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63. = 1 57. Area = 12ฯ€ square units 61. Area = 9ฯ€ square units + 65. x 2 ____ 400 y 2 ____ 144 67. Approximately 51.96 feet Section 10.2 โ€” โ€” โˆ’ โˆ’ 9. Yes x 2 ___ 52 7. Yes = 1 11. segment joining the foci. 1. A hyperbola is the set of points in a plane the difference of whose distances from two fixed points (foci) is a ...
; vertices: (0, 3), (โˆ’6, 3); foci: 2, 3); asymptotes: y = x + 6, y = โˆ’x = 1; vertices: (3, 6), (3, 2); foci: (3, 4 + 2 โˆš โ€” 5 ), 1 __ (x โˆ’ 3) + 4, y = โˆ’ 2 (x โˆ’ 3) + 4 โˆ’ 23. (y + 5)2 _______ 72 foci: (โˆ’1, โˆ’5 + 7 โˆš (x + 1)2 _______ 702 101 ), (โˆ’1, โˆ’5 โˆ’ 7 โˆš โ€” = 1; vertices: (โˆ’1, 2), (โˆ’1, โˆ’ 12); โ€” 101 ); asymptotes: โˆ’ 25. y...
(3, 7.24) Vertex (3, 6) โˆ’32 โˆ’24 โˆ’16 โˆ’8 8 16 24 32 โˆ’8 โˆ’16 โˆ’24 y Focus (3, โˆ’1.24) Vertex (โˆ’1, โˆ’2) 5 Focus (9.1, โˆ’2) 39. โˆ’20 โˆ’15 โˆ’10 โˆ’5 5 10 15 20 โˆ’5 Focus (โˆ’1.1, โˆ’2) โˆ’10 Vertex (9, โˆ’2) y 10 5 41. Vertex (โˆ’4, โˆ’4) โˆ’10 โˆ’5 43. Vertex (2, โˆ’4) 10 5 x Focus (โˆ’9.54, โˆ’4) โˆ’5 โˆ’10 Focus (7.54, โˆ’4) โˆ’16 โˆ’8 = 1 45. 47. โˆ’ y 2 x 2 ___ _...
x โˆ’ 1)2 _______ 0.25 (x โˆ’ 3)2 _______ 4 โˆ’ y 2 ____ 0.75 y 2 ___ 5 = 1 65. x 2 ____ 400 โˆ’ = 1 y 2 ____ 225 y Fountain 24 16 8 โˆ’40 โˆ’32 โˆ’24 โˆ’8โˆ’16 โˆ’8 โˆ’16 โˆ’24 8 16 24 32 40 x Section 10.3 1. A parabola is the set of points in the plane that lie equidistant from a fixed point, the focus, and a fixed line, the directrix. 3. T...
F: ๎€ข 2, _ _ 2 2 16 14 4, 1 ๎€ช ; d: x = (x โˆ’ 5), V: (5, 1); F: ๎€ข _ _ _ 29. (y โˆ’ 1)2 = 3 3 3 16 _ 5 ODD ANSWERS C-40 31. x = โˆ’2 y 8 6 4 2 โˆ’8โˆ’6 โˆ’2โˆ’4 โˆ’2 โˆ’4 โˆ’6 โˆ’8 33. y Focus (2, 0) 2 4 6 8 x 24 16 8 Focus (0, 9) โˆ’32 โˆ’24 โˆ’16 โˆ’8 8 16 24 32 y = โˆ’9 โˆ’8 โˆ’16 โˆ’24 37. โˆ’10 โˆ’7.5 โˆ’5 โˆ’2.5 y 2.5 x = 25 6 2.5 x 5 35. y 7.5 5 2.5 x โˆ’ 5 3 ...
. The conic section is a hyperbola. rotation of the axes in order to eliminate the xy term. 7. AB = 0, parabola 9. AB = โˆ’4 < 0, hyperbola 11. AB = 6 > 0, ellipse 15. B 2 โˆ’ 4AC = 0, parabola 19. 7xโ€ฒ2 + 9yโ€ฒ2 โˆ’ 4 = 0 23. ฮธ = 60ยฐ, 11xโ€ฒ2 โˆ’ yโ€ฒ2 + โˆš 25. ฮธ = 150ยฐ, 21xโ€ฒ2 + 9yโ€ฒ2 + 4xโ€ฒ โˆ’ 4 โˆš 27. ฮธ โ‰ˆ 36.9ยฐ, 125xโ€ฒ2 + 6xโ€ฒ โˆ’ 42yโ€ฒ + 1...
8 โˆ’4 4 8 x โˆ’4 โˆ’8 y (0, โˆ’2) 45. ฮธ = 30ยบ x โˆ’1.4 โˆ’1 โˆ’0.6 1.5 1 0.5 โˆ’0.2 โˆ’0.5 โˆ’1 โˆ’1.5 (0, 1) 0.2 0.6 1 1.4 x (0, โˆ’1) 49. y 8 4 โˆ’4 โˆ’8 โˆ’4 (0, 0) 4 ฮธ = 63ยบ 8 12 16 x 53. ฮธ = 60ยฐ y 3 2 1 x' 1 2 3 4 5 x y' โˆ’2โˆ’3โˆ’4โˆ’5 โˆ’1 โˆ’1 โˆ’2 โˆ’3 57. โˆ’ 59. k = 2 x x x โˆ’2โˆ’3โˆ’4 โˆ’1โˆ’1 โˆ’2 1 2 3 4 โˆ’4 47. ฮธ = 37ยบ y 4 (0, 0) โˆ’8 โˆ’4 4 8 โˆ’4 โˆ’8 51. ฮธ = 45ยฐ y y...
x 2 โˆ’ 4y 2 โˆ’ 30x + 9 = 0 25. 25x 2 โˆ’ 96y 2 โˆ’ 110y โˆ’ 25 = 0 27. 3x 2 + 4y 2 โˆ’ 2x โˆ’ 1 = 0 29. 5x 2 + 9y 2 โˆ’ 24x โˆ’ 36 = 0 31. 23. 64y 2 = 48x + 9 33. y y โ€” โ€” 1. + + x 2 ___ 52 39 ) 3. 39 ), (0, โˆ’ โˆš (x + 3)2 _ 12 y 2 ___ 82 (0, โˆ’8); foci: (0, โˆš = 1; center: (0, 0); vertices: (5, 0), (โˆ’5, 0), (0, 8), (y โˆ’ 2)2 _ 32 2 ), (โˆ’3,...
2 โˆš (y + 3)2 _ 3 ๎€ช 2 ๎€ข 2 โˆš (x โˆ’ 2)2 _______ 22 15. โˆ’ โ€” 11. Approximately 35.71 feet = 1; center: (4, โˆ’1); vertices: (4, 3), โ€” 13 ), (4, โˆ’1 โˆ’ 2 โˆš โ€” 13 ) = 1; center: (2, โˆ’3); vertices: (4, โˆ’3), 2 4 6 8 10 x (0, โˆ’3); foci: (6, โˆ’3), (โˆ’2, โˆ’3) 17. y 19. โˆ’10 โˆ’4โˆ’6โˆ’8 Focus (0, 0) โˆ’2 โˆ’2 โˆ’4 โˆ’6 37. y 3 2 1 Focus (0, 0) Vertex (โˆ’...
r = 1 _ 1 + cos ฮธ 15 _ 4 โˆ’ 3cos ฮธ 59. r = ยฑ 7 __ 8 โˆ’ 28cos ฮธ 3 _ 3 โˆ’ 3cos ฮธ 57. r = ยฑ 2 __ 1 + sin ฮธ cos ฮธ โˆš โ€” 2 __ 4cos ฮธ + 3sin ฮธ (x โˆ’ 5)2 _ 1 1 _ 23. (x + 2)2 = (y โˆ’ 1); 2 (y โˆ’ 7)2 _ โˆ’ = 1 21. 3 9 7 ๎€ช ; directrix: y = vertex: (โˆ’2, 1); focus: ๎€ข โˆ’2, _ _ 8 8 7 25. (x + 5)2 = (y + 2); vertex: (โˆ’5, โˆ’2); focus: ๎€ข โˆ’5, โˆ’ ๎€ช...
Vertex (10, 0) x Chapter 10 practice test = 1; center: (0, 0); vertices: (3, 0), (โˆ’3, 0), (0, 2), โ€” 1. + x 2 ___ 32 y 2 ___ 22 (0, โˆ’2); foci: ( โˆš 5, 0) 3. Center: (3, 2); vertices: (11, 2), (โˆ’5, 2), (3, 8), (3, โˆ’4); foci: 7, 2) (3 + 2 โˆš 5, 0), (โˆ’ โˆš โ€” โ€” โ€” 7, 2), (3 โˆ’ 2 โˆš y 15 10 5 โˆ’15 โˆ’10 โˆ’5 โˆ’5 โˆ’10 โˆ’15 x 5 10 15 5. (x ...
, 2 5 _ units to the and directrix 6 right of the pole. 1 2 3 4 5 6 x โˆ’2โˆ’3โˆ’4โˆ’5โˆ’6 โˆ’1 โˆ’1 โˆ’2 โˆ’3 โˆ’4 โˆ’5 โˆ’6 25. y 0.8 0.4 1 y = 2 ๏ฃซ ๏ฃถ 1 ๏ฃญ0, ๏ฃพ4 Vertex โˆ’1.6 โˆ’1.2 โˆ’0.8 โˆ’0.4 0.4 0.8 1.2 1.6 x Focus (0, 0) โˆ’0.4 โˆ’0.8 โˆ’1.2 โˆ’1.6 โˆ’2 ChapteR 11 Section 11.1 3. Yes, both sets go on 1. A sequence is an ordered list of numbers that can b...
187, 4 4 4 8 2 16 35. a 1 = โˆ’8, an = an โˆ’ 1 + n 891 _ 5 14 _ 5 37. a 1 = 35, an = an โˆ’ 1 + 3 4 _, 2, 10, 12, 5 39. 720 41. 665,280 33. 2, 10, 12, 27 _, 11 31. 1 _ 24, 3 1 2 _ _ _ 43. First four terms: 1,,, 2 3 2 6 24 _ _ 45. First four terms: โˆ’1, 2,, 5 11 ODD ANSWERS 47. an 6 5 4 3 2 1 โˆ’2โˆ’3โˆ’4โˆ’5โˆ’6 โˆ’1 โˆ’1 โˆ’2 โˆ’3 โˆ’4 โˆ’5 โˆ’6 5...
, an = an โˆ’ 1 โˆ’ an โˆ’ 2 71. (n + 2)! _______ = (n โˆ’ 1)! (n + 2) ยท (n + 1) ยท (n) ยท (n โˆ’ 1) ยท... ยท 3 ยท 2 ยท 1 ___________________________________ (n โˆ’ 1) ยท... ยท 3 ยท 2 ยท 1 = n(n + 1)(n + 2) = n 3 + 3n 2 + 2n Section 11.2 1. A sequence where each successive term of the sequence 3. We find increases (or decreases) by a consta...
. 49. an = 13.1 + 2.7n 47. an = 1.8n 13 ___ 12 59. an 10 5 โˆ’0.5 โˆ’5 โˆ’10 โˆ’15 โˆ’20 โˆ’25 โˆ’30 โˆ’35 65. (1, 9) (2, โˆ’1) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 (3, โˆ’11) n (4, โˆ’21) (5, โˆ’31) an 10 2โˆ’3โˆ’4โˆ’5โˆ’6 โˆ’1 โˆ’1 (4, 7) (4, 7.5) (3, 6.5) (2, 6) (1, 5.5) 1 2 3 4 5 6 n C-43 61. 1, 4, 7, 10, 13, 16, 19 63. an 14 13 12 11 10 2โˆ’3โˆ’4โˆ’5โˆ’6 โˆ’1 โˆ’1...
, 5 1 _ 125 1 _, 25 16 _ 27 19. a 4 = โˆ’ 21. a 7 = โˆ’ 1 _ 25. a = โˆ’32, an = an โˆ’ 1 2 3 1 _ _, an = 29. a 1 = an โˆ’ 1 5 6 3 3 _ _ 33. 12, โˆ’6, 3, โˆ’, 4 2 17. 800, 400, 200, 100, 50 23. 7, 1.4, 0.28, 0.056, 0.0112 2 _ 729 27. a 1 = 10, an = โˆ’0.3 an โˆ’ 1 31. a 1 =, an = โˆ’4an โˆ’ 1 1 _ 512 n โˆ’ 1 35. an = 3n โˆ’ 1 4 __ ๎€ช 39. an = โˆ’ ๎€ข...
15. S 13 = 57.2 k = 1 7 8 โˆ™ 0.5k โˆ’ 1 k = 1 5 9. โˆ‘ 4 17 __ = 1 _ 1 โˆ’ 3 k = 1 5 19. S 5 = 121 _ 9 โ‰ˆ 13.44 21. S 11 = 64(1 โˆ’ 0.211) __ = 1 โˆ’ 0.2 23. The series is defined. S = 25. The series is defined. S = โ‰ˆ 80 781,249,984 __ 9,765,625 2 _ 1 โˆ’ 0.8 โˆ’1 __________ 1 โˆ’ ๎€ข โˆ’ 1 ๎€ช __ 2 27 2000 1750 1500 1250 1000 750 500 250 10...
= 495 29. 29 = 512 31. = 6,720 8! _ 3! 27. 212 = 4,096 12! _ 3!2!3!4! 33. 35. 9 37. Yes, for the trivial cases r = 0 and r = 1. If r = 0, then C(n, r) = P(n, r) = 1. If r = 1, then r = 1, C(n, r) = P(n, r) = n. 6! ___ 2! 39. ร— 4! = 8,640 41 43. 4 ร— 2 ร— 5 = 40 45. 4 ร— 12 ร— 3 = 144 49. C(10, 3) ร— C(6, 5) ร— C(5, 2) = 7,2...
b2 15. 27a3 + 54a2b + 36ab2 + 8b3 24 ____ x2y2 8 ___ x3y 1 ___ x4 21. + + + + โ€” 37. 1,082,565a 3b 16 39. 41. f2(x) = x 4 + 12x 3 43. f4(x) = x 4 + 12x 3 + 54x 2 + 108x 31. โˆ’720x 2y 3 35. 35x 3y 4 1152y2 _ x7 7 6 5 4 3 2 1 โˆ’2โˆ’3โˆ’4โˆ’5 โˆ’1 โˆ’1 โˆ’2 โˆ’3 โˆ’4 โˆ’5 45. 590,625x 5y 2 47. k โˆ’ 1 y y f2(x) f4(x) x 1 2 90 80 70 60 50 40 30 ...
, 2) 5 (4, 2) 6 (5, 2) 7 (6, 2) 8 3 (1, 3) 4 (2, 3) 5 (3, 3) 6 (4, 3) 7 (5, 3) 8 (6, 3) 9 4 (1, 4) 5 (2, 4) 6 (3, 4) 7 (4, 4) 8 (5, 4) 9 (6, 4) 10 5 (1, 5) 6 (2, 5) 7 (3, 5) 8 (4, 5) 9 (5, 5) 10 (6, 5) 11 6 (1, 6) 7 (2, 6) 8 (3, 6) 9 (4, 6) 10 (5, 6) 11 (6, 6) 12 1 2 3 4 5 6 35. 5 ___ 12 21 ___ 26 37. 0. 4 __ 39. 9 45....
35. P(18, 4) = 73,440 1 1 ๎€ช โˆ™ ๎€ข _ _ 19. an = โˆ’ 5 3 25. S9 โ‰ˆ 23.95 15. 4, 16, 64, 256, 1024 n โˆ’ 1 37. C(15, 6) = 5,005 33. 104 = 10,000 23. S 11 = 110 29. $5,617.61 27. S = 31. 6 m = 0 5 39. 250 = 1.13 ร— 1015 41. = 3,360 43. 490,314 8! ____ 3!2! 45. 131,072a 17 + 1,114,112a 16b + 4,456,448a 15b 2 47, 1 2, 1 3, 1 4, 1 5...
26, 4) ______________ C(40, 7) โ‰ˆ 29.2% 23. 429x 14 _ 16 19. C(15, 3) = 455 4 _ 25. 7 5 _ 27. 7 ChapteR 12 9. Does not exist 7. 2 15. Answers will vary Section 12.1 1. The value of the function, the output, at x = a is f (a). When the lim f (x) is taken, the values of x get infinitely close to a but x โ†’ a never equal a....
.49999975 0.01 0.49997500 0.1 0.49750208 2sin x _ 4tan x โ†“ 1 _ 2 1 _ 2 โ†‘ 2sin x _ 4tan x x โ†’ 0+ lim โˆ’ 1 ___ x 2 e 43. lim e x โ†’ 0 = 1.0 45. lim x โ†’ โˆ’1โˆ’ |x + 1| _______ x + 1 = โˆ’(x + 1) ________ (x + 1) = โˆ’1 and lim x โ†’ โˆ’1+ |x + 1| _______ = x + 1 (x + 1) _______ (x + 1) = 1 since the right-hand limit does not equal the...
(x + h) cos (x + h) โˆ’ cos(x) __ 47. 2x + h + 4 h x 2 + 5x + 6 โˆ’1 __________ __ 55. f (x โˆš 49. 53. โ€” 57. Does not exist 59. 32 Section 12.3 1. Informally, if a function is continuous at x = c, then there is no break in the graph of the function at f (c), and f (c) is defined. 3. Discontinuous at a = โˆ’3; f (โˆ’3) does not ...
2, but not 3 limit does not exist. 47. x 3 + 6x 2 โˆ’ 7x _____________ (x + 7)(x โˆ’ 1) 49. The function is discontinuous at x = 1 because the limit as x approaches 1 is 5 and f (1) = 2. Section 12.4 1. The slope of a linear function stays the same. The derivative of a general function varies according to x. Both the slop...
p.m., the rate of change of the number of gallons in the tank is โˆ’20 gallons per minute. That is, the tank is losing 20 gallons per minute. noon, the volume of gallons in the tank is changing at the rate of 51. The height of the projectile after 30 gallons per minute. 53. The height of the projectile at t = 3 2 second...
โˆ’ 11. 1 15. f '(x) = โˆ’ 3 ____ 3 _ 2 a 2 13. Removable discontinuity at x = 3 17. Discontinuous at โˆ’2, 0, not differentiable at โˆ’2, 0, 2 19. Not differentiable at x = 0 (no limit) 21. The height of the projectile at t = 2 seconds 23. The average velocity from t = 1 to t = 2 1 _ 27. 0 29. 2 31. x = 1 33. y = โˆ’14x โˆ’ 18 2...
, 317, 880, 902, 903 B binomial 259, 1008 binomial coefficient 992 binomial expansion 993, 995, 1008 Binomial Theorem 993, 994, 1008 break-even point 769, 854 C cardioid 686, 747 carrying capacity 408, 429 Cartesian equation 676 Celsius 100 center of a hyperbola 880, 931 center of an ellipse 865, 931 central rectangle ...
static force 41 elimination 788 ellipse 721, 788, 865, 866, 867, 869, 872, 896, 923, 927, 931 ellipsis 938 end behavior 226, 287, 317 endpoint 40, 440 entry 805, 854 equation 8 Euler 697 even function 75, 113, 477, 561 even-odd identities 561, 563, 634 event 999, 1008 experiment 999, 1008 explicit formula 939, 955, 964...
, 897, 900, 902, 922, 927, 928 focus (of an ellipse) 931 D-1 D-2 focus (of a parabola) 931 formula 8 function 2, 31, 113 function notation 4 Fundamental Counting Principle 984, 1008 Fundamental Theorem of Algebra 271, 272, 317 G Gauss 697, 774, 816 Gaussian elimination 774, 819, 854 general form 209 general form of a q...
305 inverse of a rational function 307 inverse sine function 542, 552 inverse tangent function 542, 552 inverse trigonometric functions 541, 542, 544, 548 inverse variation 312, 317 invertible function 301, 317 invertible matrix 829, 843 J K Kronecker 697 L latus rectum 897, 902, 931 Law of Cosines 659, 747 Law of Sin...
344, 356, 381, 387 one-to-one function 12, 103, 113, 541 opposite side 486, 498 ordered pair 2, 23 ordered triple 774 order of magnitude 402, 429 origin 90 outcomes 999, 1008 output 2, 113 P parabola 208, 214, 720, 791, 896, 901, 903, 922, 925, 931 parallel lines 151, 152, 187 parallelograms 733 parameter 708, 747 par...
634 profit function 769, 854 properties of determinants 849 properties of limits 1029, 1070 Proxima Centauri 402 Pythagoras 697 Pythagorean identities 560, 563, 634 Pythagorean Identity 460, 461, 480, 498, 571 Pythagorean Theorem 585, 612, 658, 723 Q quadrantal angle 442, 498 quadratic 799, 801 quadratic equation 607 ...
titution method 761, 854 sum and difference formulas for cosine 572 sum and difference formulas for sine 573 sum and difference formulas for tangent 575 summation notation 969, 970, 1009 sum-to-product formula 599, 634 surface area 299 symmetry test 682 synthetic division 261, 270, 318 system of equations 817, 818, 820...
233). Use logarithmic functions to model and solve real-life problems (p. 235). 3 In Exercises 1โ€“6, evaluate the function at the indicated Review Exercises 3.1 value of Round your result to three decimal places. Section 3.3 Use the change-of-base formula to rewrite and evaluate logarithmic expressions (p. 239). Use pr...
3 โˆ’2 โˆ’1 โˆ’3 โˆ’2 1 3 2 1 2 3 x In Exercises 27โ€“30, evaluate the function given by at the indicated value of decimal places. fx ex Round your result to three x. 137โ€“142 27. 29. x 8 x 1.7 143โ€“148 28. 30. x 5 8 x 0.278 In Exercises 31โ€“34, use a graphing utility to construct a table of values for the function. Then sketch the...
to graph the function. (b) Find the value of the car 2 years after it was purchased. (c) According to the model, when does the car depreciate most rapidly? Is this realistic? Explain. 3 Chapter Test Chapter Test 275 Take this test as you would take a test in class. When you are finished, check your work against the an...
3, 4 and F E A T U R E S 15. log2 3a4 x 16. 17. In Exercises 2โ€“ 4, graph the equation without using a graphing utility. log ln 7x2 yz3 5x 6 In Exercises 18โ€“20, condense the expression to the logarithm of a single quantity. 2. x 3y 12 0 3. y x 2 9 4. y 4 x 18. 20. log3 13 log3 y 2 ln x lnx 5 3 ln y 5. Find an equation ...
Mathematics) f x 2x2, (a) Construct a table of values. Then sketch the graph of the model. gx x 6 and (b) โ‰ค x โ‰ค 6, f g 1 4 (in centimeters) of a child based is the f x x 2, where g f. 11. 12. H gx x Find the domain of each composite (b) Use the graph from part (a) to estimate the height of a four-year-old child. Then ...
below. There are several different proof methods, which you will see in later chapters. The Midpoint Formula The midpoint of the line segment joining the points given by the Midpoint Formula (p. ) x1, y1 and x2, y2 is Midpoint x1 x2 2 y1, y2 2. Proof Using the figure, you must show that y (x1, y1) d1 d2 and d1 d2 d3. ...
), create two relations: one mapping numbers onto letters, and the other mapping letters onto numbers. Are both relations functions? Explain. y (x, y) 12 ft FIGURE FOR 6 8 ft x 7. At 2:00 P.M. on April 11, 1912, the Titanic left Cobh, Ireland, on her voyage to New York City. At 11:40 P.M. on April 14, the Titanic struc...
change seem to be approaching one value? If so, what value? (g) Find the equations of the secant lines through the points x1, fx1 and x2, fx2 for parts (a)โ€“(e). (h) Find the equation of the line through the point 1, f1 using your answer from part (f ) as the slope of the line. gx x 6. f x 4x and (a) Find 9. Consider t...
x โˆ’2 โˆ’3 12. Let f x 1 1 x. (a) What are the domain and range of f? (b) Find (c) Find f f x. f f f x. What is the domain of this function? Is the graph a line? Why or why not? 14. Consider the graph of the function shown in the figure. Use this graph to sketch the graph of each function. To print an enlarged copy of th...
with online courses and content. By pairing the widely recognized tools of Blackboardยฎ with quality, text-specific content from Houghton Mifflin Company, Eduspaceยฎ makes it easy for instructors to create all or part of a course online. This online learning tool also contains ready-to-use homework exercises, quizzes, t...
. Live Tutorial Help provides real-time, one-on-one instruction. Question Submission allows students to submit questions to the tutor outside the scheduled hours and receive a reply usually within 24 hours. Independent Study Resources connects students around-the-clock to additional educational resources, ranging from ...
real numbers by points in a plane called the rectangular coordinate system, or the Cartesian plane, named after the French mathematician Renรฉ Descartes (1596โ€“1650). The Cartesian plane is formed by using two real number lines intersecting at right angles, as shown in Figure 1.1. The horizontal real number line is usua...
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 475 577 521 569 609 562 707 723 718 648 495 476 527 464 Section 1.1 Rectangular Coordinates 3 The beauty of a rectangular coordinate system is that it allows you to see relationships between two variables. It would be difficult to overestimate the i...
hypotenuse of length and sides of lengths and as shown in Figure 1.5. (The converse is also true. you have then the triangle is a right triangle.) That is, if a 2 b2 c 2, a 2 b2 c 2, a c b, d Suppose you want to determine the distance between two points x2, y2 y2 x1, y1 in the plane. With these two points, a right tri...
qxd 12/7/05 8:29 AM Page 5, 7) d1 = 45 d3 = 50 (2, 1) d2 = 5 (4, 0) x Section 1.1 Rectangular Coordinates 5 Example 4 Verifying a Right Triangle Show that the points 2, 1, 4, 0, and 5, 7 are vertices of a right triangle. Solution The three points are plotted in Figure 1.8. Using the Distance Formula, you can find the l...
quarterback for Louisiana State University threw a pass from the 28-yard line, 40 yards from the sideline. The pass was caught by a wide receiver on the 5-yard line, 20 yards from the same sideline, as shown in Figure 1.10. How long was the pass? Solution You can find the length of the pass by finding the distance bet...
of points in a coordinate plane. One type of transformation, a translation, is illustrated in Example 8. Other types include reflections, rotations, and stretches. Section 1.1 Rectangular Coordinates 7 Example 8 Translating Points in the Plane 2, 3. The triangle in Figure 1.12 has vertices at the points Shift the tria...
. To find the height of the V 200 Then, using h 200 42 200 16 and r 4, find the height. Substitute 200 for V and 4 for r. Simplify denominator. 3.98 Use a calculator. Because the value of was rounded in the solution, a check of the solution will not result in an equality. If the solution is valid, the expressions on ea...
: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1 and 2, approximate the coordinates of the points. In Exercises 11โ€“20, determine the quadrant(s) in which (x, y) is located so that the condition(s) is (are) satisfied. 1. y 6 4 2 D 2. A y 4 2 C โˆ’6 โˆ’4 B โˆ’2 โˆ’2 โˆ’4 x 2 4 C D โˆ’6...
uth, Minnesota for (Source: represents January. x 1 x, y record each month where NOAA) Month, x Temperature 10 11 12 39 39 29 5 17 27 35 32 22 8 23 34 In Exercises 31โ€“40, (a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. 32. 34. 36. 1, 12, 6...
, โˆ’2) 44. Use the result of Exercise 43 to find the coordinates of the endpoint of a line segment if the coordinates of the other endpoint and midpoint are, respectively, (a) 1, 2, 4, 1 and (b) 5, 11, 2, 4. 45. Use the Midpoint Formula three times to find the three and points that divide the line segment joining x2, y2...
) 52. y โˆ’ ( 3, 6, 3) 6 units โˆ’ ( 3, 0) โˆ’ ( 5, 3) 1 3 x 53. Original coordinates of vertices: 7, 4 2, 4, Shift: eight units upward, four units to the right 7, 2, 2, 2, 54. Original coordinates of vertices: 3, 6, Shift: 6 units downward, 10 units to the left 5, 8, 7, 6, 5, 2 Retail Price In Exercises 55 and 56, use the g...
1990 to 1995 and from 1995 to 2004. (c) Use the percent increase from 1995 to 2004 to pre- dict the minimum wage in 2008. (d) Do you believe that your prediction in part (c) is reasonable? Explain. 61. Sales The Coca-Cola Company had sales of $18,546 million in 1996 and $21,900 million in 2004. Use the Midpoint Formul...
Service) y x Year, x Pieces of mail, y 1996 1997 1998 1999 2000 2001 2002 2003 183 191 197 202 208 207 203 202 (a) Sketch a scatter plot of the data. (b) Find the entrance exam score of any student with a final exam score in the 80s. (c) Does a higher entrance exam score imply a higher final exam score? Explain. 63. V...
deterA2, 3, mine if the set of points and the set of points C6, 3 are collinear. A8, 3, C2, 1 B5, 2, B2, 6, (a) For each set of points, use the Distance Formula to find B, to from What relationship exists among these distances for and from to to C, A A B the distances from C. each set of points? (b) Plot each set of p...
0 3x 8 โ‰ฅ 1 86. 2 88. 2x 15 โ‰ฅ 11 10x 7 333202_0102.qxd 12/7/05 8:31 AM Page 14 14 Chapter 1 Function and Their Graphs 1.2 Graphs of Equations What you should learn โ€ข Sketch graphs of equations. โ€ข Find x- and y-intercepts of graphs of equations. โ€ข Use symmetry to sketch graphs of equations. โ€ข Find equations of and sketc...
-plotting method. The basic technique used for sketching the graph of an equation is the Sketching the Graph of an Equation by Point Plotting 1. If possible, rewrite the equation so that one of the variables is isolated on one side of the equation. 2. Make a table of values showing several solution points. 3. Plot thes...
2 1 1 1, 1 0 2 0, 2 1 1 1, 1 2 3 2 2, 2 7 3, 7 Next, plot the points given in the table, as shown in Figure 1.16. Finally, connect the points with a smooth curve, as shown in Figure 1.17. y 6 4 2 (โˆ’2, 2) โˆ’4 โˆ’2 (โˆ’1, โˆ’1) FIGURE 1.16 (3, 7) (2, 2) 2 (1, โˆ’1) 4 (0, โˆ’2) x (โˆ’2, 2) โˆ’4 โˆ’2 (โˆ’1, โˆ’1) FIGURE 1.17 y 6 4 2 (3, 7) y ...
ordered pair Some texts denote the x [and the y-intercept as the -intercept as the -coordinate of the point y -coordinate of the point ] rather than the point itself. Unless it is necessary to make a distinction, we will use the term intercept to mean either the point or the coordinate. -intercept can be written as th...
Knowing the symmetry of a graph before attempting to sketch it is helpful, because then you need only half as many solution points to sketch the graph. There are three basic types of symmetry, described as follows. Graphical Tests for Symmetry 1. A graph is symmetric with respect to the x-axis if, whenever is also on ...
and then reflect them to obtain the graph, as shown in Figure 1.23. x y2 1 is equivalent to x y2 1 x x x y 0 1 2 x y2 1 1 2 5 x, y 1, 0 2, 1 5, 2 Now try Exercise 37. Example 7 Sketching the Graph of an Equation Sketch the graph of y x 1. Solution This equation fails all three tests for symmetry and consequently its g...
r. FIGURE 1.25 By squaring each side of this equation, you obtain the standard form of the equation of a circle. Standard Form of the Equation of a Circle lies on the circle of radius r and center The point x, y (h, k) if and only if x h2 y k2 r 2. To find the correct h and k, from the equation of the circle in Exampl...
073x 2 6.99x 289.0, 62 โ‰ค x โ‰ค 76 is the manโ€™s height (in inches). x where Company) (Source: Metropolitan Life Insurance Height, x Weight, y 62 64 66 68 70 72 74 76 136.2 140.6 145.6 151.2 157.4 164.2 171.5 179.4 a. Construct a table of values that shows the median recommended weights for men with heights of 62, 64, 66, ...
radius ________. 6. When you construct and use a table to solve a problem, you are using a ________ approach. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1โ€“ 4, determine whether each point lies on the graph of the equation. 8. y 5 x2 Equation...
2 y 0 y x3 y x x2 1 xy 2 10 0 26. 28. 30. x y 2 0 y x4 x2 3 y 1 x2 1 32. xy 4 In Exercises 33โ€“ 44, use symmetry to sketch the graph of the equation. 33. 35. 37. 39. 41. 43. y 3x 1 y x 2 2x y x3 34. 36. 38. 40. 42. 44. y 2x 3 y x 2 2x y x3 In Exercises 45โ€“ 56, use a graphing utility to graph the equation. Use a standard...
73. Geometry A regulation NFL playing field (including the has a perimeter of and width x y end zones) of length 1040 3 yards. 346 2 3 or (a) Draw a rectangle that gives a visual representation of the problem. Use the specified variables to label the sides of the rectangle. (b) Show that the width of the rectangle is ...
30 40 50 60 70 80 90 100 x y x y (b) Use the table of values in part (a) to sketch a graph of the model. Then use your graph to estimate the resistance when x 85.5. (c) Use the model to confirm algebraically the estimate you found in part (b). (d) What can you conclude in general about the relationship between the dia...
. 85. 87. 18x 2x 70 7x 6t 2 84. 4x5 55 20 3 3y 86. 88. 333202_0103.qxd 12/7/05 8:33 AM Page 25 1.3 Linear Equations in Two Variables Section 1.3 Linear Equations in Two Variables 25 What you should learn โ€ข Use slope to graph linear equations in two variables. โ€ข Find slopes of lines. โ€ข Write linear equations in two vari...
slope and the โ€œrateโ€ at which the line rises or falls? where m 0.5, Which line falls most quickly? Use a square setting to and 4. Which line rises most quickly? Now, let 4. y mx, 333202_0103.qxd 12/7/05 8:33 AM Page 26 Chapter 1 Functions and Their Graphs 26 y 5 4 3 2 1 (3, 5) x = 3 (3, 1) 1 2 4 5 x FIGURE 1.30 Slope ...
.qxd 12/7/05 8:33 AM Page 27 Section 1.3 Linear Equations in Two Variables 27 Finding the Slope of a Line y y 2 y 1 (x 1, y 1) (x 2, y 2) y2 โˆ’ y1 x 2 โˆ’ x1 x1 x2 x FIGURE 1.34 Given an equation of a line, you can find its slope by writing the equation in slope-intercept form. If you are not given an equation, you can st...
slope of the line passing through each pair of points. a. c. 2, 0 0, 4 and 3, 1 and 1, 1 b. d. 1, 2 3, 4 and and 3, 1 2, 2 Solution a. Letting x1, y1 2, 0 and x2, y2 3, 1, you obtain a slope of m y2 x2 y1 x1 1 0 3 2 1 5. See Figure 1.35. b. The slope of the line passing through 1, 2 and 2, 2 is m 2 2 2 1 0 3 0. See Fi...
Form of the Equation of a Line The equation of the line with slope mx x1 y y1. m passing through the point x1, y1 is The point-slope form is most useful for finding the equation of a line. You should remember this form. โˆ’2 โˆ’1 y 1 โˆ’1 โˆ’2 โˆ’3 โˆ’4 โˆ’5 FIGURE 1.39 y = 3x โˆ’ 5 Example 3 Using the Point-Slope Form Find the slope...
pendicular Lines 1. Two distinct nonvertical lines are parallel if and only if their slopes are equal. That is, m1 m2. 2. Two nonvertical lines are perpendicular if and only if their slopes are negative reciprocals of each other. That is, m1 1m2. d2 (1, m2) Example 4 Finding Parallel and Perpendicular Lines Find the sl...
Multiply each side by 2. Distributive Property Write in slope-intercept form. Notice in Example 4 how the slope-intercept form is used to obtain information about the graph of a line, whereas the point-slope form is used to write the equation of a line. 333202_0103.qxd 12/7/05 8:33 AM Page 31 Section 1.3 Linear Equati...
10,000 9,000 8,000 7,000 6,000 5,000 4,000 3,000 2,000 1,000 x 50 100 150 Number of units FIGURE 1.42 Production cost Now try Exercise 101. 333202_0103.qxd 12/7/05 8:33 AM Page 32 32 Chapter 1 Functions and Their Graphs Most business expenses can be deducted in the same year they occur. One exception is the cost of pr...
a disguised form. Note how the data points are described in Example 7. 333202_0103.qxd 12/7/05 8:33 AM Page 33 Section 1.3 Linear Equations in Two Variables 33 Example 8 Predicting Sales per Share The sales per share for Starbucks Corporation were $6.97 in 2001 and $8.47 in 2002. Using only this information, write a l...
Vertical line: 3. Horizontal line: 4. Slope-intercept form: 5. Point-slope form: 6. Two-point form: y y1 Ax By C 0 x a y b y mx b y y1 mx x1 y1 x1 y2 x2 x x1 333202_0103.qxd 12/7/05 8:33 AM Page 34 34 Chapter 1 Functions and Their Graphs 1.3 Exercises VOCABULARY CHECK: In Exercises 1โ€“6, fill in the blanks. 1. The simp...
2 3 (d) Undefined (c) In Exercises 5โ€“8, estimate the slope of the line. 5. y 6 In Exercises 9โ€“20, find the slope and ble) of the equation of the line. Sketch the line. y -intercept (if possi- 9. 11. 13. 15. 17. 19. 2x 4 y 5x 3 y 1 5x 2 0 7x 6y 30 y 3 0 x 5 0 10. 12. 14. 16. 18. 20. 2x 6 y x 10 y 3 3y 5 0 2x 3y 9 y 4 0...
43. 44. 45. 46. 47. 48. 49. 50. Slope is undefined. is undefined In Exercises 51โ€“ 64, find the slope-intercept form of the equation of the line passing through the points. Sketch the line. 52. 54. 56. 58. (4, 3), (4, 4) 1, 4, 6, 4 1, 1, 6, 2 3, 4 3, 7 4, 3 3 2 4 51. 53. 55. 57. 59. 60. 61. 62. 63. 64. 5 2 4 5, 1, 5, 5...
-intercept: 83. Point on line: 2, 0 0, 3 1 6, 0 0, 2 3 1, 2 80. x -intercept: y -intercept: 82. x -intercept: y -intercept: 3, 0 0, 4 2 3, 0 0, 2 84. Point on line: 3, 4 x -intercept: y -intercept: c, 0 0, c, c 0 x -intercept: y -intercept: d, 0 0, d, d 0 Graphical Interpretation In Exercises 85โ€“88, identify any relat...
interpret any change in daily revenues for a one-day increase in time. in terms of time the slopes of x (a) The line has a slope of (b) The line has a slope of (c) The line has a slope of m 400. m 100. m 0. 95. Average Salary The graph shows the average salaries for senior high school principals from 1990 through 2002...
the data. (b) Use a straightedge to sketch the line that you think best fits the data. (c) Find an equation for the line you sketched in part (b). (d) Interpret the meaning of the slope of the line in part (c) in the context of the problem. (e) The surveyor needs to put up a road sign that indicates the steepness of t...
(Source: The Timberland Co.) 106. Number of Stores In 1999 there were 4076 J.C. Penney stores and in 2003 there were 1078 stores. Write a linear equation that gives the number of stores in terms of the year. Let represent 1999. Then predict the numbers of stores for the years 2008 and 2010. Are your answers reasonable...
Wage A microchip manufacturer pays its assembly line workers $11.50 per hour. In addition, workers receive a piecework rate of $0.75 per unit W produced. Write a linear equation for the hourly wage in terms of the number of units produced per hour. x 113. Cost, Revenue, and Profit A roofing contractor purchases a shin...
determine the increase in its perimeter. 116. Monthly Salary A pharmaceutical salesperson receives a monthly salary of $2500 plus a commission of 7% of sales. Write a linear equation for the salespersonโ€™s monthly wage W S. in terms of monthly sales 117. Business Costs A sales representative of a company using a person...
-point quizzes and 100-point exams in an algebra course. Average scores for six students, given as data y is the points 19, 96, average test score, are 16, 79, 13, 76, [Note: There are many correct answers for parts (b)โ€“(d).] is the average quiz score and 15, 82. 18, 87, 10, 55, where x, y and x (a) Sketch a scatter pl...
(b) y 6 4 2 โˆ’2 โˆ’6 โˆ’4 6 4 2 โˆ’2 x 2 x 2 โˆ’6 โˆ’4 (c) y (d) y 12 8 4 โˆ’4 โˆ’4 x 4 8 12 8 4 โˆ’4 โˆ’4 x 4 8 12 129. 130. 131. 132. y 8 3x y 8 x y 1 2 x 2 2x 1 y x 2 1 In Exercises 133โ€“138, find all the solutions of the equation. Check your solution(s) in the original equation. 133. 134. 135. 136. 137. 138. 9 4x 73 x 14x 1 4 8 2x 7 ...
A contains the range is the To help understand this definition, look at the function that relates the time of day to the temperature in Figure 1.47. Time of day (P.M.) Temperature (in degrees C) 1 2 3 6 4 5 A is the domain. Set Inputs: 1, 2, 3, 4, 5, 6 FIGURE 1.47 13 6 15 9 12 1 4 14 3 7 10 16 2 5 8 11 B contains the ...
FIGURE 1.48 Solution a. This verbal description does describe as a function of Regardless of the is always 2. Such functions are called constant the value of x. x, y y value of functions. b. This table does not describe as a function of The input value 2 is matched x. y with two different -values. y c. The graph in Fi...
Suppose you give this equation function the name โ€œ โ€ Then you can use the following function notation. y 1 x2 f. as a function of describes x. y Input x Output f x Equation f x 1 x2 f x is read as the value of The symbol y corresponds to the -value for a given mind that at For instance, the function given by x. f is t...
0, and 1. f x x2 1, x 1, x < 0 x โ‰ฅ 0 Solution Because x 1 is less than 0, use f x x2 1 to obtain For For use f1 12 1 2. fx x 1 x 0, f0 0 1 1. fx x 1 x 1, use f1 1 1 0. to obtain to obtain Now try Exercise 35. 333202_0104.qxd 12/7/05 8:35 AM Page 44 44 Chapter 1 Functions and Their Graphs Te x2 What is the Use a graphi...
d. This function is defined only for -values for which x 4 x 2 โ‰ฅ 0. By solving this inequality (see Section 2.7), you can conclude that 2 โ‰ค x โ‰ค 2. So, the domain is the interval 2, 2. Now try Exercise 59. In Example 5(c), note that the domain of a function may be implied by the physical context. For instance, from the...
1 Functions and Their Graphs Number of Alternative-Fueled Vehicles in the U.S. Example 8 Alternative-Fueled Vehicles V 500 450 400 350 300 250 200 ( FIGURE 1.51 5 6 7 8 9 10 11 Year (5 โ†” 1995) 12 t V The number (in thousands) of alternative-fueled vehicles in the United States increased in a linear pattern from 1995 t...
2 4x 4h 7 x2 4x 7 h h2x h 4 h 2xh h2 4h h 2x h 4, h 0 Summary of Function Terminology Function: A function is a relationship between two variables such that to each value of the independent variable there corresponds exactly one value of the dependent variable. Function Notation: y f x f y is the name of the function. ...
________, ________, and ________. 3. For an equation that represents as a function of y x, the domain, and the set of all values taken on by the ________ variable the set of all values taken on by the ________ variable is the range. y x is 4. The function given by f x 2x 1, x2 4, x < 0 x โ‰ฅ 0 is an example of a _______...
) (d) c, 0, b, 0, a, 3 (b) 333202_0104.qxd 12/7/05 8:35 AM Page 49 Circulation of Newspapers In Exercises 11 and 12, use the graph, which shows the circulation (in millions) of daily newspapers in the United States. (Source: Editor & Publisher Company) 50 40 30 20 10 Morning Evening ) 1992 1994 1996 1998 2000 2002 Year...