Algebra II Honors Final Exam Review

Transcription

1 Class: Date: Algebra II Honors Final Exam Review Short Answer. Evaluate the series 5n. 8 n =. Evaluate the series (n + ). n = What is the sum of the finite arithmetic series? Divide x + x + x + by x Divide x x x by x. 7. Determine which binomial is not a factor of x x 6x + 9x Is (x ) a factor of P(x)=x + x 6x? If it is, write P(x) as a product of two factors. Divide using synthetic division. 9. Divide x + x x+ 6 by (x ). 0. Divide x 80+x by (x+ ).. Use the Rational Root Theorem to list all possible rational roots of the polynomial equation x 6x + x + 9 = 0. Do not find the actual roots. Find the roots of the polynomial equation.. x x + 0x + 6 = 0. x x 5x 5=0. A polynomial equation with rational coefficients has the roots + 6, 5. Find two additional roots. 5. Find a quadratic equation with roots + i and i.

2 Find all the zeros of the equation. 6. x 5x + 5x 5x + 75 = 0 7. x x x =0 8. 7x = x Use Pascal s Triangle to expand the binomial. 9. (s + v) 5 0. (d ) 6. Find all the real square roots of Find all the real cube roots of Find the real-number root.. 5 What is a simpler form of the radical expression?. 6g x 0 y 8 6. 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.. Multiply and simplify if possible What is the simplest form of the expression? 8. 8a b 6

3 What is the simplest form of the product? 9. 7x 7 9x What is the simplest form of the quotient? x 8 x What is the simplest form of the radical expression?. x + 6 x. a 6 a What is the simplest form of the expression? What is the product of the radical expression? Ê ˆ 6. 7 Ë Á Ê Ë Á 8+ ˆ Ê ˆ 7. 5 Ë Á Ê ˆ Ë Á Ê Ë Á 0 ˆ Simplify

4 8. Write the exponential expression x in radical form.. 7 What is the simplest form of the number? What is the solution of the equation?. x = 5. x = x + 8 = 6. ( x + 6) 5 = x = 0+5x 8. x+ 8 8=x 9. Let f(x) = 5x and g(x) = 6x 7. Find f(x) + g(x). 50. Let f(x) = x 5 and g(x) = 6x. Find f(x) g(x). 5. Let f(x) = x 6 and g(x) = x. Find f g and its domain. 5. Let f(x) = x 7 and g(x) = x +. Find (f û g)( 5). 5. Let f(x) = x + 6 and g(x) = x + 8 x. Find Ê Ë Ág û f ˆ ( 7). What is the inverse of the given relation? 5. y= x For the function f(x) = x + 9, find (f û f )(5). Graph the equation. 56. y = x +

5 57. y = x 58. y = x 59. You can model the population of a certain city between the years 965 and 995 by the radical function P(x)=55000 x 950. Using this model, in what year was the population of that city 5,000? Graph the exponential function. 60. y = x 6. An initial population of 80 quail increases at an annual rate of %. Write an exponential function to model the quail population. What will the approximate population be after years? 6. The half-life of a certain radioactive material is days. An initial amount of the material has a mass of 6 kg. Write an exponential function that models the decay of this material. Find how much radioactive material remains after 5 days. Round your answer to the nearest thousandth. 6. Suppose you invest $600 at an annual interest rate of.6% compounded continuously. How much will you have in the account after years? Write the equation in logarithmic form = = 65 Evaluate the logarithm. 66. log log Graph the logarithmic equation. 68. y = log 5 (x ) Write the expression as a single logarithm. 69. log b q + 6 log b v 70. log log 5

6 Expand the logarithmic expression. 7. log d 7. log p 7. Use the Change of Base Formula to evaluate log Use the Change of Base Formula to evaluate log 7 8. Solve the exponential equation = 6x 76. x = 8 Solve the logarithmic equation. Round to the nearest ten-thousandth if necessary. 77. log x = 78. Solve log(x + 0) =. 79. log(x + 9) log x = 80. Solve log x+ log 9 = 0. Round to the nearest hundredth if necessary. Write the expression as a single natural logarithm. 8. ln + lnc 8. Simplify ln e. Use natural logarithms to solve the equation. Round to the nearest thousandth. 8. 6e x = 8. e x = 85. e x =. 6

7 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. 86. x y x 9 y Suppose that x and y vary inversely, and x = when y = 8. Write the function that models the inverse variation. Graph the function. 89. y = x Sketch the asymptotes and graph the function. 90. y = 5 x + 9. Write an equation for the translation of y = x that has the asymptotes x = 7 and y = This graph of a function is a translation of y=. What is an equation for the function? x 7

8 Find any points of discontinuity for the rational function. 9. y = (x + 6)(x + )(x + 8) (x + 9)(x + 7) 9. Describe the vertical asymptote(s) and hole(s) for the graph of y = (x + )(x + ) (x + )(x + ). 95. Find the horizontal asymptote of the graph of y = x 6 + 6x + 8x 6 + 9x +. Simplify the rational expression. State any restrictions on the variable. 96. p p p q + q + q 5q What is the product in simplest form? State any restrictions on the variable. 98. a 5 7b b a 99. z z + z + z + z + z What is the quotient in simplified form? State any restrictions on the variable. 00. x + x x + x + x 5 Simplify the sum a a 6 Simplify the difference. 0. b b 8 b + b 6 b 8

9 Simplify the complex fraction. 0. 5t t t + t Solve the equation. Check the solution x + = x + 5 6w + w = What are the solutions of the rational equation? Use a graphing calculator to solve c + c = Generate the first five terms in the sequence using the explicit formula. 07. y n = 5n What is the 5 th term in the sequence using the given formula? c n = n 09. Write a recursive formula for the sequence 7,, 9, 5,,... Then find the next term. 0. Write a recursive formula for the sequence 7,,,, 5,... Then find the next term.. Write an explicit formula for the sequence 8, 6,,, 0,... Then find a.. Suppose you drop a tennis ball from a height of 8 feet. After the ball hits the floor, it rebounds to 80% of its previous height. How high will the ball rebound after its third bounce? Round to the nearest tenth. Is the sequence arithmetic? If so, identify the common difference.., 0, 7,,....,,, 77, Find the 50th term of the sequence 5,, 9, 6,... 9

10 6. Find the nd and rd term of the sequence 7,,,, 7, Find the missing term of the arithmetic sequence,,,... Is the sequence geometric? If so, identify the common ratio. 8. 6,,, 8,... 9.,, 6, 6,... What is the fifth term of the geometric sequence? 0. 5, 5, 5,... Write the explicit formula for the geometric sequence. Then find the fifth term in the sequence.. a =, a = 8, a = 6 What is a possible value for the missing term of the geometric sequence?. 50,, 50, ,, 50,... 0

11 Algebra II Honors Final Exam Review Answer Section SHORT ANSWER x x + 59, R 6. x 8x 7, R 6 7. x + 8. yes: P(x)= (x )(x + x+ ) 9. x + 5x, R 6 0. x + 56x 0. 9,,,,, 9. ± 5i,. i 5, i 5,. 6, x + x + 7 = 0 6.,,± 5i 7. i, i, 8.,, i, i 9. s 5 + 0s v + 0s v + 80s v + 80sv + v 5 0. d 6 d d 60d + 0d 9d + 6. no real root g 5. x 5 y 6..5 in a b a 9. x 6x x 8 5x. 8 x

12 . a x x 50. x 5. ; all real numbers except x = y= x

13 f(x) = 80(.) x ; 56 Ê 6. y = 6 ˆ x Ë Á ;.95 kg 6. $,9.

14 6. log = log 5 65 = log b (q v 6 ) 70. log 7. log d log 7. log + log p ln 7c direct variation; y = x 87. neither 88. y = 6 x

15 y = x y= x 6 9. x = 9, x = 7 9. asymptote: x = and hole: x = 95. y = 96. p 8; p q q 8 ; q, q a 7b, a 0, b 0 z + z, z, 0, z + 5

16 (x + )(x + 5), x, x + 7a 9 (a 8)(a + 8) b 0 b , 5, 0, 5, a n = a n + 6, where a = 7; 7 0. a n = a n, where a = 7; 8. a n = n + 0; 8.. feet. yes; 7. no , yes; 9. no a n = ( ) n ;