept: 16. yy = xx 2 + 6xx + 7 Vertex Form: Vertex: xx-intercept(s): yy-intercept: 17. yy = xx 2 + 4xx + 7 Vertex Form: Vertex: xx-intercept(s): yy-intercept: 18. When a parabola does not cross the xx-axis, what happens algebraically when you try to solve for the xx-intercept(s)? Math 2 Unit 11 Worksheet 2

5 Math 2 Unit 11 Worksheet 3A Name: Imaginary and Complex Numbers Date: Per: [1-9] Rewrite using the imaginary number ii [10-15] Find the sum or difference. 10. (7 + 4ii) + (3 5ii) 11. (4 2ii) (8 7ii) 12. (8 + 3ii) + 2(12 8ii) 13. (3 + ii) + (12 13ii) 14. (7 + 5ii) 3(18 + 5ii) 15. (24 13ii) 2(17 + 8ii) [16-25] Multiply. 16. (4ii)(7ii) 17. (12ii)(4ii) 18. (2ii)(5ii)(8ii) 19. (2ii)(5ii)(8ii)(10ii) Math 2 Unit 11 Worksheet 3A

6 20. (1 + 5ii)(4 2ii) 21. (2 3ii)(2 + 3ii) 22. (4 + 8ii) (2 9ii)(3 + 4ii) 24. (3 7ii)(3 + 7ii) 25. (6 8ii)(4 7ii) 26. Simplify the following to include at most one ii. 27. What patterns do you notice in your answers on number 26? ii 1 = ii 2 = ii 3 = ii 4 = ii 5 = 28. Based on the patterns you noticed, simplify the following to include at most one ii. ii 6 = ii 7 = ii 12 = ii 8 = ii 23 = ii 9 = ii 28 = ii 10 = ii 46 = Math 2 Unit 11 Worksheet 3A

7 Math 2 Unit 11 Worksheet 3B Name: Using Complex Numbers Date: Per: [1-6] Solve the following equations for xx. Express answers as complex numbers in simplified radical form. 1. xx 2 + 6xx = xx 2 + 6xx + 15 = 0 3. xx 2 2xx + 10 = 0 4. xx 2 + 8xx + 24 = 0 5. xx 2 8xx = xx 2 10xx + 7 = 0 7. xx 2 20xx + 20 = 0 8. xx xx = 73 Math 2 Unit 11 Worksheet 3B

8 Review [9-18] Simplify. 9. (2 + 5ii) + (6 7ii) 10. (7 + 3ii) (5 8ii) 11. ( 3ii)(12ii) 12. (5ii) (7ii)(2ii)(5ii) 14. (3ii)(4ii)(5ii)(6ii) 15. (2ii) (6 + 4ii)(5 2ii) 17. (1 + 3ii)(1 3ii) 18. (2 + 7ii) 2 [19-21] Match the equation to what characteristics it reveals without changing the form of the equation. 19. Reveals the maximum value of mm(xx) A mm(xx) = 4(xx + 8) Reveals the zeros (xx-intercepts) of mm(xx) B mm(xx) = 4xx xx Reveals the yy-intercept when xx = 0 C mm(xx) = 4(xx + 7)(xx 5) Math 2 Unit 11 Worksheet 3B

9 Math 2 Unit 11 Worksheet 4 Name: Quadratic Formula & Mid-Unit Review Date: Per: [1-10] Solve using the Quadratic Formula. Leave answers in simplest radical form. 1. 2nn 2 7nn 3 = mm 2 = 4 6mm 3. nn 2 + 8nn 16 = xx 2 + 3xx 7 = 0 5. xx 2 + 3xx + 5 = mm 2 3mm + 1 = 2mm 7. pp 2 pp + 2 = 2xx 5 8. xx 2 9xx + 20 = kk 2 + 8kk 15 = xx 2 + 4xx + 5 = 0 Math 2 Unit 11 Worksheet 4

10 Mid-Unit Review [11-12] Solve by completing the square. Leave answers in simplest radical form. 11. xx 2 8xx + 12 = xx xx = 5 [13-14] a) Write the following functions in vertex form by completing the square. b) Find the vertex. 13. yy = xx 2 + 6xx yy = xx 2 14xx + 9 a) Vertex Form: a) Vertex Form: b) Vertex:, b) Vertex:, [15-20] Simplify the following to include at most one ii. 15. (7 + 3ii) + (5 9ii) 16. (2 + 8ii) (12 3ii) 17. (3ii)(8ii) 18. (6 + 4ii)(2 3ii) 19. (2ii)( 5ii)(3ii)(7ii)(ii 3 ) 20. (3 4ii) 7(5 + ii) Math 2 Unit 11 Worksheet 4

11 Math 2 Unit 11 Worksheet 5 Name: Quadratic Formula and Applications Date: Per: [1-4] Solve using the quadratic formula. Simplify answers in decimal form to the nearest hundredth. 1. 3xx 2 10xx 8 = pp 2 5pp + 8 = 6 3. xx 2 2xx 4 = 2xx 1 4. xx 2 10xx + 13 = 0 [5-8] Find the axis of symmetry and the vertex for each parabola. 5. yy = xx 2 8xx + 2 Vertex: 6. yy = 2xx xx 13 AOS: Vertex: AOS: 7. ff(xx) = 2xx 2 + 4xx + 5 Vertex: 8. gg(xx) = 3xx 2 12xx + 5 AOS: Vertex: AOS: [9-13] Solve the following word problems. Include the correct units in your answers. 9. The profit from a t-shirt sale fundraiser depends on the t-shirt price and can be modeled by PP = 15tt tt + 50, where tt is the price per t-shirt and PP is the profit, both measured in dollars. a) What t-shirt price yields the maximum profit? b) What is the maximum profit? Math 2 Unit 11 Worksheet 5

12 10. A ball is thrown vertically upward with an initial velocity of 48 feet per second. If the ball started its flight at a height of 8 feet, then its height h at time tt can be determined by the function h = 16tt tt + 8 where h is measured in feet above ground level and tt is the number of seconds of flight. a) What is the height of the ball at 1 second? e) Sketch the graph of the function below. Label points that correspond to answers from parts a, b, c, and d. b) What is the height of the ball at 3 seconds? c) Determine the maximum height the ball attains. f) Label the yy-intercept on the graph above. What would the yy-intercept represent in this d) When does the ball hit the ground? Round answer situation? to the nearest tenth of a second. 11. A projectile is launched vertically from the top of a tower at a velocity of 80 feet per second. The tower is 200 feet high. The height of the projectile above the ground tt seconds after launch is given by the function h(tt) = 16tt tt a) What is the maximum height achieved by the projectile? b) How long after firing does it reach its maximum height? 12. A ball is thrown vertically into the air with an initial velocity of 64 feet per second. The function yy = 16tt tt gives its height above the ground after tt seconds. a) What is its height after 1.5 seconds? c) When will the ball hit the ground? b) What is its maximum height? Math 2 Unit 11 Worksheet 5

13 Math 2 Unit 11 Worksheet 6 Name: Graphing and Solving with Different Methods Date: Per: [1-3] Solve one part by factoring, one by completing the square, and one by quadratic formula for each problem. 1. a) xx 2 + 4xx 9 = 0 b) 3xx 2 7xx 5 = 0 c) xx 2 2xx 15 = 0 2. a) 2xx 2 19xx + 9 = 0 b) xx 2 12xx 30 = 0 c) xx 2 19xx + 84 = 0 3. a) 3xx 2 11xx 7 = 0 b) xx 2 2xx 12 = 0 c) 3xx 2 + 6xx 9 = 0 4. What strategy did you use to decide which problem you should do by each method? 5. Solve the following problem three times, one with each method: 2xx 2 12xx 14 = 0 a) Solve by: factoring b) Solve by: quadratic formula c) Solve by: completing the square d) Which method did you prefer for this problem? Why? Math 2 Unit 11 Worksheet 6

14 Vertex Form yy = aa(xx h) 2 + kk Standard Form yy = aaxx 2 + bbbb + cc Intercept Form yy = aa(xx pp)(xx qq) [6-7] Graph the following parabolas written in vertex form and determine the key features. 6. yy = 2(xx 3) Vertex: Axis of Symmetry: xx-intercept(s): yy-intercept: 7. yy = 1 2 (xx + 1)2 + 2 Vertex: Axis of Symmetry: xx-intercept(s): yy-intercept: [8-9] Graph the following parabolas written in standard form and determine the key features. 8. yy = xx 2 2xx 3 Vertex: Axis of Symmetry: xx-intercept(s): yy-intercept: 9. yy = 3xx 2 + 6xx + 3 Vertex: Axis of Symmetry: xx-intercept(s): yy-intercept: [10-11] Graph the following parabolas written in intercept form and determine the key features. 10. yy = 1 (xx + 2)(xx 4) 2 Vertex: Axis of Symmetry: xx-intercept(s): yy-intercept: 11. yy = 1 (xx + 3)(xx 3) 4 Vertex: Axis of Symmetry: xx-intercept(s): yy-intercept: Math 2 Unit 11 Worksheet 6

15 Math 2 Unit 11 Worksheet 7 Name: Systems of Equations Date: Per: [1-4] Find all solutions to ff(xx) = gg(xx). a) Solve by graphing, and b) solve algebraically using substitution. 1. ff(xx) = 2xx 5 and gg(xx) = 1 xx a) ff(xx) = gg(xx) at (, ) b) Solve the system algebraically to find when ff(xx) = gg(xx). 2. ff(xx) = 2xx and gg(xx) = 2xx + 5 a) ff(xx) = gg(xx) at (, ) and (, ) b) Solve the system algebraically to find when ff(xx) = gg(xx). 3. ff(xx) = xx and gg(xx) = 3xx + 4 a) ff(xx) = gg(xx) at (, ) and (, ) b) Solve the system algebraically to find when ff(xx) = gg(xx). 4. ff(xx) = 2xx + 4 and gg(xx) = (xx 2) 2 3 a) ff(xx) = gg(xx) at (, ) and (, ) b) Solve the system algebraically to find when ff(xx) = gg(xx). Math 2 Unit 11 Worksheet 7

16 [5-12] Find all values of xx for which ff(xx) = gg(xx). Solve algebraically. 5. ff(xx) = xx 7 6. ff(xx) = xx gg(xx) = xx 2 4xx 5 gg(xx) = 7xx ff(xx) = 14xx ff(xx) = 2xx + 2 gg(xx) = xx 2 13xx + 52 gg(xx) = xx 2 + 4xx ff(xx) = 3xx ff(xx) = 5xx 1 gg(xx) = xx 2 5 gg(xx) = xx ff(xx) = ff(xx) = 2xx 4 gg(xx) = xx 2 3 gg(xx) = (xx 2) Based on your answer for problem 12, what do you know about the graphs of these two functions? Math 2 Unit 11 Worksheet 7

17 Math 2 Unit 11 Name: Review Worksheet Date: Per: [1-6] What is the value of cc that makes each trinomial a perfect square? 1. xx xx + cc 2. xx 2 8xx + cc 3. xx 2 + 6xx + cc 4. xx 2 14xx + cc 5. xx 2 + 5xx + cc 6. xx 2 24xx + cc [7-10] Solve by completing the square. 7. aa aa + 32 = 0 8. xx 2 14xx + 44 = nn 2 12nn + 24 = pp 2 8pp 60 = 0 [11-14] Solve by the quadratic formula. 11. a) Solve xx 2 9xx + 21 = 15 using the quadratic formula by completing the boxes from the number choices xx = ( ) ± ( ) 2 ( )( )( ) ( )( ) 11. b) Solve for xx in 11a, write answer in simplified radical form. 11b) 12. a) Solve nn 2 20nn + 91 = 0 using the quadratic formula by completing the boxes from the number choices xx = ( ) ± ( ) 2 ( )( )( ) ( )( ) 12. b) Solve for xx in 12a, write answer in simplified radical form. 12b) Math 2 Unit 11 Review Worksheet

18 13. 2xx xx 3 = xx 2 = 6xx 14 [15-18] Solve each equation for xx using any method. Express answers in simplified radical form and complex solutions in terms of ii. 15. xx 2 1 = 3xx 16. xx = bb 2 + 8bb 39 = aa 2 15aa = 0 [19-26] Simplify. 19. (7 + 3ii) + 3(5 9ii) 20. 4(2 + 8ii) 2(12 3ii) 21. (3ii)(9ii) 22. ( 5ii)(4ii)(7ii)(2ii)(3ii 2 ) 23. (6 + 4ii)(2 3ii) 24. (2 3ii)(5 ii) 25. (7 + 2ii) (7 + 2ii)(7 2ii) Math 2 Unit 11 Review Worksheet

19 27. Select the graph with the correct solutions for ff(xx) = gg(xx) when ff(xx) = xx 2 and gg(xx) = xx + 2. A. B. C. D. 28. Select the correct solution(s) for ff(xx) = gg(xx). 29. Select the correct solution(s) for mm(xx) = kk(xx). A. xx = 2 oooooooo A. xx = 0 oooooooo B. xx = 2 oooooooo B. xx = 5 oooooooo C. xx = 2 aaaaaa 2 C. xx = 0 aaaaaa 5 D. There are no solutions to ff(xx) = gg(xx) D. There are no solutions to mm(xx) = kk(xx) [30-31] Given a quadratic function in standard form ff(xx) = aaxx 2 + bbbb + cc, determine the equivalent equation in vertex form ff(xx) = aa(xx h) 2 + kk, where aa, h, and kk are constants. 30. ff(xx) = xx 2 12xx ff(xx) = 8xx xx 22 Math 2 Unit 11 Review Worksheet

20 32. Consider the equation (xx 1) 2 + kk = 0. Create a value for kk that gives: a. no real solutions b. two real solutions 33. Consider the equation 2(xx + 4) 2 + kk = 0. Create a value for kk that gives: a. no real solutions b. two real solutions 34. A firework is launched from a platform that is 10 feet high. It is set to explode as it reaches maximum height. The height of the firework can be modeled by the function h = 16tt tt a. How long does it take to reach the maximum height? b. What is the maximum height? 35. Aaron Judge hit his longest homerun on June 12 th, 2017 with a length of 496 feet. Bryce Harper s longest homerun was hit the following month on July 22 nd, 2017 with a length of 467 feet. If the functions below model these hits, answer the following questions. Judge: h = 16tt tt + 4 Harper: h = 16tt tt + 4 a. Whose ball reached a greater vertical height? Show work to support your answer. b. If the ball was able to travel until it hit the ground, how long would Judge s ball be in the air? c. How long would Harper s ball be in the air? Math 2 Unit 11 Review Worksheet