College Prep Math Final Exam Review Packet

Transcription

1 College Prep Math Final Exam Review Packet Name: Date of Exam: In Class 1

2 Directions: Complete each assignment using the due dates given by the calendar below. If you are absent from school, you are still responsible for the assignments! A calculator is only permitted on questions that specify that you may use one. You will NOT have access to a calculator during your final exam, so don t cheat during your review! Monday Tuesday Wednesday Thursday Friday May 14 May 15 May 16 May 17 May 18 Section I DUE Section II DUE Section III DUE Section IV DUE May 1 May May 3 May 4 May 5 Section V DUE Section VI DUE Section VII DUE Section VIII DUE NO SCHOOL May 8 May 9 May 30 May 31 June 1 NO SCHOOL Sections IX & X DUE Work on Mixed Multiple Choice in Class Mixed Multiple Choice DUE Work on Mixed Short Answer in Class June 4 June 5 June 6 June 7 June 8 Mixed Short Answer DUE No School Regents Exams FINAL EXAM FINAL EXAM MOVING UP DAY June 11 LAST DAY OF CLASSES

3 Section I: Intro Trigonometry Use the triangle below to find the following values. 1. sin A. cos A 3. tan A 4. sin B 5. cos B 6. tan B Draw each angle and determine the quadrant in which the terminal side of the angle lies Find the smallest positive angle that is coterminal with the following angles

4 Given the point P which is on the unit circle and on the terminal side of an angle, find sin, cos, and tan. 15. P 1 3, 16. P, 1,0 18. For this example, write your answer for tan as a fraction: 0.63, For each function below, state the quadrant in which the terminal side of the angle lies and whether the value of the expression would be positive or negative. cos tan sin60 0. Fill in the following chart with the appropriate special angle values. This may help you on the remainder of the review sheet. 0, sin cos tan 4

5 Find the exact value of each expression below. sin30 tan tan 45 cos tan 30 cos cos30 6. tan sin70 cos90 Find the exact value of each function below. 8. sin5 9. cos tan sin 4 3. cos tan 6 Given the function value and quadrant, find the angle. 34. tan 3, quadrant III 35. cos, quadrant II 5

6 Given the point on the unit circle and on the terminal side of angle, find ,0 36., Using what you know about the relationship between radians and degrees, follow the directions below. 38. Express 40 in radian measure. 39. Express 70 in radian measure. 40. In a circle, the length of the radius is 4 cm. Find the length of an arc intercepted by a central angle whose measure is 3 radians. 6

7 Section II: Trig Identities 4 1. If csc, findsin.. If 3 sec 3, find cos If tan, find cot If 3 sin and is in quadrant I, evaluate all remaining trigonometric functions of θ Replace with a sum or difference identity and find the value: cos15 cos10 sin15 sin10 6. Replace with a sum or difference identity and find the value: sin 00 cos10 cos 00 sin10 7. Replace with a sum or difference identity and find tan 0 the value: tan10 1 tan 0 tan10 8. Replace with a sum or difference identity and find tan 335 tan 5 the value: 1 tan 335 tan 5 7

8 9. What is the value of sin.5 cos.5? 10. What is the value of cos 15 1? 11. Find the exact value, using a double angle identity: tan 30 1tan Find the exact value, using a double angle identity: tan 45 1tan Using the sum or difference of special angles, find the exact value: sin Using the sum or difference of special angles, find the exact value: cos85 8

9 15. If A and B are acute angles, 1 sin A and 13 cos 4 5 B, what is the value of cos A B? 16. Using a difference identity, find the exact value: tan 5 if 1 tan 17. If 3 tan and is in quadrant III, what is the value of sin? 4 9

10 18. If cos, then what is cos? If sin, then what is cos? 3 0. Find tan using the double angle identity if tan 5. 10

11 Section III: Solving Triangles 1. In FWN, f 3, w 4, and N 60. Find the exact value of n.. In ABC, a 4, b 6, and c 7. Find cos A. Leave your answer written as a fraction. 3. In DEF, d 3, f 4, and me 60. Find the area. 4. In ABC, a, b 5, and c 19. Find C. 11

12 5. In a parallelogram, two adjacent sides measure 10 and 7 and the included angle is 45. Find the area of the parallelogram. 6. In DEF, d 4, e 5, and F 50. Find f to the nearest tenth. 7. In DEF, e 8, F. 1 sin E, and f 1. Find acute 3 8. In XYZ, X 37, x 6, and y 9. Find acute Y to the nearest degree. 1

13 9. Observers at J and K, 40 km apart, sight an airplane at angles of elevation 35 and 80, respectively, as shown in the diagram. How far is the plane from observer K to the nearest kilometer? 10. Nancy just bought a pool and wants to install a triangular shaped deck. Two sides of the deck will measure 9 feet and 1 feet and will meet at an angle of 35. Find what the area of the deck will be to the nearest tenth. 13

14 Section IV: Graphing Trig Functions & Trig Equations Part I: Graphing For numbers 1 &, graph each equation over the interval parent point and write the transformations that occur. 1. Graph: f x cos x 1 3 Transformations: 3 x. Be sure to place an x on each 1. Graph: f x sin x Transformations: 14

15 Part II: Solving Trig Equations For numbers 3-8, solve the equation such that 0 x tan xtan x sin x sin x cos x cos x sin x 7sin x cos x 7cos x

16 Section V: Graphing 1. State the slope and y-intercept of the line and graph: 3x y 10. State the center and radius of the circle and graph: x y

17 3. Write the equation of the circle in standard form, state the center and radius, and graph: x y 6x 10y 30 x3 y4 4. State the center of the ellipse and the values of a and b and graph: Write the equation of the ellipse shown below. 17

18 6. Write the transformations from the parent graph for the equation below and graph. x y 1 7. Write the transformations from the parent graph for the equation below and graph. y x 4 18

19 8. Write the transformations from the parent graph for the equation below and graph. 1 y x Write the transformations from the parent graph for the equation below and graph. y 3cosx Write the equation of the graph shown below. 19

20 For problems 1-3, solve for x and check. x 3 1. x 3 4 Section VI: One Variable Equations and Inequalities x 4 3. x 7 x 49 x 7 x x x 3 x 3 x 9 0

21 For problems 4-6, solve each linear inequality, write the solution in interval notation, and graph on the number line. 4. (3x1) 4x x x x

22 For problems 7-9, solve each absolute value equation and check. 7. 4x 1 8. x x 3 15

23 For problems 10-14, solve each absolute value inequality and graph the solution set. 10. x x 7 x

24 x x

25 Section VII: Solving Systems of Equations For numbers 1-4, solve the systems of equations by graphing. 1. x y 4 3x3y 9 1. y x 3 y x 3 3. y ( x 1) y 1 x 4. x y 9 y x 3 5

26 For numbers 5-7, solve the systems of equations algebraically. 5x3y x y6 x 4y 4 xy 7. x y z 4 x y z 4 x 7 y 3z 6 6

27 Section VIII: Polynomials For numbers 1-4, factor x 4x 3x 1. x 5 3. x 9x x 7 For numbers 5-8, solve for x x x x x 8x x x x A 6 th degree polynomial equation has the roots 3 i, i 5, and 3. Write the solution set for the equation. 10. A 5 th degree polynomial equation has the roots 6, 3i, and 6. Write the solution set for the equation. 7

28 11. How many roots does the following polynomial equation have? 7 4 5x 8x x How many roots does the following polynomial equation have? 5 4 6x x 3x 7x 1 0 For numbers 13-15, given the degree of the equation and some or all of the roots, write a polynomial equation in standard form. 13. quadratic equation with root 3 i rd degree equation with roots -3, 5, th degree equation with roots 1, -3, 5 8

29 16. Draw Pascal s Triangle, including 7 rows. For numbers 17 & 18, use the Binomial Theorem and Pascal s Triangle to expand the following binomials. 17. x y

30 Section IX: Rational Functions & Composition of Functions For each expression below, simplify to lowest terms. Be sure to indicate any restrictions on the domain x 9x 4x 8 x 4x4 x 9 Perform the operation, and specify which value(s) of the variable must be excluded to avoid division by ( p6)( p3) 1 x7 x8 7x 35 x 3x 40 p 4 ( p 6)( p 3) 5. x 1 x 9 x3 6. y 6 y1 y3 30

31 State whether each is a rational function and explain your answer. x 7. f( x) x x 8. f( x) 3x 5 x x 9. f( x) x 4 x 10 3x 1 x x 10. f( x) 0.4 State the domain of each rational function and identify all vertical asymptotes on the graph of each rational function. 3x 4x 11. f( x) 1. f( x) x 5 x D : Vertical Asymptotes: D : Vertical Asymptotes: x 13. f( x) x f( x) x 1 x 3 D : Vertical Asymptotes: D : Vertical Asymptotes: 31

32 Let f x x 1 and g x x 15. f g x 16. g f x. Write the rule for each composite function or evaluate the composite function. 17. f g g f f g 1 0. g f Let f x x, g x x 4, and h x 1. x 4. x. Express each of the following as a composite of these functions. x 4 3

33 For each example below, find 3. f x 3x 6 1 f in the same style that f is given. 4. For each example below, find 1 f in the same style that f is given. Then, list the domain and range for f and 1 f. 5. x f x Domain of f : Range of f : f x 1 x Domain of 1 f : Range of 1 f : 6. f,4, 5,, 1,3, 0,5, 4,7 Domain of f : Range of f : 1 f Domain of 1 f : Range of 1 f : f x x Given the function: 1 7. Graph the function, using a pencil. Select a suitable domain for f that makes it an invertible function. Erase part of your graph to reflect this new domain. Write a formula for 1 f and graph 1 f. Section X: Vectors 33

34 For examples 1-4, use the following vectors: u 3,4, v,4, s 4,,6, and t 3, 1, 1. u. t 3. u v 4. t s 5a. Write the translation map associated with the vector v, 4. d. Graph both y and y '. b. Write the equation of y ', the image of caused by the translation T v. 1 y x 4 c. What shift is caused by the translation map T v? 6a. Write the translation map associated with the vector v 4,. d. Graph both y and y '. b. Write the equation of y ', the image of x y 1 caused by the translation T v. c. What shift is caused by the translation map T v? 34

35 7a. Write the translation map associated with the vector v,5, 7. b. Write the equation of the image of the sphere x y z translation map T v after being shifted by the c. What shift is caused by the translation map T v? For examples 8-10, use the following points to write each vector in component form: A4,7, B1,1, C 6, 8. AC 9. BA 10. CB 11. If u 4 3 and w 3,5, find u w using the tip-to-tail method. 1. If s 10,6 and t,7, find s t using the parallelogram method. 35

36 13. In a video game, a robot travels from point A 8 feet due east to point B and then 1 feet due south to point C, as shown in the diagram. What is the magnitude and direction, south of east, of vector AC? Round to the nearest 10 th. 14. A boat is crossing a river and traveling due east at a speed of 8. m/s. The river's current is 4.6 m/s due north. What is the resultant velocity of the boat to the nearest tenth and the angle the boat ends up traveling, north of east? 15. If two applied forces of 16 and 8 pounds produce a resultant force of 15 pounds, find the angle (to the nearest degree) between the larger force and the resultant. 16. Two forces act on a body at an angle of 65. The forces are 4 pounds and 7 pounds. Find the magnitude of the resultant force to the nearest tenth of a pound. 36

37 37