( ) 2 + 2x 3! ( x x ) 2

Transcription

1 Review for The Final Math Rewrite as a single simplified fraction: 1. Rewrite as a single simplified fraction: ! 3. Rewrite as a single simplified fraction:! 4! ! 3 3! 4! 5 3! Rewrite as a simplified fraction: 5. Rewrite as a simplified fraction: 10a 7 b 8 c! 4a 3 b 8 c 9 ( ) 6 ab 5 c y 5 z! 4 3 y 5 z 9 ( ) 6 y 4 z 3 6. Factor completely: 75y 3! 48 3 y 7 7. Factor completely: a + a + a solve for : 6! ( + )( + 1)! (1! ) = 0 ( ) + 3! ( ) 9. Epand and rewrite as a simplified sum: Epand and rewrite as a simplified sum: ( + 1)( + )! ( + )( + 3). f ( + h)! f (! h) 11. Let f () = 3!. Simplify and write the answer as a h simplified sum. f ( + h)! f () 1. Let f () =!! 1. Simplify and write the answer as a h simplified sum. 13. Given: f () =! 3 and g() = 5! 7. Simplify: f (g())! g( f (!4)). write the answer as a polynomial in standard form. 14. Let f () = 9! 7! 5. Simplify f (! 3)! f (4) and write the answer as a polynomial in standard form. 15. Given: f () = 1! and g() =. Simplify: f (g())! g( f (!)). The answer is an integer.

2 $ 16. Sketch the graph of the functions y = f () = % 5 if! & " 3 if # 17. Find the center and radius of the circle whose equation is + y + 1! 0y + = Find the center, radius and lowest point of the circle whose equation is + y! y = Find the equation (in any correct form) of a straight line through (!5,8) which is perpendicular to the line goes through (!1,6)and (!,!13). 0. Find the equation (in any correct form) of a straight line through which is parallel to the line whose equation is + y = 4. (4,!7) 1. Find the inverse function of F() = 1+.! 3. Given: g() = 4! 7 Find the inverse function, 5! 8 g 1 ( ) 3. Solve for, answer in interval notation: 4. Solve for, answer in interval notation: 1! " 3 11! " 5. Solve for, answer in interval notation: 6. Solve for, answer in interval notation: 7. Solve for, answer in interval notation: 4 > 9! < ! 4 > 3 8. Solve for, answer in interval notation: 1 + 3! 5 < 7 9. Solve for, answer in interval notation: 7! + 5 " Sketch the graph of y =!( + 1) (! 1) 3. Label the intercepts and show the end behavior. 31. Sketch the graph of y =!(! 1) 3 ( + 3) (! ). Label the intercepts and show the end behavior.

3 3. If y = + 4! 4,First write it as a standard form of the quadratic function then Find the maimum value of y. Sketch the graph of the parabola. Find all intercepts and the coordinates of the verte. Find domain and Range. 33. Find the maimum value of y if y = 6!!. Sketch the graph of the parabola. Find all intercepts and the coordinates of the verte. Find domain and Range. 34. Given two real numbers whose difference is 14. I wish to minimize the sum of their squares. Find the value of the two numbers. (You must use algebra; NO CREDIT for guessing.) 35. In a right triangle, one of the two perpendicular legs has length 5. Write the perimeter(p) of this triangle as a function of the length()of its hypotenuse. 36. Given two real numbers whose difference is 1. I wish to minimize the sum of their squares. Find the value of the two numbers. (You must use algebra; NO CREDIT for guessing.) 37. The width of a rectangle is W and its area is A. If the length of the rectangle is twice its width, epress W as a function of A. 38. Among all rectangles with a perimeter of 4 feet, find the dimensions of the rectangle with the largest area. 39. Sketch a right triangle containing acute angle! with sin! = 1.Then find the 3 other five trig functions of!. 40. Find the eact value of 41. Find the eact value of " csc 19! % $ ' # 6 & " sin! % # $ 8 & ' 4. Find the both the tangent and secant of 330. sec! cos 43. Show that = tan is a valid trig identity. tan sec cosu secu 44. Show that = cotu is a valid trig identity. tanu

4 " 45. Find the value of sin!1! 3 % $. The answer is a single angle, ' # &! ( cos!1!1) measured in Degrees. 46. Find the value of tan!1 "!1 %. The answer is a single angle, # $ 3 & '! "!1 % cos!1 # $ & ' measured in Degrees. 47. Given: sec = and 3!. Find the value of. < <! sin 48. Suppose cos = 4 and sin < 0.Find tan. 5 # 49. For y =!7sin 3 + " &. Find the amplitude, period, phase shift, and graph $ % 6 ' ( the function. 50. For y =! tan 4 + ". Find the period, start of period, end of period and graph the function. 51. Solve for, where 0 < π. Give answer(s) in radians. 5. Solve for, where 0 < π. Give answer(s) in radians. 53. Solve for, where 0 < π. Give answer(s) in radians. cos! 4 =!3. ( ) 7 tan! 3 =!10 sin + = 3sin! $ 54. Find log 3 #. The answer is an integer or reduced fraction. " 3 3 & % ( ) 55. Find log The answer is an integer or reduced fraction. 56. Sketch the graph of y = 3! e!. On the graph, label the (, y) values of one point where and y are both integers. Also show the asymptote and give its equation. 57. Sketch the graph of y = log 3 ( + 1). On the graph, label the (, y) values of one point where and y are both integers. Also show the asymptote and give its equation and label all intercepts.

5 58. Sketch the graph of y = 7! e!. On the graph, label the (, y) values of one point where and y are both integers. Also show the asymptote and give its equation. 59. Solve for : log (! 30) = 6 The answer is an integer or reduced fraction. 60. Solve for : log ( + + 8) = 6 The answer is an integer or reduced fraction. 61. The population of a bacteria colony increases from 1000 to 000 in ten days. How many more days will be needed for the population to grow to 10000? Assume the population grows eponentially. You may leave log or ep in your answer. 6. Given: the size of a population grows eponentially. At 1 PM yesterday, the population size was 0. At 6 PM yesterday, the population size was 60. Find the population size at 10 PM yesterday. You can leave eponentials and/or logs in your answer. 63. A bacteria population that weighs grams at 10:00 A.M. grows to 3 grams at 3:00 P.M. How many grams will it weigh at 6:00 P.M.? 64. The half life of the radioactive element unobtainium 47 is 0 seconds. If 3 grams of unobtainium 47 are initially present, how many grams are present after 0 seconds? 40 seconds? 60 seconds? 80 seconds? 1000 seconds? 65. The half life of a certain tranquilizer in the bloodstream is 46 hours. How long will it take for the drug to decay to 87% of the original dosage? Ues the eponential decay model. 66. Find the center and radius of the circle whose equation is + y y 11 = Find the center and radius of the circle whose equation is + y +14! 8y + 5 = Given: f ( ) = 7 and g( ) = 5 3. Simplify: f ( g( 4)) g( f ( 6)). The answer is an integer. 69. Given: Find the inverse function, g 1 ( ) 70. Find the maimum value of y if y = 1 1. The answer is an integer or reduced fraction.

6 71. Let f () = 9! 4! 3. Simplify f (5! 6) and write the answer as a polynomial in standard form. 7. Let. Simplify and write the answer as a polynomial in standard form. 73. Let. Find and Simplify. 74. Find the equation (in any correct form) of a straight line through (4,!7) which is perpendicular to the line whose equation is 8 + 5y! 9 = Given: f () = 4! 5 and g() =! 3. Simplify: f (g(!6))! g( f (!)). The answer is an integer. 76. Given: g() = 5! 8 Find the inverse function, 7! 3 g 1 ( ) 77. Sketch the parabola with equation. On your sketch, show and label the verte and all intercepts. 78. Given. Find and simplify Factor completely: 18 y 50y Find the maimum value of y if y =13! 0!. The answer is an integer or reduced fraction. 81. Sketch the parabola with equation. On your sketch, show and label the verte and all intercepts. 8. Factor completely: 75y 3! 48 3 y Epand and rewrite as a simplified sum: 5( 3! 6)! (! 4) 84. Rewrite as a simplified fraction: 10a 7 b 8 c! 4a 3 b 8 c 9 ( ) 6 ab 5 c Find the equation (in any correct form) of a straight line through ( 5,6) which is perpendicular to the line whose equation is 4 + 7y = Rewrite as a single simplified fraction:

7 87. Rewrite as a single simplified fraction: 88. Epand and rewrite as a simplified sum: Solve for, answer in interval notation: Solve for, answer in interval notation: 7! 9! 5 + 5! w y 10w y 89. Rewrite as a simplified fraction: ( w y ) ( 4) 4 ( 3) 7! "1 9. Solve for, answer in interval notation: 93. Solve for, answer in interval notation: 94. Solve for, answer in interval notation: 95. Solve for, answer in interval notation: 96. Solve for, answer in interval notation: Solve for, answer in interval notation: 98. Solve for, answer in interval notation: Sketch the graph of y = ( 1) ( 3) ( + 1). Label the intercepts and show the end behavior Sketch the graph of y = (! 5)(! ) ( +1) 4. Label the intercepts and show the end behavior Sketch the graph of. Label the intercepts and show the end behavior. 10. Show that is a valid trig identity Find the eact value of tan 3π 6

8 104. Given: and. Find. " 105. Find the value of tan!1 (!1)! cos!1! 3 % $ '. The answer is a single angle, # & measured in Degrees Find the value of 1 1 cos sin. The answer is a single angle, measured in degrees Solve for, where 0 < π. Give answer(s) in radians Find the eact value of sin( 67.5 ) π 109. For y = 4cos 3 +. Find the amplitude, period, phase shift, and graph 4 the function Given: cot =! 8 and cos < 0. Find csc Given:. Find 11. Find the eact value of cos(11.5 ) 113. Find both tangent and secant of Find the eact value of " csc 19! % $ ' # 6 & 115. Show that is a valid trig identity. # 116. For y =!5sin + " & % (. Find the amplitude, period, phase shift, and graph $ 6 ' the function Solve for, where 0 < π. Give answer(s) in radians. 8cos! 3 =!7

9 118. Solve for, where 0 < π. Give answer(s) in radians Find. The answer is an integer or reduced fraction. 10. Solve for : log 5 (9! 5) = The answer is an integer or reduced fraction. 11. Solve for : The answer is an integer or reduced fraction. 1. Sketch the graph of y = 7 log3 ( + 5) and. On the graph, label the (, y) values of one point where and y are both integers. Also show the asymptote and give its equation. 13. Find. The answer is an integer or reduced fraction. 14. Solve for : The answer is an integer or reduced fraction. 15. Sketch the graph of y = 5! e (!3). On the graph, label the (, y) values of one point where and y are both integers. Also show the asymptote and give its equation. 16. Sketch the graph of. On the graph, label the (, y) values of one point where and y are both integers. Also show the asymptote and give its equation. 17. Given two real numbers whose difference is 1. I wish to minimize the sum of their squares. Find the value of the two numbers. (You must use algebra; NO CREDIT for guessing.) 18. Write the area (A) of a circle as a function of its circumference (C). 19. Epress the diameter D of a circle as a function of its area A The length of a rectangle is. its area is 50 square inches. Find a formula for its perimeter in the form f()= Write the perimeter P of a square as a function of its area A.

10 13. The population of a bacteria colony increases from 000 to 3000 in ten days. How many more days will be needed for the population to grow to 4000? 133. A 0 gram sample of the radioactive element mathtestium takes 10 years to decrease to15 grams. Find the half life of mathtestium. You can leave eponentials and/or logs in your answer Right now a bacteria population is Two hours ago, it was 000. How many bacteria will there be 5 hours from now? Assume that the bacteria population increases eponentially. You may leave log or ep in your answer 135. Given two real numbers whose sum is 14, find the minimum possible value for the sum of their squares. The answer is an integer or reduced fraction. (You must use algebra; NO CREDIT for guessing.) Fall a) Rewrite the inequality 5 < 0 using interval notation. b) Rewrite the interval (, 1) using inequality notation.. Rewrite 1/ 1/ as a reduced fraction. 3. Rewrite as a reduced fraction. 4. Factor completely: 3/ 1/ 1/ Let y = f ( ) = sin( ); 0 4 π. a) What are the period and amplitude of the function f? b) Sketch the graph of f on the grid below. Include - and y-aes with appropriate scales. Label all maimum and minimum points, as well as intercepts, with their coordinates Rewrite a + h a as a reduced fraction. h 7. On the grid below, draw and y aes with appropriate scales, and sketch the graph of the function

11 + for < 1 f ( ) = 3 for 1 < 1 for with domain a) Find an equation of the line through point (6,9) that is parallel to the line joining points (,4) and (,6). b) Find an equation of the line through point (6,9) that is parallel to the line joining points (,4) and (3,6). 9. If you start with the graph of the equation y 3 = + 1 and then move the graph 3 units left; and then reflect the graph through the y ais; and then move the graph units down; what is the equation of the resulting graph? Do not sketch. 10. Given 1 f ( ) = and g( ) =, find and simplify f ( g(3 b)) g( f (15)). 4 Begin your work below. If you need more space continue at the left on the back of page 5, NOT ON THE BACK OF THIS SHEET. 11. a) Sketch the graph of y = f ( ) = 9 8. Include on your graph all intercepts and the verte, each labeled with its coordinates. b) What is the maimum value of the function in part a)? 3 1. Graph the polynomial y = ( + 3) ( 1) ( + 9). Show clearly on your graph all intercepts, labeled with their coordinates, as well as the intervals in which y is positive, the intervals in which y is negative, and end behavior. 13. Simplify degrees. 3π arcsin sin 4. Write your answer as an angle, epressed in 14. Find tan a) Convert 7 π 7π radians to degrees. b) Find sin 1 1 formula for sine. by using an addition 16. Find all solutions of cos = for 0 π. 4 3

12 17. Given f ( ) = 3 +, rewrite f ( ) f ( h) h as a simplified polynomial. π 18. Find the eact value of sin 8 by using a half angle formula. 19. Sketch the graph of the equation y = 3log ( ) 9. On your graph show all intercepts (each labeled with its coordinates) and any asymptote (labeled with its equation). 0. Find all solutions for in the equation log ( + 8) + = log. 1. A sample of the radioactive element migrainium weighs 1000 pounds at 1:00 P.M. and decays eponentially. If it weighs 500 pounds at 5:00 PM, how much will it weigh at 7:00 PM?. Solve the inequality 3 5. Write your answer using interval notation. 3. Suppose fencing material costs 0 dollars per yard. Write an epression (involving ) for the total cost of a fence that goes around a rectangular field if the field s width is yards and the field s area is 40 square yards. 4. a) Find the inverse function g ( ) of the function + 3 f( ) = b) For the functions f and g in part a), find the value of f ( g( u + v)) without using algebra. Eplain your reasoning. [If you use algebra, you will waste time and get zero credit]. Spring 011 Part I. Answer any five complete questions from this part 4 1. Given f( ) = and g( ) =, write f ( g()) g( f ( )) as an integer 1 1. Find the value of a) sin and b) 1 1 cos Given f( ) =, find the inverse function f 1 ( ) 4. Find the center and radius of the circle with equation + y 8 + 6y + 4 = 0.

13 5. Solve the equation log ( + 5) = 4 log ( 1) for the eact value of Suppose that cos( ) = and tan( ) is negative. Find cos( ) 7 Part II. Answer any five complete questions from this part. 7. Let f ( ) = 6 + a) Find all intercepts and the coordinates of the verte of the graph of y = f ( ). b) Make a sketch of the graph with intercepts and verte labeled with their coordinates. cos cot 8. Simplify and rewrite the epression as a single trigonometric 1 sin function. 17π 9. Find the eact value of sec 3. f ( + h) f ( h) 10. Let f ( ) = 3 Rewrite as a single reduced h fraction. 11. In each of the following, the given rectangle s side has length, measured in feet. a) If the perimeter of the rectangle is 50 feet, write down a formula for its area. b If the area of the rectangle is 50 square feet, write down a formula for its perimeter. 1. Find an equation of the line passing through the point (0, -) that is perpendicular to the line passing through the points (3,0) and (5,-).

14 Part III. Answer any five complete questions from this part. 13. Rewrite fraction. 3 3( + ) ( 3) ( + ) ( 6) ( 3) 4 as a completely reduced Simplify and rewrite w as a completely reduced fraction. 1 1 w 15. Rewrite ( y) ( zy z) as a completely reduced fraction. 16. Find all solutions of the equation tan = 3 for in the interval [0, π ] o 17. Find an eact epression in terms of radicals for cos(15 ) 18. a) Find all intercepts and an equation of the asymptote for the graph of + y = 8 b) Sketch the graph in a). Indicate clearly all intercepts and asymptotes. Part IV. Answer any five complete questions from this part. 19. a) Find all intercepts and an equation of the asymptote for the graph of y= 4 log 3( + 1). b) Sketch the graph in a). Indicate clearly the intercepts and asymptotes on the graph. 0. Let f ( ) = ( 1)( 9). a) Find all solutions of f( ) = 0 b) In what intervals is f( ) positive? In what intervals is it negative? c) Sketch the graph of y = f ( ) using the information from a) and b). Show end behavior clearly. π 1. For the function y = 4sin + : a) Find the amplitude, period, and phase shift. b) Draw the graph of one period of the function on the interval that begins with the phase shift. Label the coordinates of intercepts, maimum, and minimum points Solve the inequality 1 for. Write your answer using interval 1 notation.

15 3. Solve the inequality 5 + > 5. Write your answer using interval notation. 4. A population of bacteria is growing eponentially. If there are 1,000 bacteria at time t = 0 hours and 6,000 bacteria at time t = 3 hours, at what time t will there be 000 bacteria? You may leave eponential or log functions in your answer Fall 010 Part I: Questions 1 6. Choose any five questions. 1. Given f( ) 4 = and g( ) =, write f ( g()) g( f ( )) as an integer.. Find the value of sin Given the function F ( ) = 3 + 1, find the inverse function F 1 ( ). 4. Find the center C and the radius r of the circle + y + = 4 8y. 5. Solve the equation log 3( 4) = for. 6. Suppose cos = and tan is negative. Find sin. 3 Part II: Questions 7 to 1. Choose any five questions. 7. Sketch the graph of the function verte and intercepts. f ( ) = Show coordinates of the 8. Simplify fraction. sec 1 sec by rewriting it as an epression that doesn t involve any 9. Find the eact value of 11π tan Let f ( ) = +. Rewrite f ( + h) f ( ) h as a single reduced fraction. 11. Rewrite 3 + ( + )( 1) ( 3) as a simplified sum.

16 1. Find an equation of the line passing through (,7) that is parallel to the line joining points (1,) and (3,1). Part III: Questions Choose any five questions. 13. Rewrite as a simplified fraction. 14. Epand and rewrite (3 )(3 + ) as a simplified sum. 15. Rewrite a c ca b c + b c 16. Solve 17. Find sec = 4 for all values in the interval [ 0, π ). o cos(75 ). You may leave a square root sign in your answer. 18. Sketch the graph of y = 3 9. Show and label all intercepts and asymptotes. Part IV: Questions 19 to 4. Choose any five questions. 19. Sketch the graph of y= 3 log ( + ). Show and label all intercepts and asymptotes. 0. Sketch the graph of 5 3 y = 4. Show all intercepts and end behavior. 3π 1. Sketch one period of the graph of y = sin + 1. Label all maimum points, minimum points, and intercepts with their coordinates.. Solve the inequality Write your answer using interval notation Solve the inequality + 7 > 1. Write your answer using interval notation. 4. The population of a bacteria colony is 000 at 3 o clock and 4000 at 4 o clock. What will the population be at 6 o clock? You may leave logs or eponents in your answer. Assume that the population grows eponentially.