30S Pre Calculus Final Exam Review Chapters 5 8

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1 30S Pre Calculus Final Exam Review Chapters 5 Name:

2 Chapter 5: Graphing Inequalites and Sstems of Equations Exam Review Multiple Choice Identif the choice that best completes the statement or answers the question. 1. Use the graph to write the solution of this quadratic inequalit: x 1 A., C., B., D.,. Represent the solution of this quadratic inequalit on a number line: A. B. C. D

3 3. Represent the solution of this quadratic inequalit on a number line: A B C D Solve the quadratic inequalit: A. 7 x < or x >, C. 7 x < or x >, B. < x < 7, D. 7 < x <, 5. Which coordinates are a solution of the inequalit? A. (, ) B. (0, 1) C. (3, 7) D. (5, ) 6. Match the inequalit x + to its graph. A. 6 C x x 6

4 B. 6 D x x 6 7. Write an inequalit to describe this graph x 3 6 A. 3x C. 3x 3 9 B. 3x 3 9 D. 3x Which ordered pair is a solution of the quadratic inequalit? A. (3, 36) B. (1, 1) C. ( 1, ) D. (, 19) 9. Which graph represents the inequalit?

5 A. 6 C x x 6 B. 6 D x x Which graph represents the inequalit? A. 1 C x x 1

6 B. 1 D x x Write the coordinates of the points of intersection. Give the solutions to the nearest tenth. A. (6, ) and (9, 1) C. (, 6) and ( 1, 9) B. (9, ) and (6, 1) D. (, 9) and ( 1, 6) 1. Two numbers are related: The sum of twice the first number and the square of the second number is 9. Twice the difference between the first number and the second number is 15. Which sstem models this relationship? A. C. B. D. 13. Solve this quadratic-quadratic sstem algebraicall. A. ( 1, 1) and (3, 7) C. (7, 1) and (1, 3) B. (1, 1) and (7, 3) D. ( 1, 1) and (7, 3) Short Answer 1. Solve this quadratic inequalit: 15. Solve this quadratic inequalit:

7 16. Graph the inequalit: 17. Graph the inequalit: 1. a) Graph the inequalit: b) Write the coordinates of 3 points that satisf the inequalit.

8 19. Write an inequalit to describe this graph x 6 0. Graph this sstem of equations. a) Sketch the graphs on the same grid.

9 b) Write the coordinates of the points of intersection. 1. Solve this quadratic-quadratic sstem algebraicall.. Solve this quadratic-quadratic sstem algebraicall. Problem 3. Consider this inequalit: a) Solve the inequalit b factoring. b) Illustrate the solution on a number line. c) What do ou notice about the solution of the inequalit. Explain wh.. A baseball is hit upward from a platform that is m high at an initial speed of 0 m/s. The approximate height of the baseball, h metres, after t seconds is given b the equation. Determine the time period for which the baseball is higher than 1 m. Explain our strateg. 5. The length of a rectangular garden is 6 m greater than its width. The area of the garden is at least 0 m. What are possible dimensions of the garden, to the nearest tenth of a metre? Show our work. Verif the solution. 6. Joe has up to $50 to spend at a sportswear store. A pair of socks costs $5 and a sweatshirt costs $13. a) Write an inequalit to describe how Joe can spend his mone. b) Use a graphing calculator to graph the inequalit. Sketch the graph.

10 c) Use the graph to describe possible was Joe can spend up to $50. Show and explain our work. 7. Two numbers are related in this wa: Twice the square of the first number minus the second number is 3. The square of the sum of the first number and 5 is equal to the second number minus. a) Create a sstem of equations to represent this relationship. b) Solve the sstem to determine the numbers. Explain the strateg ou used.

11 Chapter 5: Graphing Inequalites and Sstems of Equations Exam Review Answer Section MULTIPLE CHOICE 1. B. C 3. D. C 5. A 6. D 7. B. D 9. B 10. A 11. D 1. B 13. B SHORT ANSWER 1. The solution is:, 15. The solution is:, x 6

12 x 6 1. a) x 6 b) Sample response: Three points that satisf the inequalit have coordinates: ( 1, 7), (1, 1), (, ) 19., or

13 0. ( 3, 16) ( 7, 0) x 1. There are no real solutions.. The solutions are: (, 0) and (0, ) PROBLEM 3. a) Solve: b) When, such as, L.S. = 1 and R.S. = 0; so satisfies the inequalit. When, such as, L.S. = 1 and R.S. = 0; so satisfies the inequalit. The solution is:, x c) There is onl one number that does not satisf the inequalit. Since the square of an non-zero number is positive, an real number but.5 satisfies the inequalit.. Rearrange the equation to isolate h: An inequalit that represents the situation is: A related quadratic equation is: Solve the equation.

14 Substitute into the inequalit. When, such as, and ; so does not satisf the inequalit. When, such as, and ; so does not satisf the inequalit. When, such as, and ; so does satisf the inequalit. The solution is: So, the baseball is higher than 1 m between 1.1 s and.9 s after it is hit. 5. Write an inequalit to represent the problem. Let x m represent the width of the garden. Then its length is m. And its area, in square metres, is. The area is at least 0 m. So, an inequalit is: Use a graphing calculator to graph the corresponding quadratic function: Determine the x-intercepts: and From the graph, the inequalit is greater than or equal to 0 for Since the width of the garden is positive, the solution of the problem is: The width of the garden is greater than or equal to approximatel. m and its length is greater than or equal to approximatel, or. m. Verif the solution. The area of the garden with dimensions. m and. m is: This is greater than 0 m, so the possible dimensions are correct. 6. a) Let x represent the number of pairs of socks and represent the number of sweatshirts. An inequalit is: b) Write the corresponding equation, then solve for. Enter this equation in a graphing calculator. Use the origin as a test point. Substitute: and L.S. = 0; R.S. = 50 Since 0 < 50 is a true statement, the origin lies in the shaded region. So, shade the region that contains the origin.

15 x 1 c) Choose points with whole-number coordinates that lie in the shaded region. For example: Point A(, 3) illustrates that Joe can purchase pairs of socks and 3 sweatshirts. Point B(3, ) illustrates that Joe can purchase 3 pairs of socks and sweatshirts. 7. a) Let x represent the first number and let represent the second number. The statement that twice the square of the first number minus the second number is 3 can be modelled with the equation: The statement that the square of the sum of the first number and 5 is equal to the second number minus can be modelled with the equation: So, the sstem of equations that represents this relationship is: b) From equation, substitute in equation. So, or Substitute each value of x in equation. When : When : The numbers are: and 11; 1 and 91 Verif the solutions using the statement of the problem. For and : Two times ( ) minus 11 is 3. The square of is equal to

16 These numbers satisf the problem statement. For and : Two times (1) minus 91 is 3. is equal to 91 minus. These numbers satisf the problem statement. So, the numbers are: and 11; 1 and 91

17 Cahpter 6: Trigonometr Exam Review Multiple Choice Identif the choice that best completes the statement or answers the question. 1. Point P(9, ) is on the terminal arm of an angle in standard position. To the nearest tenth, determine the distance from the origin to P. A. 6.5 B. 9. C D Point P(x, ) is on the terminal arm of a 35 angle in standard position. The distance r between P and the origin is 7. To the nearest tenth, determine the coordinates of P. A. (5.7,.0) B. (.0, 5.7) C. (., 6.6) D. (.0, 11.5) 3. Determine the reference angle for the angle 90 in standard position. A. 90 B. 0 C. 110 D. 70. Determine the exact value of tan 10. A. B. C. D. 5. In XYZ, XY = 6 cm and X = 33. For which value of YZ is no triangle possible? A. 6 cm B. 3 cm C. cm D. 1 cm 6. For DEF, write the Sine Law equation ou would use to determine the measure of E. E 5.3 cm D 3.5 cm 77 F A. C. B. D.

18 7. For DEF, determine the measure of E to the nearest degree. E 7. cm 5 D. cm F A. 155 B. 7 C. 5 D. 5. For PQR, determine the length of QR to the nearest tenth of a centimetre. R 33 P Q.3 cm A. 6. cm B. 3. cm C. 15. cm D..6 cm 9. In PQR, determine the measure of Q to the nearest degree. P R 3.5 cm 5.5 cm Q A. 70 B. 110 C. 136 D. 15

19 10. In MNP, determine the lengths of the two unknown sides to the nearest tenth of a centimetre. P M cm 107 N A. NP = 1.0 cm; MP =. cm C. NP = 19.7 cm; MP =.7 cm B. NP = 7. cm; MP = 1.1 cm D. NP = 19.7 cm; MP =. cm 11. In XYZ, determine the length of XZ to the nearest tenth of a centimetre. X.1 cm 11 Y 6.7 cm Z A cm B. 7.7 cm C. 1.7 cm D cm 1. In ABC, AB = 6 cm, BC =.5 cm, and AC = 5. cm. Determine the measure of B to the nearest degree. C B A A. B. 0 C. 91 D. 3

20 Short Answer 13. Determine the exact value of cos Angle is in standard position in Quadrant 1 and sin. a) What are cos and tan? b) What is the measure of to the nearest degree? 15. A flagpole is 11.0 m high. At a certain point, the angle between the ground and Jon s line of sight to the top of the flagpole is 61. To the nearest tenth of a metre, how far is Jon from the flagpole? 16. What is the distance from the origin to the point P(3, )? 17. a) Determine the reference angle for the angle 3 in standard position. b) Determine the other angles between 0 and 360 that have the same reference angle. 1. To the nearest degree, which values of satisf this equation for? 19. Determine the exact value of cos Given the following information about ABC, determine how man triangles can be constructed. a = 5.6 cm, c = 7. cm, A = 3 1. Solve UVW. Give angle measures to the nearest degree and side lengths to the nearest tenth of a centimetre. U 7 cm W cm 0 V. In KLM, KL = 5.1 cm, LM = 6.9 cm, and M = 33. a) Determine how man triangles can be constructed. Justif our answer. b) Sketch a diagram to show an possible triangles. 3. In PQR, PQ = 6.5 cm, R = 6, and QR =.7 cm. a) Determine how man triangles can be constructed. Justif our answer. b) Sketch a diagram to show an possible triangles.

21 . In ABC, AB = 3.9 cm, C = 63, and BC = 5. cm. a) Determine how man triangles can be constructed. Justif our answer. b) Sketch a diagram to show an possible triangles. 5. In GHI, determine the measures of all the angles to the nearest degree. G. cm. cm H 11. cm I 6. In ABC, B = 5, AB =. cm, and BC = 5.9 cm. Determine the measure of C to the nearest degree. Problem 7. Point P( 1, 5) is a terminal point of an angle in standard position. a) Determine the ratios,, and. b) Determine the measure of to the nearest degree. Show our work.. Point P(9, ) is a terminal point of an angle in standard position. Determine to the nearest degree, then sketch the angle. Show our work. 9. In, AB = 35 cm, BC = 19 cm, and. a) Explain wh it is possible to draw two different triangles with these measures. b) Sketch a diagram to show both triangles. 30. Two firefighters want to rescue a cat stuck in a tree. The cat is at an angle of elevation of 3 with respect to one firefighter and 0 with respect to the other firefighter. The firefighters are 30 m apart, and on opposite sides of the tree. To the nearest tenth of a metre, how high off the ground is the cat? Explain our strateg. 31. Two divers are 50 m apart. Each diver sees a treasure chest on the sea floor. The treasure chest is verticall below the line between the divers. From the divers, the angles of depression to the treasure chest are 35 and 51. To the nearest metre, how far is the treasure chest from each diver? Consider possible cases and show our work.

22 3. In DEF, DE = 7.5 cm, D = 70, and EF = 9 cm. a) Determine how man triangles can be drawn. b) Solve the triangle(s). Give angle measures to the nearest degree and side lengths to the nearest tenth of a centimetre. Show our work. 33. Solve ABC. Give angle measures to the nearest degree. A B 10 cm 5 cm 9 cm C

23 Cahpter 6: Trigonometr Exam Review Answer Section MULTIPLE CHOICE 1. B. A 3. D. B 5. B 6. B 7. B. A 9. B 10. B 11. C 1. D SHORT ANSWER 13. cos 5 = 1. a) 3 b) 15. approximatel 6.1 m 16. Point P is 13 units from the origin. 17. a) The reference angle is 16. b) The other angles that have the same reference angle are: 1. and 19. cos Two triangles can be constructed. 1. W = 3 U = 66 VW = 6.5 cm. a)...

24 Since, two triangles are possible. b) L 6.9 cm 5.1 cm 5.1 cm M 33 K 1 K 3. a)... Since, one triangle can be constructed. b) Q.7 cm 6.5 cm R 6 P. a)... Since, no triangle can be constructed. b) No triangle can be constructed. 5. G = 5 H = I = 7 6. C = 5 PROBLEM 7. a) Determine the distance r from the origin to P.,

25 b) The reference angle is: Since is in Quadrant 3, the angle is approximatel:. Determine the distance r from the origin to P., Use: The reference angle, to the nearest degree, is: Since x is positive and is negative, the terminal arm is in Quadrant, and is approximatel x 9. a), or Since, there are two possible triangles with the given measures.

26 b) B 35 cm 19 cm 19 cm A 30 C C Sketch a diagram for the situation. C represents the position of the cat and h represents its height above the ground. AB represents the distance between firefighters A and B. C h a A m D B Use the angle sum in a triangle. Use the Sine Law to determine a, the length of BC. Use the sine ratio in right. The cat is approximatel 0.6 m above the ground. 31. The treasure chest could be between the two divers or on one side of both divers. Case 1: The treasure chest C is between the two divers, A and B.

27 A 50 m B b a C The treasure chest is approximatel 9 m and 39 m from the divers. Case : The treasure chest C is on one side of both divers A and B. A 50 m B b a C The treasure chest is approximatel 10 m and 11 m from the divers. 3. a) Sketch the triangle.

28 D cm E 9 cm F Use the ratio to determine the number of possible triangles. Since, one triangle is possible. b) Solve for F: Solve for E: Solve for DF: So, in DEF, the approximate measures are: E = 5, F = 5, and DF =.1 cm.

29 33. Use: Use:

30 Chapter 7: Rational Expressions and Equations Multiple Choice Identif the choice that best completes the statement or answers the question. 1. Identif the non-permissible values of x for this rational expression: A. and C. and B. and D. and. Identif the non-permissible value of the variable for this rational expression: A. C. B. D. all values are permissible 3. Simplif this rational expression and state the non-permissible values of the variable. A. ; and C. ; and B. ; and D. ; and. Determine the non-permissible values for this rational expression: A. B. and and C. and D. and 5. Simplif this expression: A. C. B. D.

31 6. Simplif this expression: A. C. B. D. 7. Simplif. A. C. B. D.. Simplif. A. C. B. D. 9. Simplif. A. C. B. D. 10. Solve. 11. Solve. A. C. B. D. no solution A. C. B. D. no solution

32 1. Solve. A. C. B. D. 13. Andres runs 6 km on pavement and then 1 km on gravel. His speed on pavement is twice as fast as his speed on gravel. If he finishes his run in 10 min, what is his speed on the gravel surface? Give the answer to 1 decimal place where necessar. A. 1.1 km/h B. km/h C. 1 km/h D. 1.9 km/h 1. Dana plants a tomato seed and a sunflower seed for a science project. She finds that on average her sunflower plant grows three times as fast as her tomato plant. It takes the tomato 0 das longer to reach a height of 1 cm. What is the growth rate, in centimetres per da, of the tomato seedling? Give the answer to the nearest hundredth where necessar. A. 0.7 cm/da B. 0.7 cm/da C..1 cm/da D..67 cm/da 15. A freight train travels 60 km. A single locomotive pulls the train for the first half of the trip, then a second locomotive is added, doubling the speed of the train. If the total time for the trip is 5 min, what is the speed of the train with one locomotive? A. 133 km/h B. 67 km/h C. 33 km/h D. 50 km/h Short Answer 16. Simplif this rational expression. State the non-permissible values of the variable. 17. Simplif this rational expression. State the non-permissible values of the variable. 1. Simplif this rational expression. State the non-permissible values of the variable. 19. Simplif this rational expression. State the non-permissible values of the variable. 0. Simplif this expression:

33 1. Simplif this expression:. Simplif. 3. Simplif.. Simplif. 5. Simplif. 6. Simplif. 7. Solve.. Solve. 9. Solve. 30. Solve. 31. Solve.

34 3. Working together, Nathaniel and his little sister Asha can rake the leaves in their backard in 1 min. Working alone, Asha can rake the leaves in 30 min. How long would it take Nathaniel to rake the leaves on his own? 33. Theo and Stefan are apprentice auto mechanics. Together, the can change the oil and filter in a car in 0 min. Working alone, Theo can change the oil and filter in 36 min. How long would it take Stefan to change the oil and filter on his own?

35 Chapter 7: Rational Expressions and Equations Answer Section MULTIPLE CHOICE 1. D. D 3. C. A 5. A 6. A 7. B. B 9. D 10. A 11. A 1. B 13. A 1. B 15. D SHORT ANSWER 16. The non-permissible values are. 17. The non-permissible values are,, and. 1. The non-permissible values are and. 19. = The non-permissible x = ,. 3.

36 The solutions are: 3. 0 min min

37 Chapter : Absolute Value and Reciprocal Functions Multiple Choice Identif the choice that best completes the statement or answers the question. 1. Solve this equation: A. and C. and B. and D. and. Solve this equation: x A. and C. and B. and D. and 3. What are the domain and range of the reciprocal function? A. domain: range B. domain: range C. domain: range D. domain: range. Without graphing, predict the number of vertical asmptotes of the graph of =. A. C. 0 B. 1 D. cannot be determined from the equation 5. Without graphing, predict the number of vertical asmptotes of the graph of =. A. 1 C. 0 B. D. cannot be determined from the equation

38 6. Which function is represented b the graph below? i) ii) iii) iv) 6 0 x 6 A. iv B. iii C. ii D. i 7. This is a graph of. Identif the vertical asmptotes of the graph of the reciprocal function x 1 A. and C. B. and D. no vertical asmptotes

39 . Here is the graph of =. Which graph below is that of its reciprocal function? x 6 A. C. 0 x 0 x B. 1 D. 0 x 1 0 x

40 9. Here is the graph of =. Which graph below is that of its reciprocal function? x 1 A. C x x 1 B. D x x 10. Identif the vertical asmptotes of the graph of the reciprocal of the quadratic function. A. and C. B. and D. no vertical asmptotes

41 Short Answer 11. Write this absolute value function in piecewise notation. 0 x 1. Write the absolute value function in piecewise notation x Solve this equation: x

42 1. Solve this equation b graphing: x 15. Solve this equation: Problem 16. Complete this table of values, then sketch the graphs of = f(x) and on the same grid. Identif the intercepts, domain, and range of the absolute value function x 1 16

43 17. Solve the equation. Use a graph to justif our answer. Graph of : x 1. Predict the number of vertical asmptotes of the graph of each reciprocal function. Explain our answer. Write the equation of each vertical asmptote ou identif. a) b)

44 19. Match each graph of a quadratic function on the left to the corresponding graph of its reciprocal function on the right. Justif our answers. a) i) x x 1 b) 1 ii) x x 1 c) 1 iii) x x 1

45 Chapter : Absolute Value and Reciprocal Functions Answer Section MULTIPLE CHOICE 1. C. B 3. A. C 5. A 6. D 7. B. C 9. C 10. C SHORT ANSWER The solutions are and x 1. The solutions are and. 15. The solutions are:,,, and PROBLEM 16.

46 16 1 = f(x) x-intercepts: -intercept: 5 Domain: x =f(x) Range: The absolute value function creates two quadratic equations: and So, the equation has two solutions: and Graph the line on the same grid as : x 1. The line intersects the graph of at two points: (3, 1) and ( 3, 1) These points satisf the equation, so and are solutions. a) Compare with. Since the value of a is positive and the value of q is negative, the related quadratic function has x-intercepts. So, the graph of the reciprocal function has vertical asmptotes. is undefined when.

47 So, the lines and are vertical asmptotes. b) Compare _ with. Since the value of a is negative and the value of q is positive, the related quadratic function has x-intercepts. So, the graph of the reciprocal function has vertical asmptotes. is undefined when. and So, the lines and are vertical asmptotes. 19. a) The graph that corresponds to graph a is graph iii. The graph of the quadratic function has two x-intercepts, so the graph of its reciprocal function has two vertical asmptotes. b) The graph that corresponds to graph b is graph i. The graph of the quadratic function has no x-intercepts, so the graph of its reciprocal function has no vertical asmptotes. c) The graph that corresponds to graph c is graph ii. The graph of the quadratic function has one x-intercept, so the graph of its reciprocal function has one vertical asmptote.