Math 3 Unit 2: Solving Equations and Inequalities

Transcription

1 Math 3 Unit 2: Solving Equations and Inequalities Unit Title Standards 2.1 Analyzing Piecewise Functions F.IF Solve and Graph Absolute Value Equations F.IF.7B F.BF Solve and Graph Absolute Value Inequalities A.CED Factoring and Solving Quadratic Equations A.SSE Solve and Graph Quadratic Equations F.IF.7A 2.6 Factoring Sum and Difference of Cubes F.IF.8, A.REI.4b 2.7 Solutions of Functions F.IF Graphing Systems of Inequalities A.CED.3 Unit 2 Review Additional Clovis Unified Resources Clovis Unified is dedicated to helping you be successful in Math 3. On the website above you will find videos from Clovis Unified teachers on lessons, homework, and reviews. Digital copies of the worksheets, as well as hyperlinks to the videos listed on the back are also available at this site.

2 Math 3 Unit 2: Online Resources 2.1 Analyzing Piecewise Functions 2.2 Solve and Graph Absolute Value Equations 2.3 Solve and Graph Absolute Value Inequalities 2.4 Factoring and Solving Quadratic Equations 2.5 Solve and Graph Quadratic Equations 2.6 Factoring Sum and Difference of Cubes 2.7 Solutions of Functions 2.8 Graphing Systems of Inequalities Patrick JMT: Find the Formula for a Piecewise Function from a Graph Patrick JMT: Finding the Domain and Range of a Piecewise Function Cool Math: Finding Relative Maximums and Minimums Virtual Nerd: Graph an Absolute Value Function Purple Math: Solving Absolute Value Equations Patrick JMT: Simple Problems Solving Absolute Value Equations ehoweducation: Solving Absolute Value Equations Khan Academy: Solving Absolute Value Inequalities Mathispower4u: Solve and Graph Absolute Value Inequalities Khan Academy: Solving Quadratics by Factoring Purple Math: Solving Quadratic Equations by Factoring Mroldridge: Factoring any Quadratic Equation Khan Academy: Graphing Quadratic Equations Purple Math: Solving Quadratic Equations by Graphing Math Planet: Use Graphing to Solve Quadratic Equations Purple Math: Solving Quadratic Equations by Taking Square Roots Solving Quadratic Equations by Taking Square Roots Purple Math: Factoring Sums & Differences of Cubes & Perfect Squares Patrick JMT: Factoring Sums and Differences of Cubes Khan Academy: Factoring Sum of Cubes Melissa Gresham: Solving Polynomial Equations Using the Sum and Difference of Cubes M-Squared Tutorials: Finding solutions from a graph for when f(x) =0 Purple Math: Finding solutions from a graph where f(x) = g(x) Bethany M: Finding values of a function from a graph ( ex: f(2.5), f(0) ) Virtual Nerd: Finding zeros of a function from a table of values Khan Academy: Graphing Linear Systems of Inequalities No Solution Patrick JMT: Graphing Linear Systems of Inequalities

3 Math 3 Unit 2 Worksheet 1 Name: Analyzing Piecewise-defined Functions Date: Per: Answer the following questions about the piecewise-defined functions below. 1. (a) State the open interval(s) on which ff is increasing. (b) State the open interval(s) on which ff is decreasing. (c) State the domain and range of ff. (d) State the coordinates of any relative minimums of ff. (e) State the coordinates of any relative maximums of ff. (f) Write a two pieced piecewise-defined function, ff, that accurately represents the graph of f shown above. 2. (a) State the open interval(s) on which ff is increasing. (b) State the open interval(s) on which ff is decreasing. (c) State the domain and range of ff. (d) State the coordinates of any relative minimums of ff. (e) State the coordinates of any relative maximums of ff. (f) Write a two pieced piecewise-defined function, ff, that accurately represents the graph of ff shown above. Math 3 Unit 2 Worksheet 1

4 3. ( 2, 2) (2, 2) (6, 2) (a) State the open interval(s) on which ff is increasing. (b) State the open interval(s) on which ff is decreasing. (c) State the domain and range of ff. ( 4, 2) (4, 2) (d) State the coordinates of any relative minimums of ff. (e) State the coordinates of any relative maximums of ff. (f) Write a three pieced piecewise-defined function, ff, that accurately represents the graph of f shown above. 4. (3, 3) (5, 3) (a) State the open interval(s) on which ff is increasing. (1, 2) (b) State the open interval(s) on which ff is decreasing. (c) State the open interval(s) on which ff is constant. (d) State the domain and range of ff. (2, 0) (e) State the coordinates of any relative minimums of ff. (f) State the coordinates of any relative maximums of ff. (g) Write a three pieced piecewise-defined function, ff that accurately represents the graph of f shown above. Math 3 Unit 2 Worksheet 1

5 Math 3 Unit 2 Worksheet 2 Name: Solving and Graphing Absolute Value Equations Date: Per: [1-4] Accurately graph ff(xx) and gg(xx) on the same set of axes. 1. ff(xx) = xx 2 and gg(xx) = 3 a) ff(xx) = gg(xx) at xx = b) Solve algebraically for xx, xx 2 = 3 2. ff(xx) = 2 xx + 3 and gg(xx) = 2 a) ff(xx) = gg(xx) at xx = b) Solve algebraically for xx, ff(xx) = 2 3. ff(xx) = 3 xx 4 and gg(xx) = 2 a) ff(xx) = gg(xx) at xx = b) Solve algebraically for xx,ff(xx) = gg(xx) 4. ff(xx) = 1 2 xx + 2 1and gg(xx) = 1 a) ff(xx) = gg(xx) at xx = b) Solve algebraically for xx, ff(xx) = gg(xx) Math 3 Unit 2 Worksheet 2

6 [5-14] Solve the following absolute value equations for xx and graph the solution(s) on a number line. If there is no solution write none and explain why = 8xx 1 6. ff(xx) = xx and gg(xx) = xx + 9 Solve: ff(gg(xx)) = xx = 3 8. ff(xx) = xx and gg(xx) = 2xx 5 Solve: 3(ff gg)(xx) = xx + 4 = ff(xx) = 5 + 2xx and gg(xx) = xx 2 Solve: 7 + gg(ff(xx)) = xx + 1 = ff(xx) = 7xx 10 and gg(xx) = xx Solve: 2(gg ff)(xx) + 1 = ff(xx) = xx and gg(xx) = 3xx xx + 7 = 1 Solve: 7 3(ff gg)(xx) = 14 Math 3 Unit 2 Worksheet 2

7 Math 3 Unit 2 Worksheet 3 Name: Solving and Graphing Absolute Value Inequalities Date: Per: [1-4] Accurately graph ff(xx) and gg(xx) on the same set of axes. 1. ff(xx) = xx and gg(xx) = 5 a) Highlight the portion of gg(xx) where ff(xx) gg(xx) and state the interval along the x-axis. b) Solve algebraically for x, xx 5 2. ff(xx) = 2 and gg(xx) = xx a) Highlight the portion of ff(xx) where gg(xx) ff(xx) and state in interval notation the interval along the x-axis. b) Solve algebraically for xx, gg(xx) ff(xx) 3. ff(xx) = xx + 1 and gg(xx) = 3 a) Highlight the portion of gg(xx) where ff(xx) gg(xx) and state the interval along the x-axis b) Solve algebraically for xx in inequality notation, ff(xx) gg(xx) 4. gg(xx) = 5 and ff(xx) = xx 2 a) Highlight the portion of gg(xx) where gg(xx) ff(xx) and state the interval along the x-axis b) Solve algebraically for xx in inequality notation, gg(xx) ff(xx) Math 3 Unit 2 Worksheet 3

8 [5-14] Solve the following absolute value inequalities for xx and graph the solution on a number line. Write the solution in interval notation. If there is no solution, explain why. 5. 3xx xx + 4 < ff(xx) = xx and gg(xx) = 4xx xx < 6 Solve: ff(gg(xx)) 14 < 5 9. ff(xx) = 7 2xx and gg(xx) = xx xx > 3 7 Solve: 2gg(ff(xx)) xx < xx + 4 < ff(xx) = xx and gg(xx) = xx xx + 4 > 2 3 Solve: ff(gg(xx)) 3 5 Math 3 Unit 2 Worksheet 3

9 [15-16] Selected Response. Select ALL answers that apply 15. What is the solution of 6xx 9 33? a. 4 xx 7 b. 7 xx 4 c. xx 4 oooo xx 7 d. xx 7 oooo xx Which inequalities are equivalent to 2 xx 3 < 8? a. xx 3 < 4 b. xx 3 > 4 c. xx 3 < 10 d. 2xx + 6 < 8 [17-24] Use the definitions of absolute value equations and inequalities to determine if the statement is True or False. 17. True or False: 3xx + 7 = 13 is equivalent to 3xx + 7 = 13 oooo 3xx + 7 = True or False: 9xx + 1 < 19 is equivalent to 9xx + 1 < 19 oooo 9xx + 1 > True or False: 2xx 4 < 12 is equivalent to 12 < 2xx 4 < True or False: 6xx 4 = 10 is equivalent to 6xx 4 = 10 oooo 6xx 4 = True or False: 2xx + 5 > 1 has no solution 22. True or False: xx < 2 or xx > 7 is equivalent to 7 < xx < True or False: xx 5 or xx > 1 is equivalent to 1 < xx True or False: xx 3 or xx < 5 is equivalent to 3 xx < An archery store carries bows that are from 40 to 52 inches long. They recommend that bows be 2 times a 3 person s arm span, ss. a) Write a compound inequality, in terms of ss, that represents the problem. b) Solve the compound inequality from part a) c) Graph the solution from part b) on a number line. d) If a person s height is equivalent to their arm span, what is the height of the tallest person that the archery store carries bows for? Math 3 Unit 2 Worksheet 3

10 26. A sporting goods store carries softball bats that are from 25 to 35 inches long. They recommend that softball bats be 5 of a person s height, h. 8 a) Write a compound inequality, in terms of h, that represents the problem. b) Solve the compound inequality from part a) c) Graph the solution from part b) on a number line. d) What is the height of the shortest person the sporting goods store carries a softball bat for? 27. A space-themed miniature golf course named Puttnik carries putters from 24 to 40 inches long. They recommend that putters be 4 times a person s height, h. 9 a) Write a compound inequality, in terms of h, that represents the problem. b) Solve the compound inequality from part a) c) Graph the solution from part b) on a number line. d) What is the height of the tallest person that the golf course carries a putter for? Math 3 Unit 2 Worksheet 3

11 Math 3 Unit 2 Worksheet 4 Name: Factoring and Solving Quadratic Equations Date: Per: [1 16] Completely Factor the Following 1. xx 2 8xx xx 2 + xx 6 3. xx 2 3xx xx xx 2 12xx 6. xx 2 + 4xx xx 2 + 5xx xx 2 + 4xx xx xx xx 2 + 7xx xx xx xx xx(xx 5) + 3(xx 5) 14. 7(xx + 1) xx(xx + 1) 15. 3xx 2 8xx xx 2 + 5xx + 2 [17 25] Solve By Factoring 17. xx 2 7xx + 10 = xx 2 4xx 12 = xx 2 + 9xx 5 = xx(3xx 1) = (xx + 2)(2xx 1) = (2xx + 3)(3xx + 1) = xx 2 8xx = xx 2 = 10xx Math 3 Unit 2 Worksheet 4

12 25. 8xx 2 + 2xx = xx 2 16 = xx(xx 1) + 9(xx 1) = xx 2 = 121 [29 34] Solve By Using Square Roots 29. (xx 5) 2 = (xx + 1) 2 = (xx + 2) 2 7 = (xx + 4) 2 = (xx 5) 2 = (2xx 3) 2 = 17 [35 38] Completely Simplify Each Expression 35. 6± ± ( 9)2 4(1)(4) 2(1) 37. 5± 52 4(5)( 1) 2(5) 38. 6± ( 6)2 4(1)( 3) 2(1) Math 3 Unit 2 Worksheet 4

13 Math 3 Unit 2 Worksheet 5 Name: Solving and Graphing Quadratic Equations Date: Per: [1-4] Accurately graph ff(xx) and gg(xx) on the same set of axes. 1. ff(xx) = (xx 1) and gg(xx) = 0 a) ff(xx) = gg(xx) at xx = b) Solve algebraically for xx: (xx 1) = 0 2. ff(xx) = 1 (xx 2 2)2 2 and gg(xx) = 0 a) ff(xx) = gg(xx) at xx = b) Solve algebraically for xx: 1 (xx 2 2)2 2 = 0 3. ff(xx) = 2(xx + 3) and gg(xx) = 0 a) ff(xx) = gg(xx) at xx = b) Solve algebraically for xx: ff(xx) = gg(xx) 4. ff(xx) = (xx 1) 2 2 and gg(xx) = 0 a) ff(xx) = gg(xx) at xx b) Solve algebraically for xx to the nearest tenth, ff(xx) = gg(xx) Math 3 Unit 2 Worksheet 5

14 [5-16] Solve the following quadratic equations for x. 5. 2(xx + 3) = xx 2 11xx 4 = 0 7. (xx + 1)(xx + 5) = 3 8. xx 2 + 4xx 6 = 0 9. xx 2 + 2xx + 5 = (xx + 4) 2 5 = xx 2 6xx + 5 = xx(xx 3) = xx 8 = xx 2 + xx 14. 3xx 2 + xx + 2 = (xx + 7) 2 = (xx 5) 2 = 88 Math 3 Unit 2 Worksheet 5

15 Math 3 Unit 2 Worksheet 6 Name: Factoring Sum and Difference of Cubes Date: Per: Factor completely: 1. xx xx 3 yy yy aa 3 125rr mm nn yy 4 + 3yy For each equation, write in factored form then determine the real and imaginary solutions. 7. xx = 0 Factored Form: Real solutions: Imaginary Solutions: 8. 0 = xx 3 Factored Form: Real solutions: Imaginary Solutions: 9. 8xx = 0 Factored Form: Real solutions: Imaginary Solutions: Math 3 Unit 2 Worksheet 6

16 xx 3 = 0 Factored Form: Real solutions: Imaginary Solutions: = 250xx 4 54xx Factored Form: Real solutions: Imaginary Solutions: xx + 81xx 4 = 0 Factored Form: Real solutions: Imaginary Solutions: 13. Is it possible to solve an equation using sum or difference of cubes factoring and have all solutions be imaginary? Please explain your thinking. Math 3 Unit 2 Worksheet 6

17 Math 3 Unit 2 Worksheet 7 Name: Solutions of Functions Date: Per: 1. The graph of yy = ff(xx) is shown in the graph below. C a) List all of the labeled points that are solutions for ff(xx) = 0. B D b) List all of the labeled points that are solutions for yy = ff(xx). A E c) List all of the labeled points that are solutions for xx = 0. d) ff(0) = 2. The graph of yy = ff(xx) and yy = gg(xx) is shown in the graph below. ff(xx) C a) List all of the labeled points that are solutions for ff(xx) = 0. B b) List all of the labeled points that are solutions for gg(xx) = ff(xx). A D c) List all of the labeled points that are solutions for xx = 0 on the graph of ff(xx). gg(xx) d) List all of the labeled points that are solutions for yy = ff(xx). E e) List all of the labeled points that are solutions for yy = gg(xx). 3. The graph of yy = ff(xx) and yy = gg(xx) is shown in the graph below. a) How many solutions are there for gg(xx) = 0? ff(xx) gg(xx) b) How many solutions are there for gg(xx) = ff(xx)? c) ff(2.5) =. d) How many times does gg(xx) = 3.5? e) On what approximate interval is gg(xx) < 1? f) How many roots does gg(xx) have? g) Is gg(xx) = 0 between xx = 4 and xx = 5? Math 3 Unit 2 Worksheet 7

18 4. State whether the following statements are correct (A) or incorrect (B) a) ff(4) > 0 b) ff( 2) < 0 c) ff(0) = 2 d) ff(2) = 0 e) ff(5.123) = 2 f) y = 0 at ff(2) g) ff(xx) = 0 between xx = 6 and xx = 8 h) ff(xx) has a relative maximum at (4, 2) i) ff(xx) equals zero five times j). ff(xx) has a relative minimum at (0, 2) 5. The table below shows several points on two continuous functions, ff(xx) and gg(xx). xx ff(xx) gg(xx) a) On the number line below, shade the interval(s) between the integers where the solution(s) to ff(xx) = gg(xx) must exist. If it is not necessary that a solution exists, explain why b) On the number line below, shade the interval(s) between the integers where the solution(s) to ff(xx) = 0 must exist. If it is not necessary that a solution exists, explain why c) On the number line below, shade the interval(s) between the integers where the solution(s) to gg(xx) = 4 must exist. If it is not necessary that a solution exists, explain why Math 3 Unit 2 Worksheet 7

19 6. The table below shows several points on two continuous functions, ff(xx) and gg(xx). x ff(xx) gg(xx) a) On the number line below, shade the interval(s) between the integers where the solution(s) to ff(xx) = gg(xx) must exist. If it is not necessary that a solution exists, explain why b) On the number line below, shade the interval(s) between the integers where the solution(s) to ff(xx) = 0 must exist. If it is not necessary that a solution exists, explain why c) On the number line below, shade the interval(s) between the integers where the solution(s) to gg(xx) = 3 must exist. If it is not necessary that a solution exists, explain why The table below shows several points on three continuous functions, ff(xx), gg(xx) and h(xx). xx ff(xx) gg(xx) h(xx) a) On the number line below, shade the interval(s) between the integers where the solution(s) to ff(xx) = gg(xx) must exist. If it is not necessary that a solution exists, explain why b) On the number line below, shade the interval(s) between the integers where the solution(s) to ff(xx) = h(xx) must exist. If it is not necessary that a solution exists, explain why c) On the number line below, shade the interval(s) between the integers where the solution(s) to gg(xx) = h(xx) must exist. If it is not necessary that a solution exists, explain why d) On the number line below, shade the interval(s) between the integers where the solution(s) to gg(xx) = 2 and h(xx) = 2 must exist. If it is not necessary that a solution exists, explain why e) On the number line below, shade the interval(s) between the integers where the solution(s) to h(xx) = 0 must exist. If it is not necessary that a solution exists, explain why Math 3 Unit 2 Worksheet 7

20 [8-15]: Solve. Show all work. 8. xx 2 + 2xx = 5xx xx 2 + 4xx = 5xx xx(xx + 3) = 12xx (xx 3) 2 = (xx + 2) 2 = xx = xx 7 = xx = 0 Hint: Quadratic formula is not needed for any of the above problems except #16. All answers may be found below; however, they are in no particular order. { 14, 6} 3 ± 2ii 11 { 2, 5} ± 5 5 ± 5ii 3 5, { 6, 20} 7, 4 2 ± Math 3 Unit 2 Worksheet 7

21 Math 3 Unit 2 Worksheet 8 Name: Graphing Systems of Inequalities Date: Per: Graph the system of inequalities on the graph provided. If the solution exists, then name 2 points in the solution set. yy 4 xx yy < xx2 + 1 yy > 2xx 3 yy 2 xx 3 3. yy xx + 1 yy < 2 xx 2 4. yy 2xx 1 yy < 1 2 xx 1 5. yy xx 2 yy 2 xx 6. yy < 2 yy (xx + 1) Math 3 Unit 2 Worksheet 8

22 yy < 4 7. yy 2 xx yy > xx 8. yy 3 xx yy > 3 xx 9. yy 2 yy > (xx 2) xx > 1 yy 2 xx yy 1 xx 11. yy < 0 yy 2 xx xx 0 yy xx yy 1 2 xx2 4 Math 3 Unit 2 Worksheet 8

23 [13-21]: Solve the following for xx xx < xx xx + 5 < xx xx + 5 > xx + 9 > xx(xx + 3) = 5(8 xx) (xx + 7) 2 = xx 3 = 0 Hint: Quadratic formula is only needed for question 21. All answers may be found below; however, they are in no particular order. (, ) { } 7 ± 3ii , , 5 (, 35] [13, ) { 10, 5 ± 5ii 3 } ( 9, 3) (, 19) (1, ) 2 Math 3 Unit 2 Worksheet 8

24 Math 3 Unit 2 Worksheet 8

25 Math 3 Unit 2 Name: Review Date: Per: 1. (a) State the open interval(s) on which ff is increasing. (b) State the open interval(s) on which ff is decreasing. (c) State the domain and range of ff. (d) State the coordinates of any relative minimums of ff. (e) State the coordinates of any relative maximums of ff. (f) Write a two pieced piecewise-defined function, ff, that accurately represents the graph of ff shown above. 2. (a) State the open interval(s) on which ff is increasing. (b) State the open interval(s) on which ff is decreasing. (c) State the domain and range of ff. (d) State the coordinates of any relative minimums of ff. (e) State the coordinates of any relative maximums of ff. (f) Write a three pieced piecewise-defined function, ff, that accurately represents the graph of ff shown above. Math 3 Unit 2 Review Worksheet

26 3. Graph ff(xx) and gg(xx) on the same set of axes. Use the graph and verify algebraically, where ff(xx) = gg(xx). ff(xx) = 3 xx + 1 and gg(xx) = 6 [4-7] Solve the following absolute value equations for x and graph the solution(s) on a number line. If there is no solution write none and explain why. 4. ff(xx) = xx and gg(xx) = 3xx xx 5 = 4 2 Solve: ff(gg(xx)) + 1 = ff(xx) = 3xx + 2 and gg(xx) = xx 7. 3 xx 10 = 0 4 Solve: 2 (gg ff)(xx) = 20 5 [8-11] Solve the following absolute value inequalities for x and graph the solution(s) on a number line. If there is no solution write none and explain why. 8. 3xx xx + 4 < 6 3 Math 3 Unit 2 Review Worksheet

27 xx < xx The graph of yy = ff(xx) and yy = gg(xx) is shown in the graph below. ff(xx) a) List all of the labeled points that are solutions for gg(xx) = 0. A B F b) List all of the labeled points that are solutions for gg(xx) = ff(xx). E D c) List all of the labeled points that are solutions for xx = 0 on the graph of ff(xx). gg(xx) C d) List all of the labeled points that are solution(s) for gg(xx) < ff(xx). e) List all of the labeled points that are solution(s) for gg(xx) > ff(xx). 13. The table below shows several points on two continuous functions, ff(xx) and gg(xx) xx ff(xx) gg(xx) a) On the number line below, shade the interval(s) between the integers where the solution(s) to ff(xx) = gg(xx) must exist. If no solutions must exist, explain why b) On the number line below, shade the interval(s) between the integers where the solution(s) to gg(xx) = 0 must exist. If no solutions must exist, explain why c) On the number line below, shade the interval(s) between the integers where the solution(s) to ff(xx) = 3 must exist. If no solutions must exist, explain why Math 3 Unit 2 Review Worksheet

28 [14-17] Graph the system of inequalities on the graph provided. 14. yy 4 yy > xx 15. yy > xx2 + 1 yy < 1 xx yy 1 xx yy xx 17. yy > 2xx 2 yy < xx + 1 Solve for x (xx 2) = ff(xx) = xx 2 and gg(xx) = xx (xx 5) 2 6 = 120 Solve: 2ff gg(xx) + 30 = xx = 24 3xx xx + 36 = 6xx xx ff(xx) = 8xx 4 and gg(xx) = 7xx 2 Solve: (ff gg)(xx) = 0 Math 3 Unit 2 Review Worksheet

29 24. Use the graph of ff(xx) below to answer the following questions. 4 y x True or False: a) ff(5) > 1 b) ff(8) < 1 c) ff(xx) = 1 between xx = 5 and xx = 6 d) ff(xx) = 1 between xx = 0 and xx = 2 e) ff(0) = 0 f) ff( 7) = 4 g) ff(3.743) = 1 h) ff(9.324) = 2 i) ff(xx) has a relative maximum at ( 2, 4) j) ff(xx) has a relative maximum at (3, 2) k) ff(xx) has a relative minimum at (9, 2) l) ff(xx) has a relative minimum at ( 2, 4) Math 3 Unit 2 Review Worksheet