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Math 161 Final Exam Version 1 (144 points)
Name ___Answers____________________
Precalculus I
2022 May 09
John Scalea
Page 1
2 answers: 4 points; #1 and #2: 8 points total each; 20 points
146 points
1. Consider y = 2 | (5/2)x – 10 | – 4
2. Consider 4x – 6y = -3
y = 2 | (5/2)(x – 4) | – 4
6y = 4x – 3 so y = (2/3)x – (1/2)
For each equation, complete the table below. Then sketch a graph of the equation on the axes below.
domain
range
x-intercept(s)
y-intercept
slope
vertex
as x → -∞, f(x) →
as x → ∞, f(x) →
0 = 2 | (5/2)x – 10 | – 4
4 = 2 | (5/2)x – 10 |
2 = | (5/2)x – 10 |
all real numbers
y ≥ -4
(16/5, 0), (24/5, 0)
(0, 16)
xxxxxxx
(4, -4)
∞
∞
all real numbers
all real numbers
(-3/4, 0)
(0, 1/2)
2/3
xxxxxxx
-∞
∞
(5/2)x – 10 = 2 or (5/2)x – 10 = -2
(5/2)x = 12 or (5/2)x = 8
x = 24/5 or x = 16/5
3. Solve -2(4 – 5x) + 3x = -1 + (3 – 2x) for x. Present your answer as a simplified fraction.
____________ x = 2/3 (self)
-8 + 10x + 3x = -1 + 3 – 2x
-8 + 13x = 2 – 2x
15x = 10
x = 10/15 = 2/3
4. Solve -5(3 – 5x) – 2x < 1 – 3(5x + 4) for x. Present your answer as a simplified fraction. ____________ x < 2/19 (self) -15 + 25x – 2x < 1 – 15x – 12 38x < 4 -15 + 23x < -15x – 11 x < 4/38 so x < 2/19 Math 161 Final Exam Version 1 (144 points) Name ___Answers____________________ Precalculus I 2022 May 09 John Scalea Page 2 2 answers: 4 points; #5: 7 points total; #6: 10 points total; 21 points this page; 41 points total 5. Consider y = x3 + 10x2 – 9x – 90 6. Consider y = x2 – 4x – 21 y = (x + 3)(x – 3)(x + 10) y = (x + 3)(x – 7) For each equation, complete the table below. Then sketch a graph of the equation on the axes below. domain range vertex (exact answer, please) x-intercept(s) y-intercept interval(s) on which y decreases interval(s) on which y increases as x → -∞, f(x) → as x → ∞, f(x) → all real numbers all real numbers xxxxxxxxx (-3, 0), (3, 0), (-10, 0) (0, -90) Xxxxxxxxx Xxxxxxxxx -∞ ∞ all real numbers [-25, ∞) (2, -25) (-3, 0), (7, 0) (0, -21) (-∞, 2) (2, ∞) ∞ ∞ 7. Solve 2x2 + 6x + 8 = -1 for x. If necessary, use simplified radicals, and use i for -1. ____________ x = [-3 ± 3i] / 2 (self) 2x2 + 6x + 9 = 0 x = [-6 ± √(36 – 4(18))] / 4 x = [-6 ± √(-36)] / 4 x = [-6 ± 6i] / 4 x = [-3 ± 3i] / 2 8. Solve 12x2 – 11x + 2 > 0 for x. ____________ x > 2/3 or x < 1/4 (self) (3x – 2)(4x – 1) = 0 x = 2/3 or x = 1/4 pos |neg | pos -----2-----+-----1-----+-----0--|--+-|---1-----+-----2-----| | Math 161 Final Exam Version 1 (144 points) Name ___Answers____________________ Precalculus I 2022 May 09 John Scalea Page 3 2 answers: 4 points; #9: 9 points total; #10: 9 points total; 22 points this page; 63 points total x3 – 2x2 – 9x + 18 x2 – 3 9. Consider y = ------------------------. 10. Consider y = ---------. x2 – 7x + 10 x+7 For each equation, complete the table below. Then sketch a graph of the equation on the axes below. factored form domain hole(s) x-intercept(s) y-intercept horizontal asymptote vertical asymptote(s) slant asymptote (x – 2)(x – 3)(x+3) -----------------------(x – 2)(x – 5) (-∞, 2)  (2, 5)  (5, ∞) (2, 5/3) (-3, 0), (3, 0) (0, 9/5) none x=5 y=x+5 x2 – 3 -------x+7 (-∞, -7)  (-7, ∞) none (-√3, 0), (√3, 0) (0, -3/7) none x = -7 y=x–7 x 14 11. Solve --------- = --------- for x. Present your answer as a simplified fraction. ____________ x+2 x + 11 x = -4 or x = 7 (self) x(x + 11) = 14(x + 2) (x + 4)(x – 7) = 0 x2 + 11x = 14x + 28 x = -4 or x = 7 x2 – 3x – 28 = 0 x2 + 10x + 16 12. Solve ------------------- ≥ 0 for x. ____________ [-8, -5)  [-2, ∞) (self: Stewart 5th: 4.RE.41) x+5 [(x + 8)(x + 2)] / (x + 5) , so critical numbers: {-8, -5, -2} neg | pos | neg | pos ---9---|---7---6---|---4---3---|---1---0--| | | Math 161 Final Exam Version 1 (144 points) Name ___Answers____________________ Precalculus I 2022 May 09 John Scalea Page 4 2 answers: 4 points; #13: 8 points total; #14: 8 points total; 20 points this page; 83 points total 13. Consider y = (-3/2)√[(3/2)x – 3] + 4. 14. Consider y = (1/2){3√[(-5/2)x – 5]} + 1 y = (-3/2)[(3/2)(x – 2)] + 4 y = (1/2){3√[(-5/2)(x + 2)]} + 1 For each equation, complete the table below. Then sketch a graph of the equation on the axes below. domain range x-intercept (exact answer, please) y-intercept (exact answer, please) initial point point of symmetry as x → -∞, f(x) → as x → ∞, f(x) → [2, ∞) (-∞, 4] (182/27, 0) none (2, 4) xxxxxxxxx does not apply -∞ all real numbers all real numbers (6/5, 0) (0, (1/2)[3√(-5)] + 1) xxxxxxxxx (-2, 1) ∞ -∞ 0 = (-3/2)√[(3/2)x – 3] + 4 0 = (1/2){3√[(-5/2)(x + 2)]} + 1 -4 = (-3/2)√[(3/2)x – 3] -1 = (1/2){3√[(-5/2)(x + 2)]} 8/3 = √[(3/2)x – 3] -2 = 3√[(-5/2)(x + 2)] 64/9 = (3/2)x – 3 -8 = (-5/2)(x + 2) 91/9 = (3/2)x 16/5 = x + 2 x = 182/27 x = 6/5 15. Solve √(3x – 2) + 4 = 1 for x. ____________ no solution (self) √(3x – 2) = -3 no solution; principal square roots are never negative 16. Solve 3√[(-5/2)(x + 5)] – 6 = 9 for x. Present your answer as a simplified fraction. ____________ -1,355 (self) 3 √[(-5/2)(x + 5)] = 15 x + 5 = -1,350 (-5/2)(x + 5) = 3,375 x = -1,355 -5(x + 5) = 6,750 Math 161 Final Exam Version 1 (144 points) Name ___Answers____________________ Precalculus I 2022 May 09 John Scalea Page 5 2 answers: 4 points; #17: 9 points total; #18: 9 points total; 20 points this page; 105 points total 17. Consider y = e(-3/2)x – 4 18. Consider y = -2 log ((3/2)x + (3/2)) + 4 y = -2 log ((3/2)(x + 1)) + 4 For each equation, complete the table below. Then sketch a graph of the equation on the axes below. domain range x-intercept (exact answer, please) y-intercept (exact answer, please) horizontal asymptote vertical asymptote as x → -∞, f(x) → as x → ∞, f(x) → all real numbers (-4, ∞) (-2/3)(ln 4) OR (-0.924, 0) (0, -3) y = -4 none ∞ -4 (-1, ∞) all real numbers (197/3, 0) OR (65.667, 0) (0, -2 log (3/2) + 4) OR (0, 3.648) none x = -1 does not apply -∞ 0 = e(-3/2)x – 4 4 = e(-3/2)x (-3/2)x = loge 4 x = (-2/3)(ln 4) 0 = -2 log ((3/2)x + (3/2))) + 4 -4 = -2 log ((3/2)x + (3/2))) 2 = log ((3/2)x + (3/2))) 102 = (3/2)x + (3/2) (3/2)x = 100 – 3/2 = 98.5 x = (2/3)(197/2) = 197/3 19. Solve log3 (2x – 1) – log3 (x – 4) = 2 for x. ____________ 5 (SLO) log3 [(2x – 1) / (x – 4)] = 2 2x – 1 = 9x – 36 (2x – 1) / (x – 4) = 9 35 = 7x 2x – 1 = 9(x – 4) x=5 20. Solve (4)22x+8 + 3 ≥ -3 for x. ____________ all real numbers (self) (4)22x+8 ≥ -6 subtract 3 from both sides 2x+8 2 ≥ -3/2 divide both sides by 4 x = all real numbers 22x+8 is always positive Math 161 Final Exam Version 1 (144 points) Name ___Answers____________________ Precalculus I 2022 May 09 John Scalea Page 6 9 answers: 12 points; 117 points total 21. Suppose you start a business making widgets. After a short period of time, you find that your total cost of producing 16 widgets is $420 and your total cost of producing 42 widgets is $680. Assume that the total cost of producing widgets varies linearly with the number of widgets your produce. (self: 1.CT.29) a. Write a linear function which models the total cost of producing widgets, y, with the number of widgets you produce, x. ____________ y = 10x + 260 b. Use your answer from part a to find the cost of producing 11 widgets. ____________ $370 c. Use your answer from part a to find the number of widgets you produce if your total cost is $560. ____________ 30 a. (16, 420) and (42, 680) y – 420 = 10x – 160 c. 560 = 10x + 260 m = (680 – 420) / (42 – 16) y = 10x + 260 10x = 300 m = 260/26 = 10 b. 10(11) + 260 x = 30 y – 420 = 10(x – 16) 110 + 260 = 370 22. A firefighter holds a hose 3 meters off the ground and directs a stream of water toward a burning building. The height of the water above the ground can be approximated by h = -0.026x2 + 0.577x + 3, where h is the height of the water, in meters, at a point x meters horizontally from the firefighter to the building. (2.1.46) a. How many meters from the firefighter will the water reach its maximum height? ____________ b. What is the maximum height of the water? ____________ 6.2 meters c. The water reaches the house after it reaches its maximum point and at a height of 6 meters above the ground. How far is the firefighter from the house? ____________ 13.9 meters a. –b/2a b. -0.026(11.1)2 + 0.577(11.1) + 3 c. 6 = -0.026x2 + 0.577x + 3 -0.577/-0.052 -0.026(123.21) + 0.577(11.1) + 3 -0.026x2 + 0.577x – 3 = 0 11.1 meters -3.20346 + 6.4047 + 3 x = [-0.577 ± √(0.332929 – 4(-0.026)(-3))] / -0.052 6.20124 x = [-0.577 ± √(0.332929 – 0.312)] / -0.052 x = [-0.577 ± √(0.020929)] / -0.052 x = (-0.577 ± 0.144669) / -0.052 x = 8.314058 or x = 13.87825 23. According to cbsnews.com, the most expensive ticket to Super Bowl I (played in 1967) was $12. The most expensive ticket to Super Bowl XLIX (played in 2015) was $1,350. When answering parts a and b, use the discrete model: F = P(1 + i)n). (self) a. Calculate the rate of return on the most expensive ticket to the Super Bowl over the time interval from 1967 to 2015. Round your answer to the nearer thousandth. ____________ 10.340% b. Formulate an equation which you can use to model the price of the most expensive ticket to the Super Bowl n years after 1967. ____________ F = 12(1.103398)n c. If most expensive ticket to the Super Bowl has appreciated at the same rate to the present year, how much should the most expensive ticket to Super Bowl XLVI (played in 2022) cost? Round your answer to the nearer dollar. ____________ $2,688 1,350 = 12(1 + i)48 F = 12(1.103398)55 48 112.5 = (1 + i) 12(244.011639) 1.103398 = 1 + i 2,688.139667 i = 0.103398 Math 161 Final Exam Version 1 (144 points) Name ___Answers____________________ Precalculus I 2022 May 09 John Scalea Page 7 4 answers: 6 points; #26: 5 points total; 11 points this page; 128 points total 24. The equation y = 3.33 + 14.9 ln x relates the diameter of a sugar maple tree, x (measured in millimeters) to the age of the tree, y (measured in years). Use this model when answering the following questions. (3.6 Ex.8, alt.) a. Estimate the age of a sugar maple tree which has a diameter of 81 millimeters. ____________ 68.807 b. Estimate the diameter of a sugar maple tree which is seven years old. ____________ 1.279 mm a. y = 3.33 + 14.9 ln (81) b. 7 = 3.33 + 14.9 ln x y = 3.33 + 14.9 (4.394449) 3.67 = 14.9 ln x y = 3.33 + 65.477292 3.67/14.9 = ln x y = 68.507292 x = e3.67/14.9 x = 1.279294 25. An administrative assistant is hired after graduating from high school and learns to type on the job. The number of words he can type per minute, N, is given by 130x + 260 N = --------------, (Young 2nd: 2.6.65) x+5 where x is the number of days he has spent working. Use this equation when answering parts a and b. a. How many words per minute would you expect him to type at the beginning of his first day of work? ____________ 52 [130(0) + 260] / [(0) + 5] = 52 b. How many words per minute would you expect him to type on his last day of work if he worked at this job until he retired? ____________ 130 horizontal asymptote at y = 130/1 = 130 26. Solve the system of equations below for x and y. Use any algebraic method you choose. Please provide exact answers. (4/3)x – (3/2)y = (3/4) (-1/4)x + (1/3)y = (2/3) a. x = ____________ 18  4/3 -3/2 -1/4 1/3 3/4  2/3   1 -9/8 -1/4 1/3 9/16 multiply row 1 by 3/4 2/3  b. y = ____________ 31/2  1 -9/8 9/16  multiply row 1 by 1/4 and add the results to row 2  0 5/96 155/192 1/4 -9/32 9/64 32/96 – 27/96 128/192 + 27/192  1 -9/8 9/16  0 1 31/2  multiply row 2 by 96/5 155/192 x 96/5 = 31 x 1/2 = 29/2  1  0 multiply row 2 by 9/8 and add the results in row 1 0 9/8 279/16 9/16 + 279/16 = 288/16 = 18 0 1 18  31/2  Math 161 Final Exam Version 1 (144 points) Name ___Answers____________________ Precalculus I 2022 May 09 John Scalea Page 8 6 answers: 12 points; #28: 6 points total; 18 points this page; 146 points total 15x + 54 27. Find the partial fraction decomposition of ----------------. ____________ [8 / (x + 10)] + [7 / (x – 2)] x2 + 8x – 20 (self: 8.CT.27) [(15x + 54) / ((x + 10)(x – 2))] = [A / (x + 10)] + [B / (x – 2)] 15x + 54 = A(x – 2) + B(x + 10) 15x + 54 = Ax – 2A + Bx + 10B A = 15 – B 23B = 84 15x + 54 = Ax + Bx – 2A + 10B -2(15 – B) + 10B = 54 B=7 15x + 54 = (A + B)x – 2A + 10B -30 + 2B + 10B = 54 A = 15 – (7) so A + B = 15 and -2A + 10B = 54 12B = 54 + 40 A=8 28. Consider f(x) = x2 and g(x) = (-2x + 14)2 – 7. State the transformations present in g(x) when compared to f(x). Use only the number of answer lines necessary. g(x) = (-2(x – 7))2 – 7 a. ____________ reflection over the y-axis a = 1; b = -2; c = -7; d = -7 b. ____________ horizontal compression 1/2 c. ____________ horizontal shift 7 right d. ____________ vertical shift 7 down e. ____________ f. ____________ Questions 29 through 33 pertain to the matrices below. (self) A -3 -4 9  -3 -9 -6  -1 2 -1  B  7 9  -9 -4  -1 -9  C -6  4 0  2  D  7 9   1 -5  E -8 -4  -4 8  29. Find E + B. no solution 32. Find CD. -42 -54   30 26  30. Find A – D. no solution 33. Find 6E. -48 -24  -24 48  31. Find BE. -92 44   88 4   44 -68  Math 161 Final Exam Version 1 (144 points) Precalculus I John Scalea Name ___Answers____________________ 2022 May 09 Page 9 34. Solve the system of equations below for x and y. Use any algebraic method you choose. Please provide exact answers. (4/3)x – (3/2)y = (3/4) (-1/4)x + (1/3)y = (2/3) a. x = ____________ b. y = ____________ Purchase answer to see full attachment

  
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