Algebra II Honors Final Exam Review

Transcription

1 Class: Date: Algebra II Honors Final Exam Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Identify the graph of the complex number 3 2i. a. c. b. d. 2. Determine which binomial is a factor of 2x x 2 24x a. x + 5 b. x + 20 c. x 24 d. x 5 1

2 3. Which function matches the graph? a. y = x c. y = x b. y = x 5 5 d. y = x Short Answer Graph the exponential function. 4. y = 4( 2) x 5. An initial population of 895 quail increases at an annual rate of 7%. Write an exponential function to model the quail population. 6. Find the annual percent increase or decrease that y = 0.35(2.3) x models. 7. Suppose you invest $1600 at an annual interest rate of 4.6% compounded continuously. How much will you have in the account after 4 years? 2

3 Write the equation in logarithmic form = 1, 296 Evaluate the logarithm. 9. log Graph the logarithmic equation. 10. y = log(x + 1) 7 Write the expression as a single logarithm log b q + 2 log b y log x 6 log (x + 2) Expand the logarithmic expression. 13. log 3 11p Use the properties of logarithms to evaluate log log 3 36 log

4 15. Solve 15 2x = 36. Round to the nearest ten-thousandth. 16. Solve 1 16 = 644x Solve log(4x + 10) = Solve log(x + 9) log x = 3. Write the expression as a single natural logarithm ln x 2 lnc 20. Solve ln(2x 1) = 8. Round to the nearest thousandth. Use natural logarithms to solve the equation. Round to the nearest thousandth e 4x 2 = 3 4

5 Is the relationship between the variables in the table a direct variation, an inverse variation, or neither? If it is a direct or inverse variation, write a function to model it. 22. x y x y Suppose that y varies directly with x and inversely with z y = 25 when x = 35, and z = 7. Write the equation that models the relationship. Then find y when x = 12 and z = The amount of oil used by a ship traveling at a uniform speed varies jointly with the distance and the square of the speed. The ship uses 28 barrels of oil in traveling 90 miles at 56 mi/h. How many barrels of oil are used when the ship travels 31 miles at 26 mi/h? Round your answer to the nearest barrel, if necessary. Describe the combined variation that is modeled by the formula or equation. 26. a = F m Graph the function. 27. y = 2 x 5

6 28. y = x y = x + 3 Factor the expression x 2 21x 31. 8x x x x x x x x 4 20x Solve by factoring. 4x x 32 = 0 6

7 Solve the equation by finding square roots x 2 = A landscaper is designing a flower garden in the shape of a trapezoid. She wants the length of the shorter base to be 3 yards greater than the height, and the length of the longer base to be 5 yards greater than the height. For what height will the garden have an area of 360 square yards? Round to the nearest tenth of a yard. 39. Simplify 175 using the imaginary number i. Write the number in the form a + bi Find the additive inverse of 7 + 5i. Simplify the expression. 42. ( 1 + 6i) + ( 4 + 2i) 43. ( 6i)( 6i) 7

8 Solve the equation x = x x + 81 = x = (3 x) 3 5 = Find the missing value to complete the square. x 2 + 2x + Rewrite the equation in vertex form. 49. y = x x The function P = h h 400 models the daily profit a barbershop makes from haircuts that include a shampoo. Here P is the profit in dollars, and h is the price of a haircut with a shampoo. Write the function in vertex form. Use the vertex form to find the price that yields the maximum daily profit and the amount of the daily profit. 8

9 Use the Quadratic Formula to solve the equation x 2 + 9x 2 = Classify 3x 5 2x 3 by degree and by number of terms. 53. Write the polynomial 6x 2 9x in standard form. 54. The table shows the number of llamas born on llama ranches worldwide since Find a cubic function to model the data and use it to estimate the number of births in Years since Llamas born (in thousands) Write 4x 3 + 8x 2 96x in factored form. 56. Find the zeros of y = x(x 3)(x 2). Then graph the equation. 57. Write a polynomial function in standard form with zeros at 5, 4, and Divide 3x 3 3x 2 4x + 3 by x

10 Divide using synthetic division. 59. (x x 3 77x x 36) (x 4) 60. 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 3 64 = 0, where s is the side length. What is the side length of the tent? 61. Use the Rational Root Theorem to list all possible rational roots of the polynomial equation x 3 + x 2 7x 4 = 0. Do not find the actual roots. Find the roots of the polynomial equation. 62. x 3 2x x = A polynomial equation with rational coefficients has the roots 5 + 1, 4 7. Find two additional roots. 64. For the equation 2x 4 5x = 0, find the number of complex roots and the possible number of real roots. 65. Find all zeros of 2x 4 5x x 2 125x + 75 = In how many different orders can you line up 8 cards on a shelf? 10

11 67. 5! Evaluate the expression P C There are 10 students participating in a spelling bee. In how many ways can the students who go first and second in the bee be chosen? 71. The Booster Club sells meals at basketball games. Each meal comes with a choice of hamburgers, pizza, hot dogs, cheeseburgers, or tacos, and a choice of root beer, lemonade, milk, coffee, tea, or cola. How many possible meal combinations are there? Use Pascal s Triangle to expand the binomial. 72. (d 5) Find all the real fourth roots of Simplify the radical expression. Use absolute value symbols if needed g 6 11

12 75. The formula for the volume of a sphere is V = 4 3 πr 3. Find the radius, to the nearest hundredth, of a sphere with a volume of 15 in Simplify 128a 13 b 6. Assume that all variables are positive. Divide and simplify Rationalize the denominator of the expression. Assume that all variables are positive A garden has width 13 and length What is the perimeter of the garden in simplest radical form? Simplify

13 Multiply. Ê ˆ Ë Á Ê Ë Á ˆ Write the exponential expression 3x in radical form. 85. Write the radical expression 7 8 x 15 in exponential form. 86. Write ( 8a 6 ) 2 3 in simplest form. 87. The area of a circular trampoline is square feet. What is the radius of the trampoline? Round to the nearest hundredth. Solve. Check for extraneous solutions x = x 89. Let f(x) = 3x 6 and g(x) = 5x + 2. Find f(x) + g(x). 13

14 90. Let f(x) = x 2 + 2x 1 and g(x) = 2x 4. Find 2f(x) 3g(x). 91. Let f(x) = 3x 6 and g(x) = x 2. Find f g and its domain. 92. Let f(x) = 2x 7 and g(x) = 4x + 3. Find (f û g)( 5). 93. Graph the relation and its inverse. Use open circles to graph the points of the inverse. x y Find the inverse of y = 7x 2 3. Sketch the asymptotes and graph the function. 95. y = 2 x Find any points of discontinuity for the rational function. 96. y = (x + 6)(x + 2)(x + 8) (x + 9)(x + 7) 14

15 97. Describe the vertical asymptote(s) and hole(s) for the graph of y = (x 5)(x 2) (x 2)(x + 4). 98. Find the horizontal asymptote of the graph of y = 6x 2 + 5x + 9 7x 2 x + 9. Simplify the rational expression. State any restrictions on the variable. 99. p 2 4p 32 p + 4 Multiply or divide. State any restrictions on the variables a 5 7b 4 2b 2 2a c + 1 c 5 c 2 c 2 7c + 10 Add or subtract. Simplify if possible a a

16 103. w 2 + 2w 24 w 2 + w w Identify the vertex, focus, and directrix of the graph of y = 1 8 (x 2) Identify the vertex, focus and the directrix of the graph of x 2 8x 28y 124 = Write an equation of a circle with center ( 5, 8) and radius Find the center and radius of the circle with equation ( x 5) 2 + Ê Ë Á y + 6ˆ 2 = Write an equation in standard form of an ellipse that has a vertex at (5, 0), a co-vertex at (0, 3), and is centered at the origin Find the equation of a hyperbola with a = 452 units and c = 765 units. Assume that the transverse axis is horizontal Write an equation of an ellipse with center (3, 3), vertical major axis of length 12, and minor axis of length Write an equation of a hyperbola with vertices (3, 2) and ( 9, 2), and foci (7, 2) and ( 13, 2). 16

17 Identify the conic section. If it is a parabola, give the vertex. If it is a circle, give the center and radius. If it is an ellipse or a hyperbola, give the center and foci x 2 + 7y x 56y = y 2 4x + 6y + 29 = The Sears Tower in Chicago is 1454 feet tall. The function y = 16t models the height y in feet of an object t seconds after it is dropped from the top of the building. a. After how many seconds will the object hit the ground? Round your answer to the nearest tenth of a second. b. What is the height of the object 5 seconds after it is dropped from the top of the Sears Tower? 115. State whether each situation involves a combination or a permutation. a. 4 of the 20 radio contest winners selected to try for the grand prize b. 5 friends waiting in line at the movies c. 6 students selected at random to attend a presentation 116. A radio station has a broadcast area in the shape of a circle with equation x 2 + y 2 = 6, 400, where the constant represents square miles. a. Graph the equation and state the radius in miles. b. What is the area of the region in which the broadcast from the station can be picked up? 17

18 Algebra II Honors Final Exam Review Answer Section MULTIPLE CHOICE 1. B 2. D 3. B SHORT ANSWER f(x) = 895(1.07) x % increase 7. $1, log 6 1, 296 = log b (q 5 y 2 ) 1

19 12. none of these 13. log log 3 p ln x 3 c , direct variation; y = 14x 23. inverse variation; y = y = 5x z ; barrels 26. a varies directly as F and inversely as m. 27. x 2

20 x(5x + 7) 31. 4( 2x 2 3x + 4) 32. (x + 6)(x + 8) 33. (4x + 5) (x + 6)(x 2 6x + 36) 35. (x 2)(x + 2)(x 4)(x + 4) 36. 8, , yards 39. 5i i i i i, 4 3 i 45. 4,

21 47. 5, y = (x + 5) P = (h 30) ; $30; $ , quintic binomial 53. 2x 2 3x L(x) = 0.5x x x + 0.2; 741,600 llamas 55. 4x(x 4)(x + 6) 56. 0, 3, f(x) = x 3 2x 2 19x x 2 12x + 32, R x x 2 x feet 61. 4, 2, 1, 1, 2, ± 5i, , complex roots; 0, 2 or 4 real roots 65. 1, 3 2, ± 5i , , ways d 6 30d d d d d and g 3 4

22 in a 4 b 2 3 2a units x x 15 7 a feet x x 2 2x ; all real numbers except x = y = ± x

23 x = 9, x = asymptote: x = 4 and hole: x = y = p 8; p a 7b, a 0, b 0 2 (c + 1)(c 2), c 5, 2 c 2 7a 49 (a 8)(a + 8) w w vertex (2, 5), focus (2, 7), directrix at y = vertex (4, 5), focus(4, 2), directrix at y = ( x + 5) 2 + Ê Ë Á y + 8ˆ 2 = (5, 6); x y 2 9 = 1 x 2 204, 304 y 2 380, 921 = 1 ( x 3) 2 9 (x + 3) Ê Ë Á y + 3 ˆ 2 36 Ê Ë Á y + 2 ˆ 2 64 = 1 = ellipse with center ( 4, 4), foci at ( 4 ± 3, 4) 113. parabola; vertex (5, 3) 6

24 114. a. 9.5 seconds b. 1,054 ft 115. a. combination b. permutation c. combination 116. a. The radius of the circle is 80 miles. b. about 20,100 square miles 7