Algebra 2 Trig Final Exam Review

Transcription

1 Name: Class: Date: Algebra Trig Final Exam Review Multiple Choice Identify the choice that best completes the statement or answers the question È 5 7 none of these. In May, Bradley bought 8 styrofoam balls and decorated them as toy figurines. In June, he sold 9 figurines. In May, Lupe bought styrofoam balls to decorate, and in June, she sold figurines. Which matrix represents all of their May purchases and their June sales? May June May June Bradley 8 9 Bradley 8 Lupe Bradley Lupe May È 8 9 June Lupe Bradley Lupe. State the dimensions of the matrix. Identify the indicated element. 9 A = 7 0, a, 8 8, 0,, 7, 7 9 May È 8 9 June Find the values of the variables.. È + t = È y + t = 8, y = t =, y = 6 t = 6, y = 8 t = 8, y = 6

2 Solve the matrix equation X + = Find the product. 6. È È The Art Department and the Homecoming Committee at a local school are ordering supplies. The supplies they need are listed in the table. Paint (bottles) Brushes Paper (reams) Glue Sticks (boxes) Art Department Homecoming Committee Tape (rolls) A bottle of paint costs $, a paint brush costs $, a ream of colored paper costs $8, a box of glue sticks costs $, and a roll of tape costs $. Find the matrix that represents the total cost of supplies for each group A manufacturer determines that the number of drills it can sell is given by the formula D = p + 80p 85, where p is the price of the drills in dollars. At what price will the manufacturer sell the maximum number of drills? What is the maximum number of drills that can be sold? $60; 85 drills $;,8 drills $0;,5 drills $90; 8,85 drills

3 9. Which is the graph of y = (x )? 0. Identify the vertex and the y-intercept of the graph of the function y = (x + ) + 5. vertex: (, 5); y-intercept: 7 vertex: (, 5); y-intercept: 7 vertex: (, 5); y-intercept: vertex: (, 5); y-intercept: 9 Factor the expression.. 5x x x( 5x ) x(5x + 7) 5x(x + 7) 5x(x + 7). x + x + 8 (x + 6)(x 8) (x 8)(x 6) (x + 8)(x 6) (x + 6)(x + 8)

4 . x x 6 (x 9)(x + 7) (x 9)(x 7) (x + 7)(x + 9) (x 7)(x + 9). 9x 6 (x + )( x ) ( x + )(x ) (x + )(x ) (x ) Solve the equation by finding square roots. 5. x = 7, 7, 7 7, 6. The function y = 6t + 86 models the height y in feet of a stone t seconds after it is dropped from the edge of a vertical cliff. How long will it take the stone to hit the ground? Round to the nearest hundredth of a secon 7.79 seconds 0.5 seconds.0 seconds 5.5 seconds 7. Simplify 75 using the imaginary number i. i 75 5i Simplify the expression. 8. ( 5i) ( + i) + 9i 9i 5 i 0i 9. ( 6i)( 6i) 6 6 6i 6i 0. ( + 5i)( + 5i) 7 + 5i + 5i + 5i + 5i Solve the equation.. 9x + 6 = 0 i, i i, i 6 6 i, 9 9 i,

5 . x + 8x + 8 = 5,,,,. Classify 7x 5 6x + x by degree and by number of terms. quartic trinomial cubic binomial quintic trinomial quadratic binomial. Zach wrote the formula w(w )(5w + ) for the volume of a rectangular prism he is designing, with width w, which is always has a positive value greater than. Find the product and then classify this polynomial by degree and by number of terms. 5w 5 w w ; quintic trinomial 0w ; quadratic monomial 5w w w; cubic trinomial 5w w w ; quartic trinomial 5. The table shows the number of hybrid cottonwood trees planted in tree farms in Oregon since 995. Find a cubic function to model the data and use it to estimate the number of cottonwoods planted in 006. Years since Trees planted (in thousands) T(x) = 0.x + 0.5x 0.x + 0.; 60. thousand trees T(x) = 0.x + 0.8x + 0.x; 60. thousand trees T(x) = 0.6x + 0.8x 0.x; 68. thousand trees T(x) = 0.6x + 0.5x + 0.x + 0.; 68. thousand trees 6. Write x + 8x 96x in factored form. 6x(x + )(x ) x(x + 6)(x + ) x(x )(x + 6) x(x + 6)(x + ) 5

6 7. Find the zeros of y = x(x )(x ). Then graph the equation.,,, 0,, 0,, 8. Write a polynomial function in standard form with zeros at 5,, and. f(x) = x x 9x 9 f(x) = x x + 60x 9 f(x) = x x 9x + 0 f(x) = x + 0x x 9 Divide using synthetic division. 9. (x + 5x 77x + x 6) (x ) x x 75x 5 x x + 9x + 9 x + 5x x 5 x + 9x x + 9 Factor the expression. 0. x + 6 (x 6)(x + 6x + 6) (x 6)(x 6x + 6) (x + 6)(x 6x + 6) (x + 6)(x + 6x + 7) 6

7 . c 5 (c 8)(c + 8c + 6) (c + 8)(c + 8c + 6) (c 8)(c + 8c + 6) (c 8)(c 8c 6). Ian designed a child s tent in the shape of a cube. The volume of the tent in cubic feet can be modeled by the equation s 6 = 0, where s is the side length. What is the side length of the tent? feet 6 feet 6 feet 8 feet. Solve x x = 5. no solution,, 5, 5, 5, Find the roots of the polynomial equation.. x + x 9x + 0 = 0 + i, i, + i i,, + i + i, i, i,, Find the real-number root The formula for the volume of a sphere is V = πr. Find the radius, to the nearest hundredth, of a sphere with a volume of 5 in...58 in in..5 in..85 in. Multiply and simplify if possible not possible Divide and simplify

8 Add if possible. 9. x + 6 x 8 x 6 x 8 x not possible to simplify 0. A garden has width and length 7. What is the perimeter of the garden in simplest radical form? units 9 units 6 units 8 units Simplify none of these Multiply. Ê ˆÊ ˆ. 7 Ë Á 8 + Ë Á Simplify Solve the equation.. x = ( x) 5 = 59 5, 5 6. Let f(x) = x 7 and g(x) = x +. Find (f û g)( 5)

9 Graph the function. 7. y = x + 9

10 8. y = x + 9. An initial population of 895 quail increases at an annual rate of 7%. Write an exponential function to model the quail population. f(x) = 895(.07) x f(x) = 895(0.07) x f(x) = 895(7) x f(x) = ( ) x 50. Write an exponential function y = ab x for a graph that includes (, 5) and (0, 6). y = 6(.5) x y =.5(6) x y = (5) x y = 5() x 5. The half-life of a certain radioactive material is 85 days. An initial amount of the material has a mass of 80 kg. Write an exponential function that models the decay of this material. Find how much radioactive material remains after 0 days. Round your answer to the nearest thousandth. y = Ê Ë Á 80 Ê y = 80 ˆ Ë Á ˆ 85 x Ê ; 0.8 kg y = 80 ˆ Ë Á 85 x Ê ; 0 kg y = Ë Á 80 ˆ 85 x ; 78.7 kg 85 x ; 0.9 kg 0

11 5. Suppose you invest $600 at an annual interest rate of.6% compounded continuously. How much will you have in the account after years? $800.6 $6,70.8 $0,8.07 $,9. 5. How much money invested at 5% compounded continuously for years will yield $80? $95.70 $88.8 $780.0 $ Write the equation in logarithmic form =, 96 log 6, 96 = log, 96 = 6 log, 96 = log, 96 = Use the properties of logarithms to evaluate log 9 + log 6 log Solve 5 x = 6. Round to the nearest ten-thousandth Solve log(x + 0) = The generation time G for a particular bacteria is the time it takes for the population to double. The bacteria t increase in population is shown by the formula G =, where t is the time period of the population. log a P increase, a is the number of bacteria at the beginning of the time period, and P is the number of bacteria at the end of the time perio If the generation time for the bacteria is 6 hours, how long will it take 8 of these bacteria to multiply into a colony of 768 bacteria? Round to the nearest hour. 77 hours 76 hours hours 85 hours 59. Solve ln(x ) = 8. Round to the nearest thousandth.,89.979, ,98.58, Simplify the rational expression. State any restrictions on the variable. 60. q + q + q 5q q + 8 q 8 ; q, q 8 q + 8 q 8 ; q, q 8 (q + 8) q 8 ; q 8 (q + 8) q 8 ; q, q 8

12 Multiply or divide. State any restrictions on the variables. 6. a 5 7b b a a 9 7b 6, a 0, b 0 7b a, a 0, b 0 a 7b, a 0, b 0 7 a9 b 6, a 0, b 0 6. x 6 x + 5x + 6 x + 5x + x x 8 (x ) (x + )(x + ) ; x, (x + ) (x + ) ; x,, (x + ) (x + ) (x ) ; x,,,, (x + )(x + ) ; x,,,, (x + )(x + ) Add or subtract. Simplify if possible a a 6 7a 9 (a 8)(a + 8) a + a 56 (a 8)(a + 8) 7a + 6 (a 8)(a + 8) Solve the equation. Check the solution. 6. g + g = g 5 g x 9 x = ± 7 or

13 66. A group of college students are volunteering for Help the Homeless during their spring break. They are putting the finishing touches on a house they built. Working alone, Irina can paint a certain room in 7 hours. Paulo can paint the same room in 6 hours. Write an equation that can be used to find how long it will take them working together to paint the room. How many hours will it take them to paint the room? If necessary, round your answer to the nearest hundredth. 7 x + 6 x = ; hours = ; 6.5 hours x x 7 + x 6 = ;. hours x 6 + x = ; 6.5 hours Write a recursive formula for the sequence 8, 0,,, 6,... Then find the next term. a n = a n +, where a = 8; 8 a n = a n +, where a = 8; 8 a n = a n, where a = 8; 8 a n = a n, where a = ; 68. Write an explicit formula for the sequence 7,,, 8,,... Then find a. a n = 5n + ; 5 a n = 5n + ; 58 a n = 5n + 7; 58 a n = 5n + 7; Find the arithmetic mean a n of a n = 5 7, a n + = A grocery clerk sets up a display of -pack cartons of col There are 5 cartons at the base of the triangle and one at the top. How many cartons of cola are needed for the complete display? 80 cartons 0 cartons 0 cartons 5 cartons Is the sequence geometric? If so, identify the common ratio. 7., 9, 7, 8 8, 6,... yes, yes, 9 yes, 6 not geometric

14 Write the explicit formula for the sequence. Then find the fifth term in the sequence. 7. a =, r = a n = ( ) n ; a n = () n ; a n = () n ; a n = ( ) n ; 79 Find the missing term of the geometric sequence. 7. 5,, 60, The sequence 5, 0, 5, 0,..., 65 has 5 terms. Evaluate the related series The sequence,, 6, 8,..., has terms. Evaluate the related series A large asteroid crashed into a moon of a planet, causing several boulders from the moon to be propelled into space toward the planet. Astronomers were able to measure the speed of one of the projectiles. The distance (in feet) that the projectile traveled each second, starting with the first second, was given by the arithmetic sequence 6,, 6, 80,.... Find the total distance that the projectile traveled in seven seconds. 5 feet 560 feet feet 6 feet 77. Use summation notation to write the series for terms. (9 + 5n) ( + 5n) n = n = ( + 5n) (9 + 5n) n = 9 n = 78. For the series (n + ), find the number of terms in the series. n = terms terms 6 terms 5 terms 79. Evaluate the series (n + ). n = Evaluate the series Justine earned $7,000 during the first year of her job at city hall. After each year she received a % raise. Find her total earnings during the first five years on the jo $,5. $7,89.89 $57,077.8 $9,077.8

15 8. Each person in a group of students was identified by year and asked when he or she preferred taking classes: in the morning, afternoon, or evening. The results are shown in the table. Find the probability that the student preferred afternoon classes given he or she is a junior. Round to the nearest thousandth. When Do You Prefer to Take Classes? Freshman Sophomore Junior Senior Morning Afternoon 7 5 Evening Find the measure of an angle between 0º and 60º coterminal with an angle of 0º in standard position. 50º 0º 0º 70º 8. Find the exact value of cos 00º and sin 00º. cos =, sin = cos =, sin = cos = cos =, sin =, sin = Write the measure in radians. Express the answer in terms of π º 6π 9 9π 6 9 6π 6 9π Write the measure in degrees π radians 5π º 7π º 5º 5.5º Use the circle below. Find the length s to the nearest tenth. 7.0 ft. ft.0 ft.0 ft 5

16 Write a cosine function for the graph. 88. y = cos θ y = cos θ y = cos θ y = cos θ 89. Suppose tanθ = 8 5. Find cotθ Find the exact value of csc 5º. If the expression is undefined, write undefine 0 undefined 9. Find the exact value of sec ( 70º). If the expression is undefined, write undefine undefined 0 Simplify the trigonometric expression. 9. secθ cosθ tanθ cotθ sinθ 6