Math Algebra Final Exam Review Answers

Chapter 8: Exponents 1. Write  x 5 3 2. Write y 7 2 in rational exponent notation. in radical notation. Simplify the expression. 3 3. 100 2 1 4. 121 2 2 5. 216 3 1 2 82 1 1 6. 27 3 7. 6  6 3 2 8. 8 1 6 Simplify. 9.  3x y   5x y  10. (2 y 2 z 3 )4  13. 2ab 2c3 3 2 4 6    12.  5a 2b 3a 4   2  15. 3 p3  6 p   2 p 2  3    11. a3  a 6 14.  x 4    4 16. 5m3  6m 3 2 17. 3 12 y 2  7 x2 y  8 y 2  11y x2 Page 3 of 17    18. x 4 x 3 21. 24. 27. 10 5 10a b 2a11b 2 15 x 3 y 2 z 4 10 x 4 y 4 z 2 19. y 3  y 3 22. 4 20.  10 x 7   8  9y x3  3x 2 5 x    23.  5 x    3 1 64 x y z 16 x  2 yz 3 3y3 25. 5 x5    2  3x 5 26.  2  10 y 0 3  5   4  x  3a 4 b 7 21a 2 b 7 c 5 28. Identify each picture as growth or decay. Then identify the graph of exponential growth y  3 2 x __________________ __________________ __________________ __________________ Page 4 of 17 Write a rule for the following functions. 29. 30. Circle whether the function represents exponential growth or decay. Graph the function. Identify its domain and range. x 1 31. y  2    Growth/Decay 5 x y -2 -1 0 32. y  3 6 x 1 x y 2 -2 Growth/Decay -1 0 1 2 D: _________ D: _________ R: _________ R: _________ CHAPTER 8 ANSWERS 5 1. 2. x3  y  or √𝑦 7 1 7. 6 2  36 8. 8 3  2 3. 7 9.  100  =1000 3  15 x 7 y 8 10. 4. 1  121 16 y 8 z 12 1 11 5. 11. a 13. 16a b c 14. x 12 15. 17 p 6  6 p 16. 125m  36m 19. 1 20. x 2 21. 5b 3 a 22. 4 25. 20 x 14 8 12 26. 27 x 27 y 13 27.  c 11 200 y 6 5 7a 6 31. Decay, D: all reals, R: y > 0 9 3  2 216 = 36 3 136 y 2  18x 2 y 23. 1 18. x 24. -2 -1 0 1 2 y 50 10 2 0.4 .08 7 3z 2 2 xy 2 30. y  1   1  29. y  1 4x 5 32. Growth, D: all reals, R: y < 0 x 27 = 3 6 17. 2 6. 12.  15a b 9 4x 6 y4z4 28. G, D, G, G  x -2 -1 y - .083-- -.5 0 1 2 -3 - -18 - -108 Page 5 of 17 x Chapter 9: Factoring Use this polynomial to answer #1: 3x  4 x  12  5x 5 3 1. (a) Is the polynomial in order? If not, put it in order. (b) What is the leading coefficient of the polynomial? (c) What is the degree of the polynomial? Simplify. 2. (h2  4h  4)  (5h2  8h  2) 3. (9b3  13b2  b)  (13b3  5b  14)    4. 5x 2  x  7  2 x 2  3x  4 5. (5x2  16x  9)  (3x2  4x  1) 6. z 2 (4 z 4  z 3  11z 2  6) 8. x  2( x  3) 9. a  6 11. 7 y  4 2 y  5 12. 2 x  4 x  2 13. 3 y  52 14. 3x  42 15. 3k  14k  9 16. (5x  9)(3x  4) 2 7.  10 pq(3 pq  4 p  5q 2 ) 10. 2 x  y  2 x  y  Page 6 of 17  Factor completely. 17. 8a 2b3  10ab2 2 2 18. 20 xy  100 x y 19. y 2  64 20. 2 x 2  200 21. x 2  5x  6 22. 9 x 2  30 x  25 23. 16 x 2  8x  1 24. 2 x 2  4 x  18 25. 6a 2  10a  4 26. a 2  12a  27 27.  2h2  5h  3 28. y 3  y  9 y 2  9 29. 5n3  4n2  25n  20 Solve the equation by factoring. 30. (3x  2)( x  5)  0 31. x 2  10 x  21  0 32. x 2  5x  14 33. 3a 2  4a 34. m2  22m  23m 35. s(s  1)  72 Page 7 of 17 f ( x)  3x 3  18x 2  24 x 36. Find the zeros of the function: 37. Find a polynomial that represents the PERIMETER of the figure. 38. Find a polynomial that represents the AREA of the figure. 2x - 7 3x + 5 CHAPTER 9 ANSWERS 1.  4 x 5  5x 3  3x  12 LC: -4, Deg: 5 5. 2 x  20 x  10 2 6 10. 4x 2  y 2 14. 3x  24 x  48 20 xy 1  5xy  19. 11. 7. 12. 2 x  8 2 31. x = 7, 3 34. m = 0, -45 35. s = -8, 9 13. 8. x  x  6 2 9 y 2  30 y  25 16. 15x  47 x  36 17. 2 20.  2h  1h  3 2  30 p 2 q 2  40 p 2 q  50 pq 3 2x  10x  10 2 27. 4. 7 x  2 x  3 2 14 y 2  27 y  20  y  8 y  8 2 ,5 3 30. x  2 23. 4 x  1 2 a  3a  9 4 2 22. 3x  5 26. 5 3 15. 12k  23k  9 2 18. 3. 22b  13b  6b  14 2 6. 4 z  z  11z  6 z 2 9. a  12a  36 2. 6h  4h  2  2 x 2  2x  9 24. 28. y 2 21.   1  y  9 x  3x  2  25. 29. n 2 23a  1a  2   5 5n  4 33. a  32. x = 7, -2. 36. x = 0, -4, -2 2ab 2 4ab  5 37. 10x + 2 Page 8 of 17 4 ,0 3 38. 6 x 2  11x  35 Chapter 10: Quadratics Graph each parabola and the parent graph y  x 2 together on the same graph and compare them. Parent y  x 2 x y 1. y  2x 2 2. y 3 2 x 8 3. y  x2  5 x x y x y y Compare:________________ Compare:________________ Fill in the following missing information and graph. 2 4. y  x  2 x  3 Compare:________________ 5. y  x  6 x  8 2 Axis of Symmetry:_____________ Axis of Symmetry:_____________ Vertex: (____, ____) Vertex: (____, ____) Max or Min Value:____________ Max or Min Value:____________ x x y y Solution(s): _______________ Solution(s): _______________ Page 9 of 17 Solve the following equations using the square root property. 6. x 2  6  13 7. 14  x 2  17 8. 6x  8  18 2 Given the following equations, find the solutions/roots/zeros by completing the square. 2 9. x 2  2 x  3 10. c 2  14c  15 11. x  8 x  9 Use complete the square to convert the quadratic to vertex form: y = a(x – h) 2 + k. Identify the vertex and axis of symmetry. 12. y  x 2  8x  15 13. y  x 2  12 x  36 14. y  x 2  2 x  3 Axis of Symmetry: _________ Axis of Symmetry: _________ Axis of Symmetry: _________ Vertex: (____, ____) Vertex: (____, ____) Vertex: (____, ____) Given the following equations: a) Find the value of the Discriminant; b) Tell whether the equation has two real solutions, one real solution, or no real solutions. 2 2 15. 3x2  6 x  2  0 16. x  1  2 x 17. 2 x  3x  5  0 A)____________________________ A)____________________________ A)____________________________ B)____________________________ B)____________________________ B)____________________________ Page 10 of 17 Solve the following equations by using the quadratic formula. 18. 5x 2  2 x  3  0 19. 2 x 2  3x  5 20. x 2  6 x  10  0 By which type of function can the data in the table be modeled: linear, exponential or quadratic? 21. x -3 -2 -1 0 1 y 13.5 6 1.5 0 1.5 22. x -2 -1 0 1 2 y 1 2 4 8 16 23. Draw the graph of a parabola which has: a) No real roots b) One real root Solve the quadratic using any method. 24. 25.  5x 2  x  13  0 x 2  14 x  0 c) Two real roots 26. 9 x 2  16  0 27. The length of Rachel’s rectangular garden is 6 yards more than its width. The area of the garden is 91 square yards. What are the dimensions of the garden? Page 11 of 17 Tell whether the table represents a linear, exponential, or quadratic function. 28. 29. x -2 -1 0 1 x 0 1 2 3 4 y -3 -1 1 3 y 5 80 40 20 2 10 5 Tell whether the graph represents a linear, exponential, or quadratic function. 30. 31. 32. CHAPTER 10 ANSWERS 1. reflection and 2. vertical shrink vertical stretch 3. translated down 5 x -1 0 1 2 3 4. a. o. s: x = 1, vertex: (1, 2), min value: 2, solutions: none y 6 3 2 3 6 5. a.o.s: x = -3 ,vertex:(-3, -1) , min value: -1, solutions: -2, -4 x -5 -4 -3 -2 -1 y 3 0 -1 0 3 6. x 7 10. c = 15, -1 7. no solution 11. x = 9, -1 8. x =  8  3 12. y  x  4  1 2 9. x = 1, -3 13. y  x  6 2 a.o.s: x = -4 vertex: (-4, -1) 15. 12, 2 real 16. 0, 1 real 17. -31, no real 21. quadratic – table is symmetric, second differences the same 23b. 23c. 14. y  ( x  1) 2  4 a.o.s: x = 1 vertex: (1, -4) a.o.s: x = 6 vertex: (6, 0) 18. x = -1, 3/5 19. no solution 20. x = 22. exponential – y’s multiplied by 2 24. x = 0, 14 25. no solution 26. x =  4 29. Exponential 30. Exponential 31. Quadratic 2 23a 27. L = 13 yd; W = 7 yd. 3 28. Linear −6 ±√76 32. Linear Page 12 of 17 Chapter 11: Radicals Graph and compare to the parent function: 𝒚 = √𝒙 1. y  2 x  1 x -1 1 2 x y 0 1 2 3 4 y 0 1 1.4 1.7 2 y  2 x 1 2. 0 x 0 1 2 y Compare to y = x : Compare to y = x : ______________________ ______________________ ______________________ ______________________ ______________________ ______________________ Simplify the following. 3. 200 4. 25x 3 5. 9a 6 6. 64 y 4 7. 27c5 8. 32r 2 s 4 t 5 9. 40 x 4 y 5 z 3 10. 3  18 11. 7 30  2 6 Page 13 of 17 3 12. 1 25 13. 121 49 14. 7 100 17. 5 48 20. 9 75 Simplify the expression by rationalizing the denominator. 15. 4 16. 5 18. 8 3n3 19. 1 3 1 5 Simplify the following expressions. 21. 3 5  5  3 5 24.  7 4 7  22. 5 18  2 32 23. 4 6  30  3 24 25. (2  6 )2 26. (2  5 )(1  2 3 ) Solve the equation. Check for extraneous solutions. 27. 5x  4  16 28. 3x  8  x  4 29. 3x  4  4 Page 14 of 17 30. 5x  6  x CHAPTER 11 ANSWERS 1. x -1 Y 0 - -2 0 x 1 2 -2.8 - 3.5 3. 10 2 4. 5 x x 5. 3a 10. 3 6 11. 84 5 12. 18. 2 6n 19. 17. 15 12 3n 2 24. 4 7  7 25. 10  4 6 35. 37 3 1 5 5 5 reflection, v. stretch, left 1 6. 8y 2 7. Y 3c 2 3c 0 -1 1 1 2 3 1.8 2.5 2. v. stretch, down 1 8. 4rs2t 2 2t 9. 2 x 2 y 2 z 10 yz 13. 11 7 14. 7 10 15. 4 5 5 16. 20. 3 5 21. 5 22. 23 2 23.  2 6  30 26. 2  4 3  5  2 15 27. x = 80 28. x = -2 29. x = 4 36. 4 2 Page 15 of 17 3 3 30. x = 2,3 Chapter 12: Rational Functions Identify the excluded values and simplify the expression. 9 x 3  18 x 4 6x 5 15m 1. 2. 3. 3x 2 48 x 7 3m  6m 2 x  _______ m  _______ x  _______ Find the product, then simplify. 8 x4 5. 2  x 4x 6. 4. x3 x  2 2 x  8 x  7 x  12 7. m3 m2  9 m  _______ 3x  6  x 2  6 x  5  x x2 2 Find the quotient, then simplify. 5 x 2 10 x 3 8.  7 21 3x  12 2 x  8 9.  4 x  18 x  4 10. x 2  25 2 x 2  12 x  16  x 2  7 x  10 x2  4 Find the sum or difference. 11. t 2t  4 4 12. x2 x6  5 5 13. 5 x  x 1 x 1 Page 16 of 17 8𝑥 14. 2𝑥 6 + 2 8  2 2 x x 15. 2𝑥 Find the LCD for: 3 1 17. and 2 20a 24ab3 16. 18. 3𝑥+5 𝑥3 − 2𝑥+5 𝑥3 10 11 and 3 49ax 2 56 x y Find the sum or difference. 19. t 2t  3 7 20. 22. 8x 1  3 5x 23. x2 x6  3 5 5 x  x 1 x 1 21. 2 8  2 3x 5x CHAPTER 12 ANSWERS 1. 3x(1  2 x) , x 0 5. 2x 6. 10. x 2( x  4) 2 x  5x  2 2x  4x  2 17. 120a b 2 3 2. 18. 11. 3t 4 392ax 3 y 5 , x  0, 1 1  2m 2 3. 1 , x0 8x 2 9. 3( x  4) 8. 3 7. 3(x+5) 4(2 x  9) 2x 12. 19. 8 5 13t 21 13. x  5 x 1 20. 8x  8 15 14. 21. 1 , x  3 , 3 m3 4. 4𝑥+3 𝑥 5 x x 1 15.  6 x2 16. 22. 40 x  3 2 15 x 23.  24 x  10 15 x 2 Page 17 of 17 1 𝑥2