College Algebra and College Algebra with Review Sample Final The final exam will consist of 25 questions (worth 4 points each); 22 of these will be multiple choice and 3 will be open-ended. This will be an in-class, closed book exam. You may use your Graphing Calculator. A formula sheet will be provided. This sample final contains a variety of both multiple choice and open-ended questions. Find the domain of the function. 1 1) f(x) = 7 x A) (7, ) B) (- , 1) C) (- , ) D) (- , 0) (0, ) Determine the intervals on which the function is increasing and decreasing. 2) A) Increasing (- , 3); Decreasing(- , -3) B) Increasing (- , 3); Decreasing (-3, ) C) Increasing (3, ); Decreasing (-3, ) D) Increasing (3, ); Decreasing (- , -3) Use the compound-interest formula to find the account balance A with the given conditions. 3) An initial investment of $480 is invested for 4 years in an account that earns 16% interest, compounded quarterly. Find the amount of money in the account at the end of the period. A) $419.03 B) $864.45 C) $899.03 D) $869.11 Find an equation for the circle. 4) Endpoints of a diameter (-1, 2), (7, -4) A) x2 + (y + 1)2 = 16 B) (x + 1)2 + (y - 3)2 = 25 C) (x - 3)2 + y2 = 9 D) (x - 3)2 + (y + 1)2 = 25 Determine the indicated asymptotes of the graph of the following function. x2 + 4x - 12 5) The vertical asymptote(s) of f(x) = x2 - 2x - 8 A) y = -2, y = 4 B) x = -2, x = 4 C) x = 4 1 D) x = 2, x = -4 Write a slope-intercept equation for a line with the given characteristics. 6) (-5, -1) and (4, -9) 8 49 A) y = - x 9 9 B) y = - 4 101 x13 13 C) y = 8 49 x9 9 D) y = 4 101 x13 13 Find the domain and the range of the function. Use interval notation. 1 7) f(x) = (x + 9)2 A) Domain: (- , 0) (0, ); Range: (- , 9) B) Domain: (- , 0) (0, ); Range: (- , -9) C) Domain: (- , 9) (9, ); Range: (- , 0) (0, ) D) Domain: (- , -9) (9, ) (-9, ) (-9, ); Range: (0, ) Determine the vertical and horizontal asymptotes of the graph of the following function. 9 8) y = + 10 x A) Vertical: x = 0; Horizontal: y = 10 B) Vertical: y = 10; Horizontal: x = 9 C) Vertical: x = 9; Horizontal: y = 10 D) Vertical: x = 0; Horizontal: y = 9 Given that the polynomial function has the given zero, find the other zeros. 9) P(x) = x3 - 3x2 + 7x - 85; 5 is a zero A) x = -1 + i, -1 - i C) x = -1 + 4, -1 - B) x = -1 + 4i, -1 - 4i 4 Given the following functions, find (f D) x = 4 + i, 4 - i g)(x). 10) f(x) = 4x + 14, g(x) = 5x - 1 A) 20x + 18 B) 20x + 69 C) 20x + 13 D) 20x + 10 C) y-axis D) No symmetry Provide an appropriate response. 11) What symmetry does the graph of y = f(x) exhibit? A) Origin B) x-axis 2 Express in terms of sums, differences, and multiples of logarithms. 5 x ) 12) log3 ( y A) log3 5 · 1 log3 x ÷ log3 y 2 B) log3 y - log3 5 - 1 log3 x 2 C) log3 5 + 1 log3 x - log3 y 2 D) log3 (5 x) - log3 y The function has a real zero between the two numbers given. Use your calculator to approximate the zero to the nearest tenth. 13) P(x) = x3 + 3x2 - 7x - 21; 1 and 4 A) x = -2.6 Solve. 14) F = B) x = 2.7 C) x = 2.6 D) x = 2.5 9 C + 32 for C 5 A) C = F - 32 9 B) C = 9 (F - 32) 5 C) C = 5 F - 32 C) x = 1 25 B) x = -10 + 19 -10 - 19 , 2 2 D) x = -5 + 31 -5 - 31 , 2 2 D) C = 5 (F - 32) 9 Solve the equation. 1 15) 5-x = 125 A) x = {3} B) x = 1 3 16) 2n 2 = -10n - 3 -5 + 19 -5 - 19 , A) x = 2 2 C) x = -5 + 19 -5 - 19 , 4 4 D) x = {-3} 17) log5 x = 2 A) x = 25 B) x = 10 18) 9x - (2 - x) = 4[3 - (4 + 2x - 3)] 7 A) x = B) x = 8 C) x = 32 5 9 C) x = - 3 D) x = 7 7 9 D) x = 5 Solve the inequality. Write the solution set in interval notation. 19) x2 + 4x - 5 > 0 A) (- , -5) B) (-5, 1) C) (1, ) D) (- , -5) B) [2, ) C) (-12, ) D) (- , -2] (1, ) 20) -12x + 10 -11x + 12 A) (- , -12] Solve and write interval notation for the solution set. 21) 8 x + 4 9 23 41 ,A) 8 8 C) - , 41 8 23 , 8 B) - , - 41 8 - 23 , 8 D) - , - 23 8 - 41 , 8 Solve. 22) Marty's Tee Shirt & Jacket Company is to produce a new line of jackets with a embroidery of a Great Pyrenees dog on the front. There are fixed costs of $630 to set up for production, and variable costs of $29 per jacket. Write an equation that can be used to determine the total cost, C(x), encountered by Marty's Company in producing x jackets, and use the equation to find the total cost of producing 76 jackets. A) $2814 B) $2834 C) $2846 D) $2826 Write the augmented matrix for the system. 23) 3x + 5z = 29 6y + 7z = 82 2x - 2y + 3z = 0 3 0 5 A) 0 6 7 2 -2 3 B) 3 5 0 29 6 7 0 82 2 -2 3 0 C) 3 0 5 29 0 6 7 82 2 -2 3 0 D) 3 0 2 29 0 6 -2 82 57 3 0 Solve. 24) Employees of a publishing company received an increase in salary of 3% plus a bonus of $1000. Find the new salary if the old salary was $25,000. A) $23,301 B) $26,750 C) $33,500 D) $26,000 Use Gaussian elimination or Gauss-Jordan elimination to solve the problem. 25) A company makes 3 types of cable. Cable A requires 3 black, 3 white, and 2 red wires. B requires 1 black, 2 white, and 1 red. C requires 2 black, 1 white, and 2 red. They used 95 black, 100 white and 90 red wires. How many of each cable were made? A) 30 cable A 5 cable B 25 cable C B) 58 cable A 30 cable B 22 cable C C) 5 cable A 30 cable B 25 cable C 4 D) 5 cable A 18 cable B 25 cable C Solve. 26) Assume that the sales of a certain appliance dealer are approximated by a linear function. Suppose that sales were $5500 in 1982 and $88,000 in 1987. Let x = 0 represent 1982. Find the equation giving yearly sales y. Solve. 27) A) y = 82,500x + 88,000 B) y = 16,500x + 5500 C) y = 16,500x + 88,000 D) y = 82,500x + 5500 5-a 3 7 + = a 4 a A) x = {-8} B) x = {8} C) x = {-4} D) x = 29 20 Simplify. Write answers in the form a + bi, where a and b are real numbers. 7 + 3i 28) 2 - 4i A) 1 17 + i 10 10 B) - 13 17 + i 6 12 C) 13 11 + i 5 5 D) - 1 17 + i 12 12 Simplify. 29) 8 + -81 A) 8 - 9i B) 8 + i C) 8 + 9i D) -1 Determine the domain and range of the relation. 30) {(1, -6), (9, 8), (12, -4), (-5, 3), (10, -1)} Write a slope-intercept equation for a line passing through the given point that is perpendicular to the given line. 31) Through (6, -3), perpendicular to 5x + 2y = 24 Find the indicated function value. 32) Find f(3) for f(x) = -4x2 - 2x + 3 Find the distance between the pair of points. Give an approximation to two decimal places. 33) P(8, 6) and Q(4, 8) Find the maximum or minimum point of the function, by using the vertex formula or by using the graphing calculator. State whether the point is a maximum or minimum. If you choose to solve this problem using a graphing calculator, you must show a graph (including the window) to support your conclusion. 34) f(x) = 3x2 - 12x + 18 Solve the inequality. Write the solution set in interval notation. 35) -8 < -4b + 4 8 If the function is one-to-one, find a formula for the inverse. 36) f(x) = 6x - 3 5 Perform the requested operation or operations. 37) Find (fg)(2) when f(x) = x + 5 and g(x) = -5x2 + 14x - 3. Graph the function. 3x - 5 38) f(x) = x-2 39) 4x + 3, y(x) = if x < 0 4x2 - 3, if x 0 6 Sketch the graph of y1 as a solid line or curve. Then sketch the graph of y2 as a dashed line or curve. 40) y1 = x2 ; y2 = (x - 3)2 - 5 Solve. 41) log5(x + 5) + log5(x - 5) = 1 42) 3(x - 1) = 14 43) x = x + 13 + 7 44) b - 6 + 1 = 7 45) 7x + 5 3x + 3 + =-2 4 3 Solve. 46) How long will it take for $1600 to grow to $28,800 at an interest rate of 12.3% if the interest is compounded continuously? Round the number of years to the nearest hundredth. 47) The number of visitors to a tourist attraction (for the first few years after its opening) can be approximated by V(x) = 50 + 10 log x, where x represents the number of months after the opening of the attraction. Find the 2 number of visitors 4 months after the opening of the attraction. 48) A ball is thrown downward from a window in a tall building. Its position at time t in seconds is s = 16t2 + 32t, where s is in feet. How long (to the nearest tenth) will it take the ball to fall 195 feet? 49) An average score of 90 for 5 exams is needed for a final grade of A. John's first 4 exam grades are 79, 89, 97, and 95. Determine the grade needed on the fifth exam to get an A in the course. 7 Solve the system of equations. 50) -6x + 7y = -7 -3x + 4y = -4 The figure below shows the graph of a function y = f(x). Use this graph to solve the problem. 51) Sketch the graph of y = f(x - 3). Using a graphing calculator, find the real zeros of the function. Round to two decimal places. 52) 0.37x3 - 5.72x2 + 5.23x + 8.25 = 0 Use synthetic division to find the quotient Q(x) and the remainder R. 53) P(x) = 4x3 - 26x2 + 28x + 10; x - 5 Write an equivalent expression in exponential form. 54) log10 10,000,000 = 7 8 College Algebra Sample Final Answer Key: Problem Number: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. Correct Answer: D D C D B A D A B D C C C D A A A B D D B B C B C B A A C Section from text: 1.2 or 4.5 2.1 5.2 1.1 4.5 1.3 or 1.4 1.2 or 4.5 4.5 4.4 2.3 2.4 5.4 4.1 or 4.2 1.5 5.5 3.2 5.5 1.5 4.6 1.6 3.5 1.5 6.3 1.5 6.3 1.4 or 1.5 3.4 3.1 3.1 Problem Number: 30. Correct Section Answer: From text: Domain = {1, 9, 12, 5, 10}; 1.2 Range = {6, 8, 4, 3, 1} 31. y = 2 x - 27 1.4 32. 33. 34. 35. f(3) = 39 d = 2 5 = 4.47 (2, 6); minimum [1, 3) 1.2 1.1 3.3 1.6 36. f - 1 (x) = x + 3 5.1 37. (fg)(2) = 35 2.2 5 5 6 38. 4.5 39. 2.1 Problem Number: Correct Answer: Section From Text: 40. 2.4 or 3.3 41. 42. 43. 44. x = 30 x = 3.402 x = 12 b = 12, 0 5.5 5.5 3.4 3.5 45. x = - 17 3.4 46. 47. 48. 49. 50. 23.50 years 70 visitors 2.6 sec. 90 (0, 1) 11 5.6 5.3 3.2 1.5 6.1 or 6.3 51. 2.4 . 52. 53. 54. x = {0.81, 1.91, 14.37} Q(x) = 4x 2 – 6x – 2; R = 0 10 7 = 10,000,000 4.1 4.3 5.3

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