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63. = 1 57. Area = 12π square units 61. Area = 9π square units + 65. x 2 ____ 400 y 2 ____ 144 67. Approximately 51.96 feet Section 10.2 — — − − 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 positive constant. 3. The foci must lie on the transverse axis and be in the interior of 5. The center must be the midpoint of the line the hyperbola. y 2 x 2 _ _ 32 = 1 62 − y 2 ___ = 1; vertices: (5, 0), (−5, 0); 62 y 2 x 2 ___ ___ 52 42 6 6 61, 0), (− √ foci: ( √ 61, 0); asymptotes: y = _ _ x — 85 ), 92 = 1; vertices: (0, 2), (0, − 2); foci: (0, √ _ _ 22 − 2 2 85 ); asymptotes: y = _ _ x, y = − x 9 9 (y − 2)2 _______ = 1; vertices: (4, 2), (−2, 2); foci: (6, 2), 42 4 4 __ __ (x − 1) + 2, y = − (−4, 2); asymptotes: y = 3 3 (y + 7)2 _______ 72 = 1; vertices: (9, −7), (−5, −7); foci: (x − 1)2 _______ 32 (x − 1) + 2 (0, − √ 15. 13. − — 2, −7); asymptotes: y = x − 9, y = −x − 5 — − 17. (x − 2)2 _______ 72 2, −7), (2 − 7 √ (2 + 7 √ (y − 3)2 _______ 32 19. − — — (x + 3)2 _______ 32 (−3 + 3 √ (y − 4)2 _ − 22 (3, 4 −2 √ 21. — — 2, 3), (−3 − 3 √ (x − 3)2 _ 42 1 __ 5 ); asymptotes: y = 2 = 1
; vertices: (0, 3), (−6, 3); foci: 2, 3); asymptotes: y = x + 6, y = −x = 1; vertices: (3, 6), (3, 2); foci: (3, 4 + 2 √ — 5 ), 1 __ (x − 3) + 4, y = − 2 (x − 3) + 4 − 23. (y + 5)2 _______ 72 foci: (−1, −5 + 7 √ (x + 1)2 _______ 702 101 ), (−1, −5 − 7 √ — = 1; vertices: (−1, 2), (−1, − 12); — 101 ); asymptotes: − 25. y = 1 ___ 10 (x + 1) − 5 (x + 1) − 5, y = − 1 ___ 10 (x + 3)2 _______ 52 (−3 + √ 2 2 2 __ __ __ y = − (x − 3) − 4 (x − 3) − 4, y = − (x + 3) + 4 27. y = 5 5 5 2 29, 4); asymptotes: y = __ 5 = 1; vertices: (2, 4), (−8, 4); foci: (y − 4)2 _______ 22 29, 4), (−3 − √ (x + 3) + 4, — — 3 __ 29. y = 4 3 __ (x − 1) + 1, y = − 4 (x − 1) + 1 ODD ANSWERS 31. y Vertex (−7, 0) 10 Vertex (7, 0) Focus (−8.06, 0) −15 −10 −20 −5 33. 5 5 −5 −10 y Focus (8.06, 0) x 10 15 20 10 5 Focus (0, 5.83) Vertex (0, 3) −20 −15 −10 −5 5 x 10 Vertex (0, −3) 20 15 35. −24 −32 Vertex (4, −2) −16 −8 −5 −10 y 16 8 −8 −16 Focus (0, −5.83) Focus (4, 0.83) 8 16 Vertex (4, −8) 24 32 37. Vertex (3, 0) −24 Focus (4, −10.83) y 24 16 8 Focus
(3, 7.24) Vertex (3, 6) −32 −24 −16 −8 8 16 24 32 −8 −16 −24 y Focus (3, −1.24) Vertex (−1, −2) 5 Focus (9.1, −2) 39. −20 −15 −10 −5 5 10 15 20 −5 Focus (−1.1, −2) −10 Vertex (9, −2) y 10 5 41. Vertex (−4, −4) −10 −5 43. Vertex (2, −4) 10 5 x Focus (−9.54, −4) −5 −10 Focus (7.54, −4) −16 −8 = 1 45. 47. − y 2 x 2 ___ ___ 9 16 (x − 6)2 _______ 25 − (y − 1)2 _______ 11 = 1 x x x y 24 16 8 Vertex (5, 15) Focus (5, 15.05) 8 x 16 Focus (5, −5.05) Vertex (5, −5) −8 −16 −24 C-39 49. (x − 4)2 _______ 25 − (y − 2)2 _______ 1 = 1 − = 1 53. y 2 ___ 9 (x + 1)2 _______ 9 — 57. y(x(x) = −3 √ x 2 + 1 y — 10 8 6 4 2 2 4 6 8 10 x −10 −8−6 −2−4 −2 −4 −6 −8 −10 61. − x 2 ___ 25 y 2 ___ 25 = 1 y − 51. y 2 ___ 16 (x + 3)2 _______ 25 − = 1 x 2 ___ 25 (y + 3)2 _______ 25 = 1 55. 59. y(x) = 1 + 2 √ y(x) = 1 − 2 √ — x 2 + 4x + 5 — x 2 + 4x + 5, y 10 8 6 4 2 2 4 6 8 10 x −10 −8−6 −2−4 −2 −4 −6 −8 −10 63. x 2 ____ 100 − y 2 ___ = 1 25 y Fountain 15 10 5 16 8 Fountain −24 −16 −8 8 16 24 x −15 −10 −5 5 10 15 x −5 −10 −15 −8 −16 67. 69. − = 1 (
x − 1)2 _______ 0.25 (x − 3)2 _______ 4 − y 2 ____ 0.75 y 2 ___ 5 = 1 65. x 2 ____ 400 − = 1 y 2 ____ 225 y Fountain 24 16 8 −40 −32 −24 −8−16 −8 −16 −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. The graph will open down. 5. The distance between the focus and directrix will increase. 7. Yes y = 4(1)x 2 1 1 x, V: (0, 0); F:  _ _ 11. y 2 = 32 8 1 y, V: (0, 0); F:  0, − _ 13. x 2 = − 4 1 1 x, V: (0, 0); F:  _ _ 144 36 9. Yes (y − 3)2 = 4(2)(x − 2) 1, 0  ; d: x = − _ 32 1 1  ; d: y = _ _ 16 16 1, 0  ; d: x = − _ 144 15. y 2 = 17. (x − 1)2 = 4(y − 1), V: (1, 1); F: (1, 2); d: y = 0 7 5 19. (y − 4)2 = 2(x + 3), V: (−3, 4); F:  −, 4  ; d: x = − _ _ 2 2 21. (x + 4)2 = 24(y + 1), V: (−4, −1); F: (−4, 5); d: y = −7 23. (y − 3)2 = −12(x + 1), V: (−1, 3); F: (−4, 3); d: x = 2 14 4  ; d: y = − (y + 3), V: (5, −3); F:  5, − _ _ 25. (x − 5)2 = 5 5 9 11  ; d: y = 27. (x − 2)2 = −2(y − 5), V: (2, 5);
F:  2, _ _ 2 2 16 14 4, 1  ; d: x = (x − 5), V: (5, 1); F:  _ _ _ 29. (y − 1)2 = 3 3 3 16 _ 5 ODD ANSWERS C-40 31. x = −2 y 8 6 4 2 −8−6 −2−4 −2 −4 −6 −8 33. y Focus (2, 0) 2 4 6 8 x 24 16 8 Focus (0, 9) −32 −24 −16 −8 8 16 24 32 y = −9 −8 −16 −24 37. −10 −7.5 −5 −2.5 y 2.5 x = 25 6 2.5 x 5 35. y 7.5 5 2.5 x − 5 3 7 3 −10 −7.5 −5 −2.5 x 2.5 39. −2.5 y 5 y = 0 −5 −20 −10 −15 Focus (−4, −2) −5 −10 −15 −20 41. x 5 x = −4 −10 −5 −2.5 −5 −7.5 −10 5 10 15 Focus (0, −5) 25 6 y 10 5 −5 −10 −15 43. y 15 10 5 −5 −10 −15 −5 x = 2 5 10 20 15 Focus (8, −1) x — — 3 )2 = −4 √ 45. x 2 = −16y 47. (y − 2)2 = 4 √ 49. (y + √ 51. x 2 = y 1 __ 53. (y − 2)2 = 4 3 )2 = 4 √ 55. (y − √ 57. y 2 = −8x 59. (y + 1)2 = 12(x + 3) 61. (0, 1) (x + 2) — — 2 (x − 2) — 2 (x − √ — 2 ) 5 (x + √ — 2 ) 63. At the point 2.25 feet above the vertex 67. x 2 = −125(y − 20), height is 7.2 feet 65. 0.5625 feet 69. 0.2304 feet Section 10.4 5. It gives the angle of 1. The xy term causes a rotation of the graph to occur. 3
. The conic section is a hyperbola. rotation of the axes in order to eliminate the xy term. 7. AB = 0, parabola 9. AB = −4 < 0, hyperbola 11. AB = 6 > 0, ellipse 15. B 2 − 4AC = 0, parabola 19. 7x′2 + 9y′2 − 4 = 0 23. θ = 60°, 11x′2 − y′2 + √ 25. θ = 150°, 21x′2 + 9y′2 + 4x′ − 4 √ 27. θ ≈ 36.9°, 125x′2 + 6x′ − 42y′ + 10 = 0 29. θ = 45°, 3x′2 − y′2 − √ 3 x′ + y′ − 4 = 0 3 y′ − 6 = 0 2 y′ + 1 = 0 2 x′ + √ 13. B 2 − 4AC = 0, parabola — — — — 17. B 2 − 4AC = −96 < 0, ellipse 21. 3x′2 + 2x′y′ − 5y′2 + 1 = 0 31. — 2 √ ____ 2 1 __ (x′ + y′ ) = (x′ − y′ )2 2 y 10 5 x −10 −5 5 10 x −5 −10 35. (x′ + y′ )2 ________ − 2 y (x′ − y′ )2 ________ 2 = 1 5 2.5 −7.5 −5 −2.5 2.5 5 7.5 −2.5 −5 39. θ = 45º (3, 3) y 8 4 −8 −4 4 8 x x x −4 −8 (−3, −3) y θ = 30º 43. 4 3 2 1 33. (x′ − y′ )2 ________ 8 + (x′ + y′ )2 ________ 2−3−4−5 −1 −1 −2 −3 37. — √ 3 ____ 2 1  __ x′ + 2 y y′ = 1 __ x′ − 2 — 3 √ y′ − 1  ____ 2 2 10 5 −5 −10 5 10 15 20 x 41. θ = 45º y 8 4 (0, 2) −
8 −4 4 8 x −4 −8 y (0, −2) 45. θ = 30º x −1.4 −1 −0.6 1.5 1 0.5 −0.2 −0.5 −1 −1.5 (0, 1) 0.2 0.6 1 1.4 x (0, −1) 49. y 8 4 −4 −8 −4 (0, 0) 4 θ = 63º 8 12 16 x 53. θ = 60° y 3 2 1 x' 1 2 3 4 5 x y' −2−3−4−5 −1 −1 −2 −3 57. − 59. k = 2 x x x −2−3−4 −1−1 −2 1 2 3 4 −4 47. θ = 37º y 4 (0, 0) −8 −4 4 8 −4 −8 51. θ = 45° y y' 3 2 1 x' −2−3−4−5 −1 −1 −2 −3 55. θ ≈ 36.9° y y' 3 2 1 −2−3−4−5 −1 −1 −2 −3 1 2 3 4 5 x' 1 2 3 4 5 ODD ANSWERS Section 10.5 Chapter 10 Review exercises C-41 11. Parabola 1. If eccentricity is less than 1, it is an ellipse. If eccentricity is equal to 1, it is a parabola. If eccentricity is greater than 1, it is a 3. The directrix will be parallel to the polar axis. hyperbola. 5. One of the foci will be located at the origin. 7. Parabola with 3 _ e = 1 and directrix units below the pole. 4 9. Hyperbola 5 _ with e = 2 and directrix units above the pole. 2 3 _ with e = 1 and directrix 10 units to the right of the pole. 2 _ 13. Ellipse with e = and directrix 2 units to the right of the pole. 7 5 11 _ _ 15. Hyperbola with e = and directrix 5 3 7 8 _ _ 17. Hyperbola with e = units to the right of and directrix 7 8 units above the pole. 19. 25x 2 + 16y 2 − 12y − 4 = 0 the pole. 21. 21
x 2 − 4y 2 − 30x + 9 = 0 25. 25x 2 − 96y 2 − 110y − 25 = 0 27. 3x 2 + 4y 2 − 2x − 1 = 0 29. 5x 2 + 9y 2 − 24x − 36 = 0 31. 23. 64y 2 = 48x + 9 33. y y — — 1. + + x 2 ___ 52 39 ) 3. 39 ), (0, − √ (x + 3)2 _ 12 y 2 ___ 82 (0, −8); foci: (0, √ = 1; center: (0, 0); vertices: (5, 0), (−5, 0), (0, 8), (y − 2)2 _ 32 2 ), (−3, 2); (−2, 2), (−4, 2), (−3, 5), (−3, −1); (−3, 2 + 2 √ (−3, 2 − 2 √ 2 ) 5. Center: (0, 0); vertices: (6, 0), (−6, 0), (0, 3), (0, −3); 3, 0), (−3 √ foci: (3 √ 7. Center: (−2, −2); vertices: (2, −2), (−6, −2), (−2, 6), (−2, −10); foci: (−2, − 2 + 4 √ 3 ) (−2, − 2 − 4 √ 3, 0 ), y 7.5 5 2.5 −10 −7.5 −5 −2.5 −2.5 −5 −7.5 y 15 10 5 2.5 5 7.5 10 x −20 −15 −10 −5 −5 −10 −15 5 10 15 20 x Vertex (−5, 0) 5 2.5 −7.5 −5 −2.5 −2.5 Vertex (−1.67, −2.89) 35. Vertex (−1.67, 2.89) Focus (0, 0) x 5 7.5 Vertex 5 3 y 15 10 5 16 Vertex (0, 10) 14 12 10 8 6 4 2 9. + = 1 y 2 x 2 ___ ___ 25 16 (y + 1)2 (x − 4)2 _ _ − 13. 42 62 (4, −5); foci: (4, −1 +
2 √ (y + 3)2 _ 3  2  2 √ (x − 2)2 _______ 22 15. − — 11. Approximately 35.71 feet = 1; center: (4, −1); vertices: (4, 3), — 13 ), (4, −1 − 2 √ — 13 ) = 1; center: (2, −3); vertices: (4, −3), 2 4 6 8 10 x (0, −3); foci: (6, −3), (−2, −3) 17. y 19. −10 −4−6−8 Focus (0, 0) −2 −2 −4 −6 37. y 3 2 1 Focus (0, 0) Vertex (−1, 8) Focus (−1, 8.28) 20 15 10 5 −2−3−4−5 −1 −1 −2 −3 x 1 2 3 4 5 Vertex (0, −1) y = −2 Vertex (−1, −6) −20 −15 −10 −5 −5 −10 −15 −20 5 10 15 20 x Focus (−1, −6.28) y Focus (0, 9) Vertex (0, 6.46) x 5 10 15 20 Focus (0, −3) Vertex (0, −0.46) −20 −15 −10 20 15 10 5 −5 −5 −10 −15 −20 −20 −15 −10 −5 Focus (0, 0) 15 10 5 x Vertex (−8, 0) −5 −10 39. Focus (0, 0) −16 −14 −12 −10 −8−6 −15 y 14 12 10 8 6 4 2 −2−4 −2 −4 −6 −8 −10 −12 −14 41. Vertex (0, 6) Vertex (−2.68, 2.4) y 7.5 5 2.5 2 4 6 x −7.5 −5 −2.5 −2.5 Vertex (2.68, 2.4) Focus (0, 0) 2.5 5 7.5 43. r = 4 _________ 5 + cos θ x = 5 45. r = 51. r = 4 _ 1 + 2sin θ 12 _ 2 + 3sin θ 47. r = 49. r = 53. r = 55.
r = 1 _ 1 + cos θ 15 _ 4 − 3cos θ 59. r = ± 7 __ 8 − 28cos θ 3 _ 3 − 3cos θ 57. r = ± 2 __ 1 + sin θ cos θ √ — 2 __ 4cos θ + 3sin θ (x − 5)2 _ 1 1 _ 23. (x + 2)2 = (y − 1); 2 (y − 7)2 _ − = 1 21. 3 9 7  ; directrix: y = vertex: (−2, 1); focus:  −2, _ _ 8 8 7 25. (x + 5)2 = (y + 2); vertex: (−5, −2); focus:  −5, −  ; _ 4 9 _ directrix: y = − 4 27. x = −25 8 x y 3 2 1 29. y x = 5 4 10 5 x x 5 10 −2−3−4−5 −1 1 2 3 4 5 −15 −10 −5 1 4 −5 −1 −2 −3 Vertex (−3, 1)  (y − 1) 31. (x − 2)2 =  1 __ 2 33. B 2 − 4AC = 0, parabola 35. B 2 − 4AC = − 31 < 0, ellipse 37. θ = 45°, x′ 2 + 3y′ 2 − 12 = 0 −15 −10 1 2 ODD ANSWERS C-42 39. θ = 45° y (−2, 0) −2−3−4 −1 4 3 2 1 −1 −2 −3 −4 41. Hyperbola with e = 5 and directrix 2 units to the left of the pole. 3 1 _ _ 43. Ellipse with e = and directrix 4 3 unit above the pole. (2, 0) 1 2 3 4 x 45. y Focus (4, 0) 7.5 5 2.5 −10 −7.5 −5 y = −3 −2.5 −2.5 −5 −7.5 2.5 5 7.5 10 Vertex 49. r = 3 _ 1 + cos θ 47. x Focus (0, 0) y 24 16 8 −32 −24 −16 −8 8 16 24 32 −8 −16 −24
Vertex (10, 0) x Chapter 10 practice test = 1; center: (0, 0); vertices: (3, 0), (−3, 0), (0, 2), — 1. + x 2 ___ 32 y 2 ___ 22 (0, −2); foci: ( √ 5, 0) 3. Center: (3, 2); vertices: (11, 2), (−5, 2), (3, 8), (3, −4); foci: 7, 2) (3 + 2 √ 5, 0), (− √ — — — 7, 2), (3 − 2 √ y 15 10 5 −15 −10 −5 −5 −10 −15 x 5 10 15 5. (x − 1)2 _______ 36 + (y −2)2 ______ 27 = 1 7. y 2 x 2 _ _ 92 = 1; center: (0, 0); 72 − — vertices (7, 0), (−7, 0); foci: ( √ 130, 0); 130, 0), (− √ 9 __ asymptotes: y = ± x 7 — 9. Center: (3, −3); vertices: (8, −3), (−2, −3); foci: (3 + √ (3 − √ 1 26, −3); asymptotes: y = ± _ (x − 3) − 3 5 — y Vertex (−2, −3) 5 Center (3, −3) Focus — 26, −3), −20 −15 −10 −5 5 10 15 20 x Focus −5 −10 Vertex (8, −3) − 11. (x − 1)2 (y − 3)2 _______ _______ 8 1 11 ___ (2, −1); focus:  2, − 12 15. y = 1 1 _ 13. (x − 2)2 = (y + 1); vertex: 3  ; directrix: y = − 13 ___ 12 17. Approximately 8.48 feet 19. Parabola; θ ≈ 63.4° Vertex (−3, 4) Focus (−5, 4) −20 −15 −10 −5 15 10 5 −5 −10 x 5 10 x = −1 21. x′ 2 − 4x′ + 3y2, 3 3 _ 23. Hyperbola with e =
, 2 5 _ units to the and directrix 6 right of the pole. 1 2 3 4 5 6 x −2−3−4−5−6 −1 −1 −2 −3 −4 −5 −6 25. y 0.8 0.4 1 y = 2   1 0, 4 Vertex −1.6 −1.2 −0.8 −0.4 0.4 0.8 1.2 1.6 x Focus (0, 0) −0.4 −0.8 −1.2 −1.6 −2 ChapteR 11 Section 11.1 3. Yes, both sets go on 1. A sequence is an ordered list of numbers that can be either finite or infinite in number. When a finite sequence is defined by a formula, its domain is a subset of the non-negative integers. When an infinite sequence is defined by a formula, its domain is all positive or all non-negative integers. indefinitely, so they are both infinite sequences. 5. A factorial is the product of a positive integer and all the positive integers below it. An exclamation point is used to indicate the operation. Answers may vary. An example of the benefit of using factorial notation is when indicating the product It is much easier to write than it is to write out 13 ∙ 12 ∙ 11 ∙ 10 ∙. 16 _ 5 7. First four terms: −8, − 9. First four terms:, − 4, − 16 _ 3 1 _, 2, 2 11. First four terms: 1.25, −5, 20, −80 8 1 _ _, 4 27 9 16 1 4 _ _ _ _ 13. First four terms 17.,,, 7 5 3 19. −0.6, −3, −15, −20, −375, −80, −9375, −320 4 _ −, 4, −20, 100 5, 31, 44, 59 25 _ 11 16 _ 9, 15. First four terms: n − 1 2n _ 2n 2n − 1 _ n or −1, 1, −9, 23. an = 21. an = n 2 + 3 1  25. an =  − _ 2 27. First five terms: 3, −9, 27, −81, 243 29. First five terms: 531,441 9 81 2
187, 4 4 4 8 2 16 35. a 1 = −8, an = an − 1 + n 891 _ 5 14 _ 5 37. a 1 = 35, an = an − 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: −1, 2,, 5 11 ODD ANSWERS 47. an 6 5 4 3 2 1 −2−3−4−5−6 −1 −1 −2 −3 −4 −5 −6 51. an 36 30 24 18 12 6 −2−3−4−5 −1 −6 −12 −18 −24 −30 −36 (1, 0) 1 2 3 4 5 6 n 49. an 6 5 4 3 2 1 −2−3−4−5−6 −1 −1 −2 −3 −4 −5 −6 (1, 2) (2, 1) (4, 1) n 1 2 3 4 5 6 (5, 0) (3, 0) (5, 30) (4, 20) (3, 12) (2, 6) 2 3 4 5 6 7 1 (1, 2) n 53. an = 2n − 2 55. a 1 = 6, an = 2an − 1 − 5 57. First five terms: 3188 716 29 _____ ____ ___,,, 999 333 37 152 ____, 111 13724 ______ 2997 59. First five terms: 2, 3, 5, 17, 65537 61. a 10 = 7,257,600 63. First six terms: 0.042, 0.146, 0.875, 2.385, 4.708 65. First four terms: 5.975, 32.765, 185.743, 1057.25, 6023.521 67. If an = −421 is a term in the sequence, then solving the equation −421 = −6 − 8n for n will yield a non-negative integer. However, if −421 = −6 − 8n, then n = 51.875 so an = −421 is not a term in the sequence. 69. a 1 = 1, a 2 = 0
, an = an − 1 − an − 2 71. (n + 2)! _______ = (n − 1)! (n + 2) · (n + 1) · (n) · (n − 1) ·... · 3 · 2 · 1 ___________________________________ (n − 1) ·... · 3 · 2 · 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 constant value. whether the difference between all consecutive terms is the same. This is the same as saying that the sequence has a common 5. Both arithmetic sequences and linear functions difference. have a constant rate of change. They are different because their domains are not the same; linear functions are defined for all real numbers, and arithmetic sequences are defined for natural numbers or a subset of the natural numbers. 7. The common 21. a 1 = 5 9. The sequence is not arithmetic because 1 _ difference is 2 8 4 2 _ _ _ 13. 0, −5, −10, −15, −20 16 − 4 ≠ 64 − 16. 11. 0,, 2,, 3 3 3 19. a 1 = 2 17. a 6 = 41 15. a 4 = 19 23. a 1 = 6 25. a 21 = −13.5 27. −19, −20.4, −21.8, −23.2, −24.6 29. a 1 = 17; an = an − 1 + 9; n ≥ 2 31. a 1 = 12; an = an − 1 + 5; n ≥ 2 1 1 __ __ 33. a 1 = 8.9; an = an − 1 + 1.4; n ≥ 2 35. a 1 = ; n ≥ 2 ; an = an − 1 + 5 4 1 __ ; n ≥ 2 39. a 1 = 4; an = an − 1 + 7; a 14 = 95 ; an = an − 1 − 37. a 1 = 6 43. an = 1 + 2n 41. First five terms: 20, 16, 12, 8, 4 45. an = −105 + 100n 1 1 _ _ 51. an = n − 3 3 55. There are 6 terms in the sequence. 57. The graph does not represent an arithmetic sequence. 53. There are 10 terms in the sequence
. 49. an = 13.1 + 2.7n 47. an = 1.8n 13 ___ 12 59. an 10 5 −0.5 −5 −10 −15 −20 −25 −30 −35 65. (1, 9) (2, −1) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 (3, −11) n (4, −21) (5, −31) an 10 2−3−4−5−6 −1 −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 2−3−4−5−6 −1 −1 (5, 13) (4, 10) (3, 7) (2, 4) (1, 1) 1 2 3 4 5 6 n 67. Answers will vary. Examples: an = 20.6n and an = 2 + 20.4n 69. a 11 = −17a + 38b 71. The sequence begins to have negative values at the 13th 1 _ term, a 13 = − 3 sequence is arithmetic. Example: recursive formula: a 1 = 3, an = an − 1 − 3. First 4 terms: 3, 0, −3, −6; a 31 = −87 73. Answers will vary. Check to see that the Section 11.3 1. A sequence in which the ratio between any two consecutive 3. Divide each term in a sequence by terms is constant. the preceding term. If the resulting quotients are equal, then 5. Both geometric sequences the sequence is geometric. and exponential functions have a constant ratio. However, their domains are not the same. Exponential functions are defined for all real numbers, and geometric sequences are defined only for positive integers. Another difference is that the base of a geometric sequence (the common ratio) can be negative, but the base of an 7. The common ratio exponential function must be positive. is −2 1 _ 11. The sequence is geometric. The common ratio is −. 2 13. The sequence is geometric. The common ratio is 5. 9. The sequence is geometric. The common ratio is 2. 1 _ 15. 5, 1,
, 5 1 _ 125 1 _, 25 16 _ 27 19. a 4 = − 21. a 7 = − 1 _ 25. a = −32, an = an − 1 2 3 1 _ _, an = 29. a 1 = an − 1 5 6 3 3 _ _ 33. 12, −6, 3, −, 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 = −0.3 an − 1 31. a 1 =, an = −4an − 1 1 _ 512 n − 1 35. an = 3n − 1 4 __  39. an = −  5 1 ________ 43. a 12 = 177, 147 37. an = 0.8 ∙ (−5)n − 1 1 __ 41. an = 3 ∙  −  3 45. There are 12 terms in the sequence. 47. The graph does not represent a geometric sequence. 49. n − 1 an 60 48 36 24 12 −0.5 −1 (5, 48) (4, 24) (3, 12) (1, 3) (2, 6) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n 51. Answers will vary. Examples: a 1 = 800, an = 0.5an − 1 and a 1 = 12.5, an = 4an − 1 53. a 5 = 256b 55. The sequence exceeds 100 at the 14th term, a 14 ≈ 107. ODD ANSWERS C-44 32 ___ is the first non-integer value 3 57. a 4 = − 59. Answers will vary. Example: explicit formula with a decimal common ratio: an = 400 ∙ 0.5n − 1; first 4 terms: 400, 200, 100, 50; a 8 = 3.125 Section 11.4 3. A geometric series is the sum of the terms in a 1. An nth partial sum is the sum of the first n terms of a sequence. geometric sequence. equal payments that earn a constant compounded interest. 11. ∑ 7. ∑ 5. An annuity is a series of regular 13. S 5 = 8k + 2 5n 20 4 25 _ 2 n = 0
15. S 13 = 57.2 k = 1 7 8 ∙ 0.5k − 1 k = 1 5 9. ∑ 4 17 __ = 1 _ 1 − 3 k = 1 5 19. S 5 = 121 _ 9 ≈ 13.44 21. S 11 = 64(1 − 0.211) __ = 1 − 0.2 23. The series is defined. S = 25. The series is defined. S = ≈ 80 781,249,984 __ 9,765,625 2 _ 1 − 0.8 −1 __________ 1 −  − 1  __ 2 27 2000 1750 1500 1250 1000 750 500 250 10 11 12 Month x 31. 49 33. 254 35. S 7 = 39. S 7 = 5208.4 45. S = 9.2 51. ak = 30 − k 57. $400 per month 41. S 10 = − 47. $3,705.42 53. 9 terms 29. Sample answer: The graph of Sn seems to be approaching 1. This makes sense because ∑ 1   _ 2 defined infinite geometric k is a k = 1 ∞ series with S = 1 __ 2 _________ = 1. 1 −  1  __ 2 55 _ 2 4 _ 43. S = − 3 37. S 11 = 147 _ 2 1023 _ 256 49. $695,823.97 4 _ 55. r = 5 59. 420 feet 61. 12 feet Section 11.5 1. There are m + n ways for either event A or event B to occur. 3. The addition principle is applied when determining the total possible of outcomes of either event occurring. The multiplication principle is applied when determining the total possible outcomes of both events occurring. The word “or” usually implies an addition problem. The word “and” usually implies a multiplication problem. C(n, r) = 5. A combination; 9. 5 + 4 + 7 = 16 7. 4 + 2 = 6 n! _ (n − r)!r! 13. 103 = 1,000 11. 2 × 6 = 12 17. P(3, 3) = 6 19. P(11, 5) = 55,440 23. C(7, 6) = 7 25. 210 = 1,024 15. P(5, 2) = 20 21. C(12, 4)
= 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. × 4! = 8,640 41 43. 4 × 2 × 5 = 40 45. 4 × 12 × 3 = 144 49. C(10, 3) × C(6, 5) × C(5, 2) = 7,200 47. P(15, 9) = 1,816,214,400 51. 211 = 2,048 53. 20! ______ 6!6!8! = 116,396,280 Section 11.6 n k = 0 n r 7. 35 9. 10 11. 12,376 n! _________. r!(n − r)!  x n − ky k and 1. A binomial coefficient is an alternative way of denoting the combination C(n, r). It is defined as   = C(n, r) =  n 3. The Binomial Theorem is defined as (x + y)n = ∑ k can be used to expand any binomial. 5. 15 13. 64a3 − 48a2b + 12ab2 − b3 17. 1024x5 + 2560x4y + 2560x3y2 + 1280x2y3 + 320xy4 + 32y5 19. 1024x5 − 3840x4y + 5760x3y2 − 4320x2y3 + 1620xy4 − 243y5 16 ___ y4 32 ___ xy3 25. a15 − 30a14b + 420a13b2 27. 3,486,784,401a20 + 23,245,229,340a19b + 73,609,892,910a18b2 29. x 24 − 8x 21 √ y + 28x 18y 33. 220,812,466,875,000y 7 23. a17 + 17a16b + 136a15
b2 15. 27a3 + 54a2b + 36ab2 + 8b3 24 ____ x2y2 8 ___ x3y 1 ___ x4 21. + + + + — 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. −720x 2y 3 35. 35x 3y 4 1152y2 _ x7 7 6 5 4 3 2 1 −2−3−4−5 −1 −1 −2 −3 −4 −5 45. 590,625x 5y 2 47. k − 1 y y f2(x) f4(x) x 1 2 90 80 70 60 50 40 30 20 10 x 1 2 −2−3−4−5−6−7−8−9 −1 −10 −20 −30 −40 −50 −60 −70 −80 −90 49. The expression (x3 + 2y2 − z)5 cannot be expanded using the Binomial Theorem because it cannot be rewritten as a binomial. Section 11.7 1. Probability; the probability of an event is restricted to values between 0 and 1, inclusive of 0 and 1. 3. An experiment is an activity with an observable result. 5. The probability of the union of two events occurring is a number that describes the likelihood that at least one of the events from a probability model occurs. In both a union of sets A and B and a union of events A and B, the union includes either A or B or both. The difference is that a union of sets results in another set, while the union of events is a probability, so it is always a numerical value between 0 and 1. 5 __ 9. 8 1 __ 21. 8 5 __ 11. 8 15 ___ 16 23. 3 __ 13. 8 5 __ 25. 8 1 __ 15. 4 27. 1 __ 7. 2 3 3 __ __ 17. 19. 4 8 1 ___ 26 29. 31. 1 ___ 13 12 ___ 13 ODD ANSWERS 33. 1 (1, 1) 2 (2, 1) 3 (3, 1) 4 (4, 1) 5 (5, 1) 6 (6, 1) 7 2 (1, 2) 3 (2, 2) 4 (3
, 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. 47. C(12, 5) ________ = C(48, 5) 1 _____ 2162 3 __ 43. 4 1 __ 41. 4 C(12, 3)C(36, 2) ______________ = C(48, 5) 49. 175 _____ 2162 51. C(20, 3)C(60, 17) _______________ C(80, 20) ≈ 12.49% 53. C(20, 5)C(60, 15) _______________ C(80, 20) ≈ 23.33% 57. 55. 20.50 + 23.33 − 12.49 = 31.34% C(40000000, 1)C(277000000, 4) ___________________________ C(317000000, 5) C(40000000, 4)C(277000000, 1) ___________________________ C(317000000, 5) 59. = 36.78% = 0.11% Chapter 11 Review exercises 13 _ 24 13. r = 2 3. 13, 103, 1003, 10003 9. a 1 = −20, an = an − 1 + 10 1. 2, 4, 7, 11 5 _ 5. The sequence is arithmetic. The common difference is d =. 3 7. 18, 10, 2, −6, −14 1 _ 11. an = n + 3 17. 3, 12, 48, 192, 768 21. ∑ 1 __ m + 5   2 135 ____ 4
35. P(18, 4) = 73,440 1 1  ∙  _ _ 19. an = − 5 3 25. S9 ≈ 23.95 15. 4, 16, 64, 256, 1024 n − 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 × 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, 1 6, 1 1, 2 2, 2 3, 2 4, 2 5, 2 6, 2 1, 3 2, 3 3, 3 4, 3 5, 3 6, 3 1, 4 2, 4 3, 4 4, 4 5, 4 6, 4 5 1, 5 2, 5 3, 5 4, 5 5, 5 6, 5 6 1, 6 2, 6 3, 6 4, 6 5, 6 6, 6 4 __ 53. 9 5 __ 51. 9 1 __ 49. 6 C(150, 3)C(350, 5) ________________ C(500, 8) 57. ≈ 25.6% 55. 1 − C(350, 8) ________ C(500, 8 ) ≈ 94.4% Chapter 11 practice test 1. −14, −6, −2, 0 common difference is d = 0.9. 3 _ ; a 22 = − 5. a 1 = −2, an = an − 1 − 2 67 _ 2 3. The sequence is arithmetic. The C-45 1 _ 7. The sequence is geometric. The common ratio is r =. 2 1 _ 9. a 1 = 1, an = − ∙ an −1 2 13. S 7 = −2,604.2 earned: $14,355.75 5 __ k   3k 2 − 6 15. Total in account: $140,355.75; Interest 17 = 180 11. ∑ k = −3 15 21. 29. = 151,200 10! _ 2!3!2! C(14, 3)C(
26, 4) ______________ C(40, 7) ≈ 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 → a never equal a. As the values of x approach a from the left and right, the limit is the value that the function is approaching. 3. −4 5. −4 11. 4 13. Does not exist 17. Answers 19. Answers will vary 21. Answers will vary will vary 23. 7.38906 25. 54.59815 27. e 6 ≈ 403.428794, e 7 ≈ 1096.633158, en _________ _ ≈ 0.83 6 x 2 − 9   = −2.00 35. lim 10 − 10x 2  = 20.00 __ x 2 − 3x + 2  does not exist. Function values decrease  x 2 − 1 __ x 2 − 3x + 2 x  __ 37. lim 4x 2 + 4x + 1 1 _ x → − 2 29. lim x → −2 33. lim x → 1 31. lim f (x) = 1 x → 3 x → 1 without bound as x approaches −0.5 from either left or right. 39. lim x → 0 7tan x _ 3x 7 _ = 3 x f(x) −0.1 −0.01 7tan x _ 2.34114234 lim x → 0− 3x ↓ 2.33341114 7 _ 3 −0.001 2.33333411 0 Error 0.001 2.33333411 0.01 2.33341114 0.1 2.34114234 7 _ 3 ↑ 7tan x _ 3x x → 0+ lim 41. lim x → 0 2sin x _ 4tan x 1 _ = 2 x −0.1 f(x) x → 0− 0.49750208 lim −0.01 0.49997500 −0.001 0.49999975 0 Error 0.001 0
.49999975 0.01 0.49997500 0.1 0.49750208 2sin x _ 4tan x ↓ 1 _ 2 1 _ 2 ↑ 2sin x _ 4tan x x → 0+ lim − 1 ___ x 2 e 43. lim e x → 0 = 1.0 45. lim x → −1− \|x + 1\| _______ x + 1 = −(x + 1) ________ (x + 1) = −1 and lim x → −1+ \|x + 1\| _______ = x + 1 (x + 1) _______ (x + 1) = 1 since the right-hand limit does not equal the left-hand limit, lim does not exist. \|x + 1\| _______ x + 1 x → −1 does not exist. The function increases without 47. lim x → −1 1 _______ (x + 1)2 bound as x approaches −1 from either side. 49. lim x → 0 5 _______ 1 − e 2 _ x does not exist. Function values approach 5 from the left and approach 0 from the right. 51. Through examination of the postulates and an understanding of relativistic physics, as v → c, m → ∞. Take this one step v → c− m = lim v → c − further to the solution, lim mo __ = ∞ √ _________ 1 −  v 2  _ c2 ODD ANSWERS C-46 Section 12.2 1. If f is a polynomial function, the limit of a polynomial function 3. It could mean either as x approaches a will always be f (a). (1) the values of the function increase or decrease without bound as x approaches c, or (2) the left and right-hand limits are not equal. 5. − 10 _ 3 13. Does not exist 7. 6 1 _ 9. 2 15. −12 17. − 19. −108 — 11. 6 5 √ _ 10 23. 6 21. 1 31. 6 + √ — 5 25. 1 3 _ 33. 5 27. 1 29. Does not exist 35. 0 37. −3 39. Does not exist; right-hand limit is not the same as the left-hand limit. 41. 2 43. Limit does not exist; limit approaches infinity. 45. 4x + 2h 51. −1 ________ x
(x + h) cos (x + h) − cos(x) __ 47. 2x + h + 4 h x 2 + 5x + 6 −1 __________ __ 55. f (x √ 49. 53. — 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 = −3; f (−3) does not exist 5. Removable discontinuity at a = −4; f (−4) is not defined 7. Discontinuous at a = 3; lim x → 3 not equal to the limit. f (x) = 3, but f (3) = 6, which is 9. lim x → 2 f (x) does not exist 11. lim x → 1− 13. lim x → 1− f (x) = 4; lim f (x) = 1, therefore, lim x → 1 x → 1+ f (x) does not exist. f (x) = 5 ≠ lim f (x) = −1, thus lim f (x) does not exist. x → 1+ x → 1 15. lim x → −3− f (x) = −6, lim x → −3+ 1 _ f (x) = −, therefore, lim 3 x → −3 f (x) does 23. Continuous on ( −∞, ∞) not exist. 17. f (2) is not defined 19. f (−3) is not defined. 21. f (0) is not defined. 25. Continuous on ( − ∞, ∞) and x = 2 on (0, ∞) ( −∞, ∞) 41. f (0) is undefined 45. At x = −1, the limit does not exist. At x = 1, f (1) does not exist. At x = 2, there appears to be a vertical asymptote, and the 29. Discontinuous at x = 0 33. Continuous on [4, ∞) 37. 1, but not 2 or 3 27. Discontinuous at x = 0 31. Continuous 35. Continuous on 43. ( −∞, 0) ∪ (0, ∞) 39. 1 and
2, but not 3 limit does not exist. 47. x 3 + 6x 2 − 7x _____________ (x + 7)(x − 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 slope of a line and the derivative at a point measure the rate of change of 3. Average velocity is 55 miles per hour. The the function. instantaneous velocity at 2:30 p.m. is 62 miles per hour. The instantaneous velocity measures the velocity of the car at an instant of time whereas the average velocity gives the velocity of 5. The average rate of change of the the car over an interval. amount of water in the tank is 45 gallons per minute. If f (x) is the function giving the amount of water in the tank at any time t, then the average rate of change of f (x) between t = a and t = b is f (a) + 45(b − a). 9. f ′(x) = 4x + 1 7. f ′(x) = −2 11. f ′(x) = 17. f ′(x) = 0 1 _______ (x − 2)2 13. − 1 _ 19. − 3 16 ________ (3 + 2x)2 21. Undefined 15. f ′(x) = 9x 2 − 2x + 2 33. Discontinuous at 31. Discontinuous at x = −2 and 23. f ′(x) = −6x − 7 25. f ′(x) = 9x 2 + 4x + 1 27. y = 12x − 15 29. k = −10 or k = 2 x = 0. Not differentiable at −2, 0, 2. x = 5. Not differentiable at −4, −2, 0, 1, 3, 4, 5. 35. f (0) = −2 37. f (2) = −6 39. f ′(−1) = 9 41. f ′(1) = −3 43. f ′(3) = 9 45. Answers vary. The slope of the tangent line near x = 1 is 2. 47. At 12:30
p.m., the rate of change of the number of gallons in the tank is −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 seconds is 96 feet. 55. The height of the projectile is zero at seconds is 96 feet. t = 0 and again at t = 5. In other words, the projectile starts on 57. 36π the ground and falls to earth again after 5 seconds. 59. $50.00 per unit, which is the instantaneous rate of change of revenue when exactly 10 units are sold. 63. $36 49. At 200 minutes after 65. f ′(x) = 10a − 1 61. $21 per unit 4 _ (3 − x)2 67. Chapter 12 Review exercises f (x) = 0 3. Does not exist 5. Discontinuous at x = −1 f (x) does not exist), x = 3 (jump discontinuity), and x = 7 7. lim a f (x) does not exist). x → 1 3 _ 11. − 17. 500 13. 1 5 21. At x = 4, the function has a vertical asymptote. 1. 2 ( lim x → a ( lim x → a 9. Does not exist 6 _ 19. − 7 23. At x = 3, the function has a vertical asymptote. 25. Removable discontinuity at a = 9 discontinuity at x = 5 discontinuity at x = 1 29. Removable discontinuity at x = 5, 31. Removable discontinuity at x = −2, 27. Removable 15. 6 discontinuity at x = 5 33. 3 37. e 2x + 2h − e 2x __ h 39. 10x − 3 35. 1 __ (x + 1)(x + h + 1) 41. The function would not be differentiable at however, 0 is not in its domain. So it is differentiable everywhere in its domain. Chapter 12 practice test 1. 3 3. 0 5. −1 7. lim 5 _ f (x) = − a and lim 2 x → 2+ f (x) = 9 x → 2− thus, the limit of the function as x approaches 2 does not exist. 1 _ 50 9.
− 11. 1 15. f '(x) = − 3 ____ 3 _ 2 a 2 13. Removable discontinuity at x = 3 17. Discontinuous at −2, 0, not differentiable at −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 = −14x − 18 25. 3 35. The graph is not differentiable at x = 1 (cusp). 37. f '(x) = 8x 39. f '(x) = − 41. f '(x) = −3x 2 43. f '(x) = 1 _______ (2 + x)2 1 _ — x − 1 2 √ ODD ANSWERS Index A AAS (angle-angle-side) 644 absolute maximum 47, 113 absolute minimum 47, 113 absolute value 31, 89 absolute value equation 92, 113 absolute value function 29, 89, 92 absolute value inequality 94, 113 addition method 762, 767, 854 Addition Principle 982, 1008 adjacent side 486, 498 altitude 644, 747 ambiguous case 646, 747 amplitude 509, 510, 552, 617, 629 angle 440, 458, 498 angle of depression 492, 498 angle of elevation 492, 498, 644 angle of rotation 913, 931 angular speed 453, 498 annual interest 977 annual percentage rate (APR) 336, 429 annuity 977, 1008 apoapsis 922 arc 443 arc length 444, 450, 458, 498 arccosine 552 arccosine function 542 Archimedes’ spiral 693, 747 arcsine 552 arcsine function 542 arctangent 552 arctangent function 542 area of a circle 224 area of a sector 451, 498 argument 700, 747 arithmetic sequence 951, 952, 954, 955, 970, 971, 1008 arithmetic series 971, 1008 arrow notation 278, 317 ASA (angle-side-angle) 644 asymptote 880 augmented matrix 816, 820, 821, 833, 854 average rate of change 38, 113, 1052, 1070 axis of symmetry 208, 211
, 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 880 change-of-base formula 387, 429 circle 787, 789 circular motion 517 circumference 443 coefficient 224, 268, 317 coefficient matrix 816, 818, 835, 854 cofunction 578 cofunction identities 491, 578 column 805, 854 column matrix 806 combination 987, 992, 1008 combining functions 52 common base 391 common difference 951, 970, 1008 common logarithm 358, 429 common ratio 961, 973, 1008 commutative 53 complement of an event 1003, correlation coefficient 180, 187 cosecant 473, 498, 528 cosecant function 529, 562 cosine 561, 596, 597 cosine function 458, 498, 507, divisor 258 domain 2, 10, 22, 23, 113 domain and range 22 domain and range of inverse functions 104 508, 510, 517, 528 domain of a composite function cost function 51, 768, 854 cotangent 473, 498, 534 cotangent function 534, 562 coterminal angles 448, 450, 498 co-vertex 865, 867, 880 Cramer’s Rule 843, 846, 850, 854 cube root 225 cube root function 30 cubic function 29, 302 curvilinear path 708 D damped harmonic motion 625, 634 decompose a composite function 59 decomposition 795 decreasing function 43, 113, 128 decreasing linear function 129, 187 degenerate conic sections 909 degree 229, 317, 441, 498 De Moivre 697, 702 De Moivre’s Theorem 703, 704, 747 58 domain of a rational function 283 dot product 739, 747 double-angle formulas 584, 585, 634 doubling time 401, 405, 429 Dürer 688 E eccentricity 923, 931 electro
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, 1008 dependent system 759, 767, 1008 complex conjugate 202, 317 Complex Conjugate Theorem 272 complex number 198, 317, 697 complex plane 199, 317, 697 composite function 51, 52, 53, 113 compound interest 336, 429 compression 146, 348, 370 conic 864, 879, 928 conic section 714, 931 conjugate 1032 conjugate axis 880, 931 consistent system 758, 854 constant function 29 constant of variation 311, 317 constant rate of change 162 continuity 1040 continuous 239, 1040 continuous function 234, 317, 1037, 1070 convex limaçon 687, 747 coordinate plane 897 779, 854 dependent variable 2, 113 derivative 1053, 1054, 1055, 1056, 1060, 1070 Descartes 697 Descartes’ Rule of Signs 273, 317 determinant 843, 845, 846, 854 difference formula 571 difference quotient 1053 differentiable 1060, 1070 dimpled limaçon 687, 747 directrix 897, 900, 902, 923, 927, 928, 931 direct variation 311, 317 discontinuou 1040 discontinuous function 1038, 1070 displacement 452 distance formula 881, 897 diverge 974, 1008 dividend 258 Division Algorithm 258, 259, 266, 317 exponential 344 exponential decay 328, 334, 343, 401, 403, 406, 416 exponential equation 390 exponential function 328 exponential growth 328, 331, 401, 405, 407, 416, 429 exponential regression 417 extraneous solution 394, 429 extrapolation 177, 178, 187 F factorial 946 Factor Theorem 267, 317 Fahrenheit 100 feasible region 791, 854 finite arithmetic sequence 956 finite sequence 939, 1008 foci 865, 867, 880, 931 focus 865
, 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 quadratic function 209, 317 Generalized Pythagorean Theorem 658, 747 geometric sequence 961, 963, 973, 1008 geometric series 973, 1008 global maximum 251, 252, 317 global minimum 251, 252, 317 Graphical Interpretation of a Linear Function 145 gravity 724 growth 344 H half-angle formulas 589, 634 half-life 397, 401, 403, 429 harmonic motion 624 Heaviside method 797 Heron of Alexandria 663 Heron’s formula 663 horizontal asymptote 280, 281, 286, 317 horizontal compression 79, 113, 611 horizontal line 150, 151, 187 horizontal line test 15, 113 horizontal reflection 71, 72, 113 horizontal shift 67, 113, 346, 367, 507 horizontal stretch 79, 113 hyperbola 879, 882, 883, 884, 887, 888, 891, 897, 924, 926, 931 hypotenuse 486, 498 I identities 468, 479, 498 identity function 29 identity matrix 829, 833, 854 imaginary number 198, 317 inconsistent system 759, 766, 778, 854 increasing function 43, 113, 128 increasing linear function 129, 187 independent variable 2, 113 index of summation 969, 970, 1008 inequality 790 infinite geometric series 974 infinite sequence 939, 1008 infinite series 974, 1008 initial point 729, 732, 747 initial side 441, 498 inner-loop limaçon 688, 747 input 2, 113 instantaneous rate of change 1053, 1070 instantaneous velocity 1065, 1070 Intermediate Value Theorem 249, 317 interpolation 177, 187 intersection 1001 interval notation 22, 26, 43, 113 inverse cosine function 542, 552 inverse function 101, 113, 299, 302 inversely proportional 312, 317 inverse matrix 833, 835 inverse of a radical function
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 Sines 645, 659, 747 leading coefficient 229, 317 leading term 229, 318 least common denominator (LCD) 1032 least squares regression 178, 187 left-hand limit 1021, 1040, 1070 lemniscate 689, 747 limaçon 687, 688 limit 1018, 1019, 1023, 1029, 1030, 1031, 1070 linear function 126, 143, 147, nth root of a complex number 162, 187 704 linear growth 328 linear model 163, 175 linear relationship 175 linear speed 452, 453, 498 local extrema 42, 113 local maximum 42, 113, 252 local minimum 42, 113, 252 logarithm 356, 429 logarithmic equation 395 logarithmic model 419 logistic growth model 408, 429 logistic regression 422 long division 257 lower limit 1008 lower limit of summation 969, 970 M magnitude 31, 66, 698, 729, 731, 747 main diagonal 818, 854 major and minor axes 867 major axis 865, 869, 931 marginal cost 1059 matrix 805, 806, 810, 816, 854 matrix multiplication 810, 830, 835 matrix operations 806 maximum value 208 measure of an angle 441, 498 midline 509, 510, 552, 617 minimum value 208 minor axis 865, 869, 931 model breakdown 177, 187 modeling 162 modulus 31, 700, 747 Multiplication Principle 983, 984, 1008 matrix 829, 854 multiplicity 243, 318 mutually exclusive events 1002, 1008 N natural logarithm 360, 393, 429 natural numbers 2 negative angle 441, 448, 467, 498 nth term of a sequence 939, 1008 O oblique triangle 644, 747 odd function 75, 113, 477, 561 one-loop limaçon 687, 747 one-to-one 299,
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 parametric equation 709, 719, 721 parametric form 722 parent function 367 partial fraction 795, 854 partial fraction decomposition 795, 854 Pascal 688 Pascal’s Triangle 994 periapsis 922 period 481, 498, 507, 523, 525, 603, 617 periodic function 481, 507, 552 periodic motion 617, 624 permutation 984, 1008 perpendicular lines 152, 153, 187 pH 380 phase shift 511, 552 piecewise function 31, 113, 942, 1041, 1044, 1046 point-slope form 131, 187 point-slope formula 885 polar axis 670, 747 polar coordinates 670, 672, 673, 674, 681, 747 Newton’s Law of Cooling 406, polar equation 676, 682, 683, 429 n factorial 946, 1008 nominal rate 336 nondegenerate conic section 909, 911, 931 nonlinear inequality 790, 854 non-right triangles 644 nth partial sum 969, 1008 747, 923, 931 polar form 698 polar form of a complex number 699, 747 polar form of a conic 928 polar grid 670 pole 670, 747 polynomial 268, 1044 joint variation 314, 317 jump discontinuity 1040, 1070 multiplicative inverse 831 multiplicative inverse of a independent system 758, 759, Linear Factorization Theorem 854 272, 318 INDEX polynomial function 228, 239, 246, 250, 318 position vector 729, 731 positive angle 441, 448, 498 power function 224, 318 power rule for logarithms 383, 387, 429 probability 999, 1008 probability model 999, 1008 product of two matrices 810 product rule for logarithms 381, 383, 429 product-to-sum formula 596, 598,
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 quadratic formula 219, 607 quadratic function 29, 211, 213 quotient 258 quotient identities 562, 563, 634 quotient rule for logarithms 382, 429 R radian 444, 445, 498 radian measure 445, 450, 498 radical functions 301 radiocarbon dating 404 range 2, 113, 542 rate of change 38, 114, 126 rational expression 795, 801 rational function 282, 289, 292, 318, 1028 Rational Zero Theorem 268, 318 ray 440, 498 reciprocal 101, 225 reciprocal function 30, 278 reciprocal identities 562, 563, 634 reciprocal identity 528, 534 reciprocal squared function 30 rectangular coordinates 670, 672, 673, 674 rectangular equation 676, 713 rectangular form 699, 722 recursive formula 944, 954, 963, 1009 D-3 upper limit of summation 969, 970, 1009 upper triangular form 774 V varies directly 311, 318 varies inversely 312, 318 vector 729, 748 vector addition 733, 748 velocity 1053, 1065 vertex 208, 318, 440, 498, 865, 897, 903 vertex form of a quadratic function 210, 318 vertical asymptote 280, 283, 287, 318, 542 vertical compression 76, 114 vertical line 151, 187 vertical line test 13, 114 vertical reflection 71, 72, 114 vertical shift 64, 65, 114, 146, 345, 368, 406, 511 vertical stretch 76, 114, 146, 370 vertical tangent 1062 vertices 865, 867 volume of a sphere 224 reduction formulas 588, 634 reference angle 448, 466, 467, SSA (side-side-angle) 644 SSS (side-side-side) triangle 468, 476, 498 reflection 349, 372 regression analysis 416, 419, 422 relation 2, 114 relative 42 remainder 258 Remain
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, 821, 835 system of linear equations 168, X 758, 760, 761, 855 system of nonlinear equations 785, 855 system of nonlinear inequalities 791, 855 x-intercept 149, 187 Y y-intercept 127, 128, 145, 187 System of Three Equations in Three Variables 846 Z zeros 209, 240, 243, 268, 318, 684 T tangent 473, 498, 523 tangent function 524, 525, 526, 536, 562 tangent line 1051, 1070 term 938, 952, 1009 terminal point 729, 732, 748 terminal side 441, 498 term of a polynomial function 228, 318 sine 561, 597, 598 sine function 458, 477, 498, 506, 511, 516, 518 sinusoidal function 508, 552, transformation 64, 146 translation 901 transverse axis 880, 931 trigonometric equations 713, 617 714 slope 127, 128, 130, 187 slope-intercept form 127, 187 slope of the curve 1051 slope of the tangent 1051 smooth curve 234, 318 solution set 775, 854 solving systems of linear equations 760, 762 special angles 464, 489, 570 square matrix 806, 843 square root function 30 trigonometric functions 486 trigonometric identities 659 turning point 232, 247, 318 two-sided limit 1021, 1053, 1070 U union of two events 1001, 1009 unit circle 445, 458, 461, 465, 477, 486, 498, 604 unit vector 736, 748 INDEXponential functions to model and solve real-life problems (p. 223). Section 3.2 Recognize and evaluate logarithmic functions with base a (p. 229). Graph logarithmic functions (p. 231). Recognize, evaluate, and graph natural logarithmic functions (p.
233). Use logarithmic functions to model and solve real-life problems (p. 235). 3 In Exercises 1–6, 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 properties of logarithms to evaluate or rewrite logarithmic expressions (p. 240). 19. Use properties of logarithms to expand or condense logarithmic expressions (p. 241). 21. Use logarithmic functions to model and solve real-life problems (p. 242). Function f x 6.1x f x 30x f x 20.5x f x 1278x5 f x 70.2x f x 145x Value Section 3.4 x 2.4 Solve simple exponential and logarithmic equations (p. 246). x 3 2. Solve more complicated exponential equations (p. 247). x 3. Solve more complicated logarithmic equations (p. 249). x 1 4. Use exponential and logarithmic equations to model and solve x 11 x 0.8 5. real-life problems (p. 251). 6. 25. 23. 1. x. f x 5 x2 4 79–94 f x 1 x 3 95, 96 2 x. 97–104 3x2 1 105–118 9 119–134 e5x7 e15 135, 136 In Exercises 23–26, use the One-to-One Property to solve the equation for 81 x2 1 3 e82x e3 24. 26. Section 3.5 Recognize the five most common types of models involving exponential In Exercises 7–10, match the function with its graph. [The graphs are labeled (a), (b), (c), and (d).] and logarithmic functions (p. 257). y (a) real-life problems (p. 258). y Use exponential growth and decay functions to model and solve 5 4 3 2 Use Gaussian functions to model and solve real-life problems (p. 261). −1 Use logistic growth functions to model and solve real-life problems (p. 262). −2 −3 Use logarithmic functions to model and solve real-life problems (p. 263). −4 −5 (b) −
3 −2 −1 −3 −2 1 3 2 1 2 3 x In Exercises 27–30, evaluate the function given by at the indicated value of decimal places. fx ex Round your result to three x. 137–142 27. 29. x 8 x 1.7 143–148 28. 30. x 5 8 x 0.278 In Exercises 31–34, use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. 149 150 x 31. 33. hx ex2 f x e x2 151, 152 32. 34. hx 2 ex2 st 4e2t, t > 0 (c) y (d3 −2 −1 1 2 3 x −3 −2 −1 21 3 x 7. 9. f x 4x f x 4x 8. 10. f x 4x f x 4x 1 In Exercises 11–14, use the graph of transformation that yields the graph of f g. to describe the 11. 12. 13. 14. f x 5x, f x 4x gx 5x1 gx 4x 3 gx 1 x2 x gx 8 2 2 3 In Exercises 15–22, use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. 15. 17. f x 4x 4 f x 2.65x1 16. 18. f x 4x 3 f x 2.65x1 Compound Interest table to determine the balance rate In Exercises 35 and 36, complete the for dollars invested at n P times per year. for years and compounded A r t 1 2 4 12 365 Continuous n A 35. 36. P $3500, r 6.5%, t 10 years P $2000, r 5%, t 30 years 37. Waiting Times The average time between incoming calls at a switchboard is 3 minutes. The probability of waiting t less than minutes until the next incoming call is approxiFt 1 et 3. mated by the model A call has just come in. Find the probability that the next call will be within F (a) minute. 1 2 (b) 2 minutes. (c) 5 minutes. 38. Depreciation After t years, the value originally cost $14,000 is given by V Vt 14,0003 of a car that t. 4 (a) Use a graphing utility
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 answers given in the back of the book. In Exercises 1– 4, evaluate the expression. Approximate your result to three decimal places. 1. 12.42.79 2. 432 3. e710 4. e3.1 • Chapter Tests and Cumulative Tests Chapter Tests, at the end of each chapter, and periodic Cumulative Tests offer students frequent opportunities for self-assessment and to develop strong study- and test-taking skills. 276 3 y 4 2 −2 2 4 12,000 10,000 8,000 6,000 4,000 2,000 −4 Exponential Growth y FIGURE FOR 6 (9, 11,277) (0, 2745) 2 4 6 8 10 t FIGURE FOR 27 In Exercises 5–7, construct a table of values. Then sketch the graph of the function. f x 6 x2 f x 1 e2x f x 10x 6. 7. 5. 8. Evaluate (a) log7 70.89 and (b) 4.6 ln e2. Chapter 3 Exponential and Logarithmic Functions In Exercises 9–11, construct a table of values. Then sketch the graph of the function. Identify any asymptotes. f x log x 6 9. 10. Cumulative Test for Chapters 1–3 In Exercises 12–14, evaluate the logarithm using the change-of-base formula. Round your result to three decimal places. f x 1 lnx 6 f x lnx 4 11. 13. 12. log7 44 In Exercises 15–17, use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. 14. log24 68 Take this test to review the material from earlier chapters. When you are finished, check your work against the answers given in the back of the book. 1, 1. segment joining the points and the distance between the points. Find the coordinates of the midpoint of the line 1. Plot the points log25 0.9
3, 4 and F E A T U R E S 15. log2 3a4 x 16. 17. In Exercises 2– 4, graph the equation without using a graphing utility. log ln 7x2 yz3 5x 6 In Exercises 18–20, 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 of the line passing through 19. 4 ln x 4 ln y 1 2, 1 and 3, 8. 6. Explain why the graph at the left does not represent as a function of x. In Exercises 21– 26, solve the equation algebraically. Approximate your result to x 2 three decimal places. f s 2 7. Evaluate (if possible) the function given by f 2 (b) (c) for each value. y f x x 5x 1 25 5 1025 8 e4x 18 4 ln x 7 21. 23. 25. (a) 22. f 6 3e5x 132 necessary to sketch the graphs.) 24. (a) 26. ln x 1 r x 1 2 3x 2 log x log8 5x 2 (b) hx 3x 2 (c) gx 3x 2 8. Compare the graph of each function with the graph of y 3x. (Note: It is not (b) In Exercises 9 and 10, find (a) 27. Find an exponential growth model for the graph shown in the figure. is the domain of f x x 3, is 21.77 years. What percent of a present gx 4x 1 amount of radioactive actinium will remain after 19 years? 28. The half-life of radioactive actinium f/g? 227Ac f x x 1, 10. 9. f gx, f gx, (c) fgx, and (d) f/gx. What gx x2 1 29. A model that can be used for predicting the height In Exercises 11 and 12, find (a) H 70.228 5.104x 9.222 ln x, function. on his or her age is age of the child in years. x (Source: Snapshots of Applications in
Mathematics) f x 2x2, (a) Construct a table of values. Then sketch the graph of the model. gx x 6 and (b) ≤ x ≤ 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 hx 5x 2 has an inverse function. If so, find the inverse 13. Determine whether calculate the actual height using the model. function. 14. The power produced by a wind turbine is proportional to the cube of the wind speed A wind speed of 27 miles per hour produces a power output of 750 kilowatts. Find P S. the output for a wind speed of 40 miles per hour. 15. Find the quadratic function whose graph has a vertex at the point 4, 7. 8, 5 and passes through In Exercises 16–18, sketch the graph of the function without the aid of a graphing utility. 16. hx x 2 4x 17. f t 1 4tt 22 18. gs s2 4s 10 In Exercises 19–21, find all the zeros of the function and write the function as a product of linear factors. 19. 20. 21. f x x3 2x2 4x 8 f x x4 4x3 21x2 f x 2x4 11x3 30x2 62x 40 333200__SE_FM.qxd 12/7/05 10:20 AM Page xvi xvi Textbook Features and Highlights • Proofs in Mathematics At the end of every chapter, proofs of important mathematical properties and theorems are presented as well as discussions of various proof techniques. • P.S. Problem Solving Each chapter concludes with a collection of thought-provoking and challenging exercises that further explore and expand upon the chapter concepts. These exercises have unusual characteristics that set them apart from traditional text exercises. Proofs in Mathematics What does the word proof mean to you? In mathematics, the word proof is used to mean simply a valid argument. When you are proving a statement or theorem, you must use facts, definitions, and accepted properties in a logical order. You can also use previously proved theorems in your proof. For instance, the Distance Formula is used in the proof of the Midpoint Formula
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. d1 d 3 ( x1 + x2 2, y1 + y2 2 ) d 2 (x 2, y 2) x By the Distance Formula, you obtain x1 d1 x2 2 x12 y1 y2 2 y12 1 2 x2 x1 2 y2 y1 2 x2 d2 x1 x2 2 2 y2 y1 y2 2 2 x1 2 y2 y1 2 x2 1 2 x2 d3 So, it follows that x1 d1 2 y2 d2 y1 and d1 2 d2 d3. The Cartesian Plane The Cartesian plane was named after the French mathematician René Descartes (1596–1650). While Descartes was lying in bed, he noticed a fly buzzing around on the square ceiling tiles. He discovered that the position of the fly could be described by which ceiling tile the fly landed on. This led to the development of the Cartesian plane. Descartes felt that a coordinate plane could be used to facilitate description of the positions of objects. 124 P.S. Problem Solving This collection of thought-provoking and challenging exercises further explores and expands upon concepts learned in this chapter. 1. As a salesperson, you receive a monthly salary of $2000, plus a commission of 7% of sales. You are offered a new job at $2300 per month, plus a commission of 5% of sales. (a) Write a linear equation for your current monthly wage W1 in terms of your monthly sales S. W2 (b) Write a linear equation for the monthly wage S. new job offer in terms of the monthly sales of your (c) Use a graphing utility to graph both equations in the same viewing window. Find the point of intersection. What does it signify? (d) You think you can sell $20,000 per month. Should you change jobs? Explain. 2. For the numbers 2 through 9 on a telephone keypad (see figure
), 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 struck an iceberg and sank, having covered only about 2100 miles of the approximately 3400-mile trip. (a) What was the total duration of the voyage in hours? (b) What was the average speed in miles per hour? (c) Write a function relating the distance of the Titantic from New York City and the number of hours traveled. Find the domain and range of the function. (d) Graph the function from part (c). 3. What can be said about the sum and difference of each of the following? (a) Two even functions (b) Two odd functions (c) An odd function and an even function 4. The two functions given by gx x f x x and are their own inverse functions. Graph each function and explain why this is true. Graph other linear functions that are their own inverse functions. Find a general formula for a family of linear functions that are their own inverse functions. 5. Prove that a function of the following form is even. y a2nx2n a2n2x2n2... a2x2 a0 6. A miniature golf professional is trying to make a hole-inone on the miniature golf green shown. A coordinate plane is placed over the golf green. The golf ball is at the point 2.5, 2 The professional wants to bank the ball off the side wall of the green at the point Then Find the coordinates of the point write an equation for the path of the ball. and the hole is at the point 9.5, 2. x, y. x, y. 8. Consider the function given by the average rate of change of the function from fx x2 4x 3. x2. to Find x1 1.5 (b) x1 1, x2 (a) (c) (d) (e) x1 x1 x1 x1 1, x2 1, x2 1, x2 1, x2 2 1.25 1.125 1.0625 (f) Does the average rate of
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)–(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 the functions given by f gx. f g1x. f 1x and g1 f 1x (b) Find (c) Find (d) Find g1x. and compare the result with that of part (b). (e) Repeat parts (a) through (d) for gx 2x. f x x3 1 and (f) Write two one-to-one functions and parts (a) through (d) for these functions. f g, and repeat (g) Make a conjecture about f g1x and g1 f 1x. 125 126 10. You are in a boat 2 miles from the nearest point on the coast. You are to travel to a point 3 miles down the coast and 1 mile inland (see figure). You can row at 2 miles per hour and you can walk at 4 miles per hour. Q, 13. Show that the Associative Property holds for compositions of functions—that is, f g hx f g hx. 2 mi x 3 − x 1 mi 3 mi Q Not drawn to scale. (a) Write the total time of the trip as a function of T x. (b) Determine the domain of the function. (c) Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. (d) Use the zoom and trace features to find the value of x that minimizes T. 11. The Heaviside function (e) Write a brief paragraph interpreting these values. Hx is widely used in engineering applications. (See figure.) To print an enlarged copy of the graph, go to the website www.mathgraphs.com. Hx 1, 0, x ≥ 0 x < 0 Sketch the graph of each function by hand. Hx Hx 2 2 Hx 2 Hx (b) (d) (e) (a) (c) (f) Hx 2 2 Hx 1 y 3 2 1 −3 −2 −1 1 2 3
x −2 −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 the graph, go to the website www.mathgraphs.com. f (a) (e) f x 1 f x (b) (f) f x 1 f x (c) (g) 2f x f x (d) f x y 4 2 −4 −2 2 4 x −2 −4 15. Use the graphs of function values. f and f1 to complete each table of y 4 2 −2 −2 −1 −2 −2 −4 (a) x 4 2 0 4 f f 1x (b) x 3 2 0 1 (c) (d) f f 1x x f f 1x x f 1x 3 2 0 1 4 3 0 4 333200__SE_FM.qxd 12/7/05 10:20 AM Page xvii Supplements Supplements for the Instructor Precalculus, Seventh Edition, has an extensive support package for the instructor that includes: Instructor’s Annotated Edition (IAE) Online Complete Solutions Guide Online Instructor Success Organizer Online Teaching Center: This free companion website contains an abundance of instructor resources. HM ClassPrep™ with HM Testing (powered by Diploma™): This CD-ROM is a combination of two course management tools. • HM Testing (powered by Diploma™) offers instructors a flexible and powerful tool for test generation and test management. Now supported by the Brownstone Research Group’s market-leading Diploma™ software, this new version of HM Testing significantly improves on functionality and ease of use by offering all the tools needed to create, author, deliver, and customize multiple types of tests—including authoring and editing algorithmic questions. Diploma™ is currently in use at thousands of college and university campuses throughout the United States and Canada. • HM ClassPrep™ also features supplements and text-specific resources for the instructor. Eduspace®: Eduspace®, powered by Blackboard®, is Houghton Mifflin’s customizable and interactive online learning tool. Eduspace® provides instructors
with online courses and content. By pairing the widely recognized tools of Blackboard® with quality, text-specific content from Houghton Mifflin Company, Eduspace® 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, tests, tutorials, and supplemental study materials. Visit www.eduspace.com for more information. Eduspace ® with eSolutions: Eduspace® with eSolutions combines all the features of Eduspace® with an electronic version of the textbook exercises and the complete solutions to the odd-numbered exercises, providing students with a convenient and comprehensive way to do homework and view course materials xvii 333200__SE_FM.qxd 12/7/05 10:20 AM Page xviii xviii Supplements Supplements for the Student Precalculus, Seventh Edition, has an extensive support package for the student that includes: Study and Solutions Guide Online Student Notetaking Guide Instructional DVDs Online Study Center: This free companion website contains an abundance of student resources. HM mathSpace® CD-ROM: This tutorial CD-ROM provides opportunities for self-paced review and practice with algorithmically generated exercises and stepby-step solutions. Eduspace®: Eduspace®, powered by Blackboard®, is Houghton Mifflin’s customizable and interactive online learning tool for instructors and students. Eduspace® is a text-specific, web-based learning environment that your instructor can use to offer students a combination of practice exercises, multimedia tutorials, video explanations, online algorithmic homework and more. Specific content is available 24 hours a day to help you succeed in your course. Eduspace® with eSolutions: Eduspace® with eSolutions combines all the features of Eduspace® with an electronic version of the textbook exercises and the complete solutions to the odd-numbered exercises. The result is a convenient and comprehensive way to do homework and view your course materials. Smarthinking®: Houghton Mifflin has partnered with Smarthinking® to provide an easy-to-use, effective, online tutorial service. Through state-of-theart tools and whiteboard technology, students communicate in real-time with qualified e-instructors who can help the students understand difficult concepts and guide them through the problem-solving process while studying or completing homework. Three levels of service are offered to the students
. 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 interactive websites to Frequently Asked Questions. Visit smarthinking.com for more information. *Limits apply; terms and hours of SMARTHINKING ® service are subject to change. 333202_0100.qxd 12/7/05 8:28 AM Page 1 Functions and Their Graphs 11 Rectangular Coordinates Graphs of Equations Linear Equations in Two Variables Functions Analyzing Graphs of Functions 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 A Library of Parent Functions 1.9 Inverse Functions Transformation of Functions 1.10 Mathematical Modeling and Variation Combinations of Functions: Composite Functions Functions play a primary role in modeling real-life situations. The estimated growth in the number of digital music sales in the United States can be modeled by a cubic function AT I O N S Functions have many real-life applications. The applications listed below represent a small sample of the applications in this chapter. • Data Analysis: Mail, Exercise 69, page 12 • Population Statistics, Exercise 75, page 24 • College Enrollment, Exercise 109, page 37 • Cost, Revenue, and Profit, • Fuel Use, Exercise 97, page 52 Exercise 67, page 82 • Digital Music Sales, Exercise 89, page 64 • Fluid Flow, Exercise 70, page 68 • Consumer Awareness, Exercise 68, page 92 • Diesel Mechanics, Exercise 83, page 102 1 333202_0101.qxd 12/7/05 8:29 AM Page 2 2 Chapter 1 Functions and Their Graphs 1.1 Rectangular Coordinates What you should learn • Plot points in the Cartesian plane. • Use the Distance Formula to find the distance between two points. • Use the Midpoint Formula to find the midpoint of a line segment. • Use a coordinate plane and geometric formulas to model and solve real-life problems. Why you should learn it The Cartesian plane can be used to represent relationships between two variables. For instance, in Exercise 60 on page 12, a graph represents the minimum wage in the United States from 1950 to 2004. The Cartesian Plane Just as you can represent real numbers by points on a real number line, you can represent ordered pairs of
real numbers by points in a plane called the rectangular coordinate system, or the Cartesian plane, named after the French mathematician René Descartes (1596–1650). 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 usually called the x-axis, and the vertical real number line is usually called the y-axis. The point of intersection of these two axes is the origin, and the two axes divide the plane into four parts called quadrants. y-axis y-axis Quadrant II Origin −3 −2 −1 3 2 1 Quadrant I (Vertical number line) 11 2 3 (Horizontal number line) −1 −2 Directed distance x x-axis (x, y) y Directed distance x-axis Quadrant III −3 Quadrant IV FIGURE 1.1 FIGURE 1.2 y, and Each point in the plane corresponds to an ordered pair of real numbers x called coordinates of the point. The x-coordinate represents the directed distance from the -axis to the point, and the y-coordinate represents the directed distance from the -axis to the point, as shown in Figure 1.2. (x, y) y x Directed distance from y-axis x, y Directed distance from x-axis © Ariel Skelly/Corbis The notation x, y denotes both a point in the plane and an open interval on the real number line. The context will tell you which meaning is intended. (3, 4) Example 1 Plotting Points in the Cartesian Plane Plot the points (1, 2), (3, 4), (0, 0), (3, 0), and (2, 3). (0, 0) (3, 0) x 1 2 3 4 Solution on the -axis and a To plot the point horizontal line through 2 on the -axis. The intersection of these two lines is the point The other four points can be plotted in a similar way, as shown in Figure 1.3. imagine a vertical line through 1, 2. (1, 2), 1 y x Now try Exercise 3. y 4 3 1 −1 −2 − ( 1, 2) −4 −3 −1 − − ( 2, 3) −4 FIGURE 1.3 333202_0101.qxd 12/7/05 8:29 AM Page 3 Year, t Amount, A
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 importance of Descartes’s introduction of coordinates in the plane. Today, his ideas are in common use in virtually every scientific and business-related field. Example 2 Sketching a Scatter Plot From 1990 through 2003, the amounts equipment in the United States are shown in the table, where year. Sketch a scatter plot of the data. Association) (in millions of dollars) spent on skiing represents the (Source: National Sporting Goods t A Solution To sketch a scatter plot of the data shown in the table, you simply represent each pair of values by an ordered pair and plot the resulting points, as shown in Figure 1.4. For instance, the first pair of values is represented by the ordered pair 1990, 475. t Note that the break in the -axis indicates that the numbers between 0 and 1990 have been omitted. (t, A) Amount Spent on Skiing Equipment 800 700 600 500 400 300 200 100 1991 1995 1999 2003 t Year FIGURE 1.4 Now try Exercise 21. In Example 2, you could have let t 1 represent the year 1990. In that case, the horizontal axis would not have been broken, and the tick marks would have been labeled 1 through 14 (instead of 1990 through 2003). Te c h n o l o g y The scatter plot in Example 2 is only one way to represent the data graphically. You could also represent the data using a bar graph and a line graph. If you have access to a graphing utility, try using it to represent graphically the data given in Example 2. The HM mathSpace® CD-ROM and Eduspace® for this text contain additional resources related to the concepts discussed in this chapter. 333202_0101.qxd 12/7/05 8:29 AM Page 4 4 a FIGURE 1.5  FIGURE 1.6 Chapter 1 Functions and Their Graphs a2 + b2 = c2 The Pythagorean Theorem and the Distance Formula The following famous theorem is used extensively throughout this course. c b (x, y ) 1 1 d Pythagorean Theorem For a right triangle with
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 triangle can be formed, as y1, x1. By the Pythagorean Theorem, and shown in Figure 1.6. The length of the vertical side of the triangle is and the length of the horizontal side is you can write d 2 x2 d x2 x12 y2 x12 y2 2 y2 x2 y12 y12 This result is the Distance Formula. y1 x1 x2 2. (x, y ) 2 1 (x, y ) 2 2 x1 x x2 x − x  1 2 The Distance Formula x1, y1 d The distance between the points 2. d x2 y1 x1 2 y2 and x2, y2 in the plane is Example 3 Finding a Distance Find the distance between the points 2, 1 and 3, 4. Algebraic Solution 2, 1 x1, y1 Let Formula. and x2, y2 3, 4. Then apply the Distance 2 x1 y1 2 y2 d x2 3 22 4 12 52 32 34 5.83 Distance Formula Substitute for x1, y1, x2, and y2. Simplify. Simplify. Use a calculator. So, the distance between the points is about 5.83 units. You can use the Pythagorean Theorem to check that the distance is correct. d2? 34 2? 32 52 32 52 34 34 Pythagorean Theorem Substitute for d. Distance checks. ✓ Now try Exercises 31(a) and (b). Graphical Solution Use centimeter graph paper to plot the points A2, 1 B3, 4. Carefully sketch the line and A B. to Then use a centimeter segment from ruler to measure the length of the segment. cm 1 2 3 4 5 6 7 FIGURE 1.7 The line segment measures about 5.8 centimeters, as shown in Figure 1.7. So, the distance between the points is about 5.8 units. 333202_0101.
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 lengths of the three sides as follows. 5 2 2 7 12 9 36 45 4 2 2 0 12 4 1 5 5 4 2 7 02 1 49 50 d1 d2 d3 1 2 3 4 5 6 7 FIGURE 1.8 Because d1 2 d2 2 45 5 50 d3 2 you can conclude by the Pythagorean Theorem that the triangle must be a right triangle. Now try Exercise 41. The Midpoint Formula To find the midpoint of the line segment that joins two points in a coordinate plane, you can simply find the average values of the respective coordinates of the two endpoints using the Midpoint Formula. The Midpoint Formula The midpoint of the line segment joining the points given by the Midpoint Formula x1, y1 and x2, y2 is Midpoint x1 x2 2 y1, y2 2. For a proof of the Midpoint Formula, see Proofs in Mathematics on page 124. Example 5 Finding a Line Segment’s Midpoint Find the midpoint of the line segment joining the points 5, 3 and 9, 3. y (9, 3) (2, 0) 3 6 9 x Midpoint 6 3 −3 −6 −6 −3 − − ( 5, 3) FIGURE 1.9 Solution x1, y1 Let Midpoint x1 5, 3 9, 3. x2, y2 and y2 y1 2 3 3 2, x2, 2 5 9 2 2, 0 Simplify. Midpoint Formula Substitute for x1, y1, x2, and y2. The midpoint of the line segment is 2, 0, as shown in Figure 1.9. Now try Exercise 31(c). 333202_0101.qxd 12/7/05 8:29 AM Page 6 6 Chapter 1 Functions and Their Graphs Applications Example 6 Finding the Length of a Pass Football Pass (40, 28) During the third quarter of the 2004 Sugar Bowl, the
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 between the points 40, 28 35 30 25 20 15 10 20, 5) 5 10 15 20 25 30 35 40 Distance (in yards) FIGURE 1.10 e u n e v e R 26 25 24 23 22 21 20 ) ( FedEx Annual Revenue (2004, 24.7) (2003, 22.65) Midpoint (2002, 20.6) 2002 2003 2004 Year x1 2 y1 2 y2 20, 5. and d x2 40 20 2 28 52 400 529 929 30 Distance Formula Substitute for x1, y1, x2, and y2. Simplify. Simplify. Use a calculator. So, the pass was about 30 yards long. Now try Exercise 47. In Example 6, the scale along the goal line does not normally appear on a football field. However, when you use coordinate geometry to solve real-life problems, you are free to place the coordinate system in any way that is convenient for the solution of the problem. Example 7 Estimating Annual Revenue FedEx Corporation had annual revenues of $20.6 billion in 2002 and $24.7 billion in 2004. Without knowing any additional information, what would you estimate the 2003 revenue to have been? (Source: FedEx Corp.) Solution One solution to the problem is to assume that revenue followed a linear pattern. With this assumption, you can estimate the 2003 revenue by finding the midpoint of the line segment connecting the points 2004, 24.7. 2002, 20.6 and Midpoint x1, y1 y2 x2 2 2 2002 2004 2, Midpoint Formula 20.6 24.7 2 Substitute for x1, y1, x2, and y2. 2003, 22.65 Simplify. So, you would estimate the 2003 revenue to have been about $22.65 billion, as shown in Figure 1.11. (The actual 2003 revenue was $22.5 billion.) FIGURE 1.11 Now try Exercise 49. 333202_0101.qxd 12/7/05 8:30 AM Page Much of computer graphics, including this computer-generated goldfish tessellation, consists of transformations
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 triangle three units to the right and two units upward and find the vertices of the shifted triangle, as shown in Figure 1.13. 1, 2, 1, 4, and y 5 4 − ( 1, 2) (2, 3) y 5 4 3 2 1 −2 −2 −1 1 2 3 5 6 7 x −2 −3 −4 (1, 4)− −2 −3 −4 FIGURE 1.12 FIGURE 1.13 Solution To shift the vertices three units to the right, add 3 to each of the -coordinates. To shift the vertices two units upward, add 2 to each of the -coordinates. x y Original Point 1, 2 1, 4 2, 3 Translated Point 1 3, 2 2 2, 4 1 3, 4 2 4, 2 2 3, 3 2 5, 5 Now try Exercise 51. The figures provided with Example 8 were not really essential to the solution. Nevertheless, it is strongly recommended that you develop the habit of including sketches with your solutions—even if they are not required. The following geometric formulas are used at various times throughout this course. For your convenience, these formulas along with several others are also provided on the inside back cover of this text. Common Formulas for Area A, Perimeter P, Circumference C, and Volume V Rectangle Rectangular Solid Circular Cylinder Circle A lw P 2l 2w A r2 C 2r Triangle A 1 2 P a b c bh V lwh V r 2h h l w r h Sphere V 4 3 r3 r w l r a h c b 333202_0101.qxd 12/7/05 8:30 AM Page 8 8 Chapter 1 Functions and Their Graphs Example 9 Using a Geometric Formula 4 cm h FIGURE 1.14 A cylindrical can has a volume of 200 cubic centimeters 4 centimeters (cm), as shown in Figure 1.14. Find the height of the can. and a radius of cm3 Solution The formula for the volume of a cylinder is can, solve for h V r 2 h. V r2h
. 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 each side of the equal sign will be approximately equal to each other. h V r2 h 200? 423.98 200 200.06 Write original equation. Substitute 200 for V, 4 for r, and 3.98 for h. Solution checks. ✓ You can also use unit analysis to check that your answer is reasonable. 200 cm3 16 cm2 3.98 cm Now try Exercise 63. W RITING ABOUT MATHEMATICS Extending the Example Example 8 shows how to translate points in a coordinate plane. Write a short paragraph describing how each of the following transformed points is related to the original point. Original Point x, y x, y x, y Transformed Point x, y x, y x, y 333202_0101.qxd 12/7/05 8:30 AM Page 9 1.1 Exercises The HM mathSpace® CD-ROM and Eduspace® for this text contain step-by-step solutions to all odd-numbered exercises. They also provide Tutorial Exercises for additional help. Section 1.1 Rectangular Coordinates 9 VOCABULARY CHECK 1. Match each term with its definition. (a) x -axis (b) y -axis (c) origin (d) quadrants (e) x -coordinate (f) y -coordinate (i) point of intersection of vertical axis and horizontal axis (ii) directed distance from the x-axis (iii) directed distance from the y-axis (iv) four regions of the coordinate plane (v) horizontal real number line (vi) vertical real number line In Exercises 2–4, fill in the blanks. 2. An ordered pair of real numbers can be represented in a plane called the rectangular coordinate system or the ________ plane. 3. The ________ ________ is a result derived from the Pythagorean Theorem. 4. Finding the average values of the representative coordinates of the two endpoints of a line segment in a coordinate plane is also known as using the ________ ________. PREREQUISITE SKILLS REVIEW
: 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 11–20, 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 −6 −4 B −2 −2 −4 x 2 4 C D −6 −4 −2 x 2 B A −2 −4 In Exercises 3–6, plot the points in the Cartesian plane. 3. 4. 5. 6. 4, 2, 0, 0, 3, 8, 1, 1 3 3, 1, 0.5, 1,, 4, 3, 3 3, 6, 0, 5, 1, 4 2, 4, 1, 1 5, 6, 3, 4, 2, 2.5 3, 3 2 4 In Exercises 7–10, find the coordinates of the point. 7. The point is located three units to the left of the -axis and y four units above the -axis. x 8. The point is located eight units below the -axis and four x units to the right of the -axis. y 9. The point is located five units below the -axis and the x coordinates of the point are equal. 10. The point is on the -axis and 12 units to the left of the x y -axis. and y < 0 and y > 0 and y > 0 11. 13. 15. 17 19. xy > 0 12. x < 0 and 14. x > 2 and y < 0 y 3 16. 18. 20. x > 4 x > 0 xy < 0 and y < 0 In Exercises 21 and 22, sketch a scatter plot of the data shown in the table. 21. Number of Stores The table shows the number of from 1996 through 2003. y x Wal-Mart stores for each year (Source: Wal-Mart Stores, Inc.) Year, x Number of stores, y 1996 1997 1998 1999 2000 2001 2002 2003 3054 3406 3599 3985 4189 4414 4688 4906 333202_0101.qxd 12/7/05 8:30 AM Page 10 10 Chapter 1 Functions and Their Graphs 22. Meteorology The table shows the lowest temperature on (in degrees Fahrenheit) in Dul
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 31–40, (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, 0 7, 4, 2, 8 2, 10, 10, 2 31. 33. 35. 37. 38. 39. 40. 1, 1, 9, 7 4, 10, 4, 5 1, 2, 5, 4 2, 1, 5 1 2, 4, 1 1 6, 1 3, 1 6.2, 5.4, 3.7, 1.8 16.8, 12.3, 5.6, 4.9 3 3 2 In Exercises 41 and 42, show that the points form the vertices of the indicated polygon. 41. Right triangle: 4, 0, 2, 1, 1, 5 42. Isosceles triangle: 1, 3, 3, 2, 2, 4 43. A line segment has x1, y1 as one endpoint and its midpoint. Find the other endpoint segment in terms of and x1, y1, xm, ym. xm, ym as of the line x2, y2 23. In Exercises 23–26, find the distance between the points. (Note: In each case, the two points lie on the same horizontal or vertical line.) 6, 3, 6, 5 1, 4, 8, 4 3, 1, 2, 1 3, 4, 3, 6 24. 25. 26. In Exercises 27–30, (a) find the length of each side of the right triangle, and (b) show that these lengths satisfy the Pythagorean Theorem. 27. y 28. y 5 4 3 2 1 (4, 5) (0, 2) (4, 2) 1 2 3 4 5 29. y 6 4 2 (9, 4) (9, 1) (−1, 1) 6 8 x x 8 4 (1, 0) (13, 5) x 4 8 (13, 0) 30. y (1, 5) 4 2 −2 (5, −2) x 6 (1
, −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 into four parts. x1, y1 46. Use the result of Exercise 45 to find the points that divide the line segment joining the given points into four equal parts. (a) 1, 2, 4, 1 (b) 2, 3, 0, 0 47. Sports A soccer player passes the ball from a point that is 18 yards from the endline and 12 yards from the sideline. The pass is received by a teammate who is 42 yards from the same endline and 50 yards from the same sideline, as shown in the figure. How long is the pass? 50 40 30 20 10 ) 50, 42) (12, 18) 10 20 30 40 50 Distance (in yards) 60 48. Flying Distance An airplane flies from Naples, Italy in a straight line to Rome, Italy, which is 120 kilometers north and 150 kilometers west of Naples. How far does the plane fly? 333202_0101.qxd 12/7/05 8:30 AM Page 11 Sales In Exercises 49 and 50, use the Midpoint Formula to estimate the sales of Big Lots, Inc. and Dollar Tree Stores, Inc. in 2002, given the sales in 2001 and 2003. Assume that the sales followed a linear pattern. (Source: Big Lots, Inc.; Dollar Tree Stores, Inc.) 55. Approximate the highest price of a pound of butter shown in the graph. When did this occur? 56. Approximate the percent change in the price of butter from the price in 1995 to the highest price shown in the graph. Section 1.1 Rectangular Coordinates 11 49. Big Lots Year 2001 2003 Sales (in millions) $3433 $4174 50. Dollar Tree Year 2001 2003 Sales (in millions) $1987 $2800 In Exercises 51–54, the polygon is shifted to a new position in the plane. Find the coordinates of the vertices of the polygon in its new position. 51. y 4 − − ( 1, 1) −4 − units (2, 3)− − − ( 2, 4
) 52. y − ( 3, 6, 3) 6 units − ( 3, 0) − ( 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 graph below, which shows the average retail price of 1 pound of butter from 1995 to 2003. (Source: U.S. Bureau of Labor Statistics.50 3.25 3.00 2.75 2.50 2.25 2.00 1.75 1.50 1995 1997 1999 Year 2001 2003 Advertising In Exercises 57 and 58, use the graph below, which shows the cost of a 30-second television spot (in thousands of dollars) during the Super Bowl from 1989 to 2003. (Source: USA Today Research and CNN ( 2400 2200 2000 1800 1600 1400 1200 1000 800 600 1989 1991 1993 1995 1997 1999 2001 2003 Year 57. Approximate the percent increase in the cost of a 30-second spot from Super Bowl XXIII in 1989 to Super Bowl XXXV in 2001. 58. Estimate the percent increase in the cost of a 30-second spot (a) from Super Bowl XXIII in 1989 to Super Bowl XXVII in 1993 and (b) from Super Bowl XXVII in 1993 to Super Bowl XXXVII in 2003. 59. Music The graph shows the numbers of recording artists who were elected to the Rock and Roll Hall of Fame from 1986 to 2004. 16 14 12 10 1987 1989 1991 1993 1995 1997 1999 2001 2003 Year (a) Describe any trends in the data. From these trends, predict the number of artists elected in 2008. (b) Why do you think the numbers elected in 1986 and 1987 were greater in other years? 333202_0101.qxd 12/7/05 8:30 AM Page 12 12 Chapter 1 Functions and Their Graphs Model It 60. Labor Force Use the graph below, which shows the minimum wage in the United States (in dollars) from 1950 to 2004. (Source: U.S. Department of Labor 1950 1960 1970 1980 1990 2000 Year (a) Which decade shows the greatest increase in minimum wage? (b) Approximate the percent increases in the minimum wage from
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 Formula to estimate the sales in 1998, 2000, and 2002. Assume that the sales followed a linear pattern. (Source: The Coca-Cola Company) 62. Data Analysis: Exam Scores The table shows the mathand the final examination in an algebra course for a sample of 10 students. ematics entrance test scores scores x y x y x y 22 53 48 90 29 74 53 76 35 57 58 93 40 66 65 83 44 79 76 99 64. Length of a Tank The diameter of a cylindrical propane gas tank is 4 feet. The total volume of the tank is 603.2 cubic feet. Find the length of the tank. 65. Geometry A “Slow Moving Vehicle” sign has the shape of an equilateral triangle. The sign has a perimeter of 129 centimeters. Find the length of each side of the sign. Find the area of the sign. 66. Geometry The radius of a traffic cone is 14 centimeters and the lateral surface of the cone is 1617 square centimeters. Find the height of the cone. 67. Dimensions of a Room A room is 1.5 times as long as it is wide, and its perimeter is 25 meters. (a) Draw a diagram that represents the problem. Identify the length as and the width as l w. (b) Write w in terms of w. perimeter in terms of l and write an equation for the (c) Find the dimensions of the room. 68. Dimensions of a Container The width of a rectangular storage container is 1.25 times its height. The length of the container is 16 inches and the volume of the container is 2000 cubic inches. (a) Draw a diagram that represents the problem. Label the height, width, and length accordingly. (b) Write w volume in terms of in terms of h. h and write an equation for the (c) Find the dimensions of the container. 69. Data Analysis: Mail The table shows the number of pieces of mail handled (in billions) by the U.S. Postal Service for each year from 1996 through 2003. (Source: U.S. Postal
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. Volume of a Billiard Ball A billiard ball has a volume of 5.96 cubic inches. Find the radius of a billiard ball. (a) Sketch a scatter plot of the data. (b) Approximate the year in which there was the greatest decrease in the number of pieces of mail handled. (c) Why do you think the number of pieces of mail handled decreased? 333202_0101.qxd 12/7/05 8:30 AM Page 13 70. Data Analysis: Athletics The table shows the numbers of men’s M and women’s W college basketball teams for each year (Source: National Collegiate Athletic Association) from 1994 through 2003. x Year, x Men’s teams, M Women’s teams, W 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 858 868 866 865 895 926 932 937 936 967 859 864 874 879 911 940 956 958 975 1009 (a) Sketch scatter plots of these two sets of data on the same set of coordinate axes. (b) Find the year in which the numbers of men’s and women’s teams were nearly equal. (c) Find the year in which the difference between the numbers of men’s and women’s teams was the greatest. What was this difference? 2, 1, 3, 5, 71. Make a Conjecture Plot the points and 7, 3 on a rectangular coordinate system. Then change the sign of the -coordinate of each point and plot the three new points on the same rectangular coordinate system. Make a conjecture about the location of a point when each of the following occurs. x (a) The sign of the -coordinate is changed. x (b) The sign of the -coordinate is changed. y (c) The signs of both the - and -coordinates are changed. y x 72. Collinear Points Three or more points are collinear if they all lie on the same line. Use the steps below to
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 points in the Cartesian plane. Do all the points of either set appear to lie on the same line? (c) Compare your conclusions from part (a) with the conclusions you made from the graphs in part (b). Make a general statement about how to use the Distance Formula to determine collinearity. Section 1.1 Rectangular Coordinates 13 Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 73 and 74, determine whether 73. In order to divide a line segment into 16 equal parts, you 74. The points would have to use the Midpoint Formula 16 times. 2, 11, 8, 4, vertices of an isosceles triangle. 5, 1 and represent the 75. Think About It When plotting points on the rectangular coordinate system, is it true that the scales on the - and y -axes must be the same? Explain. x 76. Proof Prove that the diagonals of the parallelogram in the figure intersect at their midpoints. y y )x, y ) 0 0 x (0, 0) a (, 0) x FIGURE FOR 76 FIGURE FOR 77–80 In Exercises 77–80, use the plot of the point in the figure. Match the transformation of the point with the correct plot. [The plots are labeled (a), (b), (c), and (d).] x0, y0 (a) (c) y y y y (b) (d) x x x x 77. 79. x0, y0 x0, 1 2 y0 Skills Review 78. 80. 2x0, y0 x0, y0 In Exercises 81– 88, solve the equation or inequality. 81. 83. 85. 87. 2x 1 7x 4 x2 4x 7 0 3x 1 < 22 x x 18 < 4 1 82. 84. 3x 2 5 1 6x 2x2 3x 8
0 3x 8 ≥ 1 86. 2 88. 2x 15 ≥ 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 • Sketch graphs of equations. • Find x- and y-intercepts of graphs of equations. • Use symmetry to sketch graphs of equations. • Find equations of and sketch graphs of circles. • Use graphs of equations in solving real-life problems. Why you should learn it The graph of an equation can help you see relationships between real-life quantities. For example, in Exercise 75 on page 24, a graph can be used to estimate the life expectancies of children who are born in the years 2005 and 2010. The Graph of an Equation In Section 1.1, you used a coordinate system to represent graphically the relationship between two quantities. There, the graphical picture consisted of a collection of points in a coordinate plane. Frequently, a relationship between two quantities is expressed as an equay 7 3x y. and An if the is substituted for For instance, is a solution or solution point of an equation in and b is substituted for and tion in two variables. For instance, ordered pair a equation is true when 1, 4 y 7 3x is a solution of x is an equation in x y. is a true statement. x because 4 7 31 a, b y In this section you will review some basic procedures for sketching the graph of an equation in two variables. The graph of an equation is the set of all points that are solutions of the equation. Example 1 Determining Solutions Determine whether (a) y 10x 7. 2, 13 and (b) 1, 3 are solutions of the equation Solution a. y 10x 7 13? 13 13 102 7 Write original equation. Substitute 2 for x and 13 for y. is a solution. ✓ 2, 13 Because the substitution does satisfy the original equation, you can conclude that the ordered pair y 10x 7 is a solution of the original equation. Write original equation. 2, 13 b. 101 7 3? 3 17 Substitute 1, 3 1 for x and 3 for y. is not a solution. Because the substitution does not satisfy the original equation, you can conis not a solution of the original equation. clude that the ordered pair 1, 3 Now try Exercise 1. © John Griffin/The Image Works point
-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 these points on a rectangular coordinate system. 4. Connect the points with a smooth curve or line. 333202_0102.qxd 12/7/05 8:31 AM Page 15 Section 1.2 Graphs of Equations 15 Example 2 Sketching the Graph of an Equation Sketch the graph of y 7 3x. Solution Because the equation is already solved for y, construct a table of values that x 1, consists of several solution points of the equation. For instance, when y 7 31 10 which implies that 1, 10 is a solution point of the graph 3x 10 7 4 1 2 5 x, y 1, 10 0, 7 1, 4 2, 1 3, 2 4, 5 From the table, it follows that 1, 10, 0, 7, 1, 4, 2, 1, 3, 2, and 4, 5 are solution points of the equation. After plotting these points, you can see that they appear to lie on a line, as shown in Figure 1.15. The graph of the equation is the line that passes through the six plotted points. y (− 1, 10) 8 6 4 2 (0, 7) (1, 4) (2, 1) x 2 6 4 (3, − 2) 8 10 (4, − 5) −4 −2 −2 −4 −6 FIGURE 1.15 Now try Exercise 5. 333202_0102.qxd 12/7/05 8:31 AM Page 16 16 Chapter 1 Function and Their Graphs One of your goals in this course is to learn to classify the basic shape of a graph from its equation. For instance, you will learn that the linear equation in Example 2 has the form y mx b and its graph is a line. Similarly, the quadratic equation in Example 3 has the form y ax2 bx c and its graph is a parabola. Example 3 Sketching the Graph of an Equation Sketch the graph of y x 2 2. Solution Because the equation is already solved for values. y, begin by constructing a table of x y x2 2 x, y 2 2 2,
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 (−2, 2) −4 −2 (−1, −1) FIGURE 1.16 (3, 7) (2, 2) 2 (1, −1) 4 (0, −2) x (−2, 2) −4 −2 (−1, −1) FIGURE 1.17 y 6 4 2 (3, 7) y = x2 − 2 (2, 2) x 2 (1, −1) 4 (0, −2) Now try Exercise 7. The point-plotting method demonstrated in Examples 2 and 3 is easy to use, but it has some shortcomings. With too few solution points, you can misrepresent the graph of an equation. For instance, if only the four points 2, 2, 1, 1, 1, 1, and 2, 2 in Figure 1.16 were plotted, any one of the three graphs in Figure 1.18 would be reasonable2 2 x −2 2 x −2 2 x FIGURE 1.18 333202_0102.qxd 12/7/05 8:31 AM Page 17 Section 1.2 Graphs of Equations 17 Te c h n o l o g y To graph an equation involving x and y on a graphing utility, use the following procedure. 1. Rewrite the equation so that y is isolated on the left side. 2. Enter the equation into the graphing utility. 3. Determine a viewing window that shows all important features of the graph. 4. Graph the equation. For more extensive instructions on how to use a graphing utility to graph an equation, see the Graphing Technology Guide on the text website at college.hmco.com. Intercepts of a Graph y It is often easy to determine the solution points that have zero as either the x -coordinate or the -coordinate. These points are called intercepts because they are the points at which the graph intersects or touches the - or -axis. It is possible for a graph to have no intercepts, one intercept, or several intercepts, as shown in Figure 1.19. Note that an and a y -intercept can be written as the
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 the ordered pair 0, y. 0, b a, 0 x, 0 x x y x Finding Intercepts x 1. To find -intercepts, let be zero and solve the equation for y x. 2. To find -intercepts, let be zero and solve the equation for y. x y Example 4 Finding x- and y-Intercepts Find the - and -intercepts of the graph of y x y x3 4x. y x No x-intercepts; one y-intercept y x Three x-intercepts; one y-intercept y x One x-intercept; two y-intercepts y x No intercepts FIGURE 1.19 y = x − 4x 3 y 4 Then Solution y 0. Let 0 x3 4x xx2 4 x 0 has solutions and x ±2. 0, 0, 2, 0, 2, 0 (−2, 0) − 4 (0, 0) (2, 0) 4 x Let -intercepts: x x 0. Then y 03 40 −2 − 4 has one solution, y -intercept: y 0. 0, 0 See Figure 1.20. FIGURE 1.20 Now try Exercise 11. 333202_0102.qxd 12/7/05 8:31 AM Page 18 18 Chapter 1 Function and Their Graphs Symmetry Graphs of equations can have symmetry with respect to one of the coordinate axes or with respect to the origin. Symmetry with respect to the -axis means that if the Cartesian plane were folded along the -axis, the portion of the graph above the -axis would coincide with the portion below the -axis. Symmetry with y respect to the -axis or the origin can be described in a similar manner, as shown in Figure 1.21. x x x x y y (−x, y) (x, y) (x, y) x (x, −y) y (x, y) x x (−x, −y) x-axis symmetry FIGURE 1.21 y-axis symmetry Origin symmetry
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 the graph. the graph, x, y x, y is on 2. A graph is symmetric with respect to the y-axis if, whenever is also on the graph. the graph, x, y x, y is on 3. A graph is symmetric with respect to the origin if, whenever is also on the graph. x, y the graph, x, y is on Example 5 Testing for Symmetry The graph of x, y point below confirms that the graph is symmetric with respect to the -axis. is symmetric with respect to the -axis because the (See Figure 1.22.) The table y x2 2 is also on the graph of y x2 2. y y x y x, y 3 2 7 3, 7 2 2, 2 1 1 1, 1 1 1 1, 1 2 3 2 2, 2 7 3, 7 Now try Exercise 23. (−3, 7) (−2, 23, 7) (2, 2) x −4 −3 −2 (−1, −1) −3 3 4 5 2 (1, −1) 2 y = x − 2 FIGURE 1.22 y-axis symmetry 333202_0102.qxd 12/7/05 8:31 AM Page 19 Section 1.2 Graphs of Equations 19 Algebraic Tests for Symmetry 1. The graph of an equation is symmetric with respect to the -axis if x replacing with y y yields an equivalent equation. 2. The graph of an equation is symmetric with respect to the -axis if y replacing with x x yields an equivalent equation. 3. The graph of an equation is symmetric with respect to the origin if yields an equivalent equation. replacing with y and with x y x Example 6 Using Symmetry as a Sketching Aid (5, 2) Use symmetry to sketch the graph of x y 2 1. Solution Of the three tests for symmetry, the only one that is satisfied is the test for -axis symmetry because. So, the graph is symmetric with respect to the -axis. Using symmetry, you only need to find the solution points above the -axis
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 graph is not symmetric with respect to either axis or to the origin. The absolute value sign indiis always nonnegative. Create a table of values and plot the points as cates that shown in Figure 1.24. From the table, you can see that So, the y -intercept is y 1. So, the -intercept is x 0 x Similarly, x 1. 0, 1. 1, 0. y 0 when when y x y x 1 x, 3 2 1, 2 1 0, 1 0 1, 0 1 2, 1 2 3, 2 3 4, 3 Now try Exercise 41. y 2 1 (1, 0) x − y = 1 2 (2, 1) 2 3 4 5 x −1 −2 FIGURE 1.23 Notice that when creating the table in Example 6, it is easier to choose y-values and then find the corresponding x-values of the ordered pairs. y 6 5 4 3 2 (−2, 3) (−1, 2) y  −  x= 1 (4, 3) (3, 2) (0, 1) (2, 1) −3 −2 −1 (1, 0) 2 3 4 5 x −2 FIGURE 1.24 333202_0102.qxd 12/7/05 8:31 AM Page 20 20 y Chapter 1 Function and Their Graphs Throughout this course, you will learn to recognize several types of graphs from their equations. For instance, you will learn to recognize that the graph of a second-degree equation of the form y ax 2 bx c Center: (h, k) is a parabola (see Example 3). The graph of a circle is also easy to recognize. Radius: r Circles Point on circle: (x, y) x Consider the circle shown in Figure 1.25. A point if its distance from the center x, y is r. By the Distance Formula, h, k is on the circle if and only x h2 y k2
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 Example 8, it may be helpful to rewrite the quantities and using subtraction. x 12 y 22, x 12 x 12, y 22 y 22 h 1 and k 2. So, From this result, you can see that the standard form of the equation of a h, k 0, 0, circle with its center at the origin, is simply x2 y 2 r 2. Circle with center at origin Example 8 Finding the Equation of a Circle 1, 2, The point 1.26. Write the standard form of the equation of this circle. lies on a circle whose center is at 3, 4 as shown in Figure y 6 4 (−1, 2) (3, 4) −6 −2 2 4 x −2 −4 Solution The radius of the circle is the distance between r x h2 y k2 3 12 4 22 42 22 16 4 20 h, k 1, 2 and 1, 2 and 3, 4. Distance Formula Substitute for x, y, h, and k. Simplify. Simplify. Radius Using r 20, the equation of the circle is x h2 y k2 r2 x 12 y 22 202 x 12 y 22 20. Equation of circle Substitute for h, k, and r. Standard form FIGURE 1.26 Now try Exercise 61. 333202_0102.qxd 12/7/05 8:31 AM Page 21 You should develop the habit of using at least two approaches to solve every problem. This helps build your intuition and helps you check that your answer is reasonable. Section 1.2 Graphs of Equations 21 Application In this course, you will learn that there are many ways to approach a problem. Three common approaches are illustrated in Example 9. A Numerical Approach: Construct and use a table. A Graphical Approach: Draw and use a graph. An Algebraic Approach: Use the rules of algebra. Example 9 Recommended Weight The median recommended weight y (in pounds) for men of medium frame who are 25 to 59 years old can be approximated by the mathematical model y 0.
073x 2 6.99x 289.0, 62 ≤ x ≤ 76 is the man’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, 68, 70, 72, 74, and 76 inches. b. Use the table of values to sketch a graph of the model. Then use the graph to estimate graphically the median recommended weight for a man whose height is 71 inches. c. Use the model to confirm algebraically the estimate you found in part (b). Solution a. You can use a calculator to complete the table, as shown at the left. b. The table of values can be used to sketch the graph of the equation, as shown in Figure 1.27. From the graph, you can estimate that a height of 71 inches corresponds to a weight of about 161 pounds. Recommended Weight 180 170 160 150 140 130 FIGURE 1.27 62 64 66 68 70 72 74 76 Height (in inches) x c. To confirm algebraically the estimate found in part (b), you can substitute 71 for x in the model. y 0.073(71)2 6.99(71) 289.0 160.70 So, the graphical estimate of 161 pounds is fairly good. Now try Exercise 75. 333202_0102.qxd 12/7/05 8:31 AM Page 22 22 Chapter 1 Function and Their Graphs 1.2 Exercises VOCABULARY CHECK: Fill in the blanks. a, b 1. An ordered pair b is substituted for and y. x is a ________ of an equation in and x y if the equation is true when a is substituted for 2. The set of all solution points of an equation is the ________ of the equation. 3. The points at which a graph intersects or touches an axis are called the ________ of the graph. x, y 4. A graph is symmetric with respect to the ________ if, whenever is on the graph, x, y is also on the graph. 5. The equation x h2 y k2 r2 is the standard form of the equation of a ________ with center ________ and
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 1– 4, determine whether each point lies on the graph of the equation. 8. y 5 x2 Equation y x 4 y x 2 3x 2 y 4 x 2 y 1 3x3 2x2 1. 2. 3. 4. Points (a) (a) (a) (a) 0, 2 2, 0 1, 5 2, 16 3 (b) (b) (b) (b) 5, 3 2, 8 6, 0 3, 9 In Exercises 5–8, complete the table. Use the resulting solution points to sketch the graph of the equation. 5. y 2x, y 6. y 3 4x, y 7. y x2 3x 1 0 1 2 3 x y x, y In Exercises 9–20, find the x- and y-intercepts of the graph of the equation. 9. y 16 4x2 10. y x 32 y 20 8 4 y 10 8 6 −1 1 3 x −6 −4 −2 2 4 x 12. 11. 13. 14. 15. y 5x 6 y 8 3x y x 4 y 2x 1 y 3x 7 y x 10 y 2x3 4x2 y x4 25 y2 6 x 19. 20. y2 x 1 17. 16. 18. 333202_0102.qxd 12/7/05 8:31 AM Page 23 In Exercises 21–24, assume that the graph has the indicated type of symmetry. Sketch the complete graph of the equation. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. 21. 23. y 4 2 −4 x 2 4 22. y 4 2 −2 −4 2 4 6 8 y -axis symmetry x -axis symmetry y 4 2 24. y 4 2 −4 −2 2 4 x −4 −2 2 4 −2 −4 −2 −4 Origin symmetry y -axis symmetry x x In Exercises 25–32, use the algebraic tests to check for symmetry with respect to both axes and the origin. 25. 27. 29. 31. x
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 33– 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 45– 56, use a graphing utility to graph the equation. Use a standard setting. Approximate any intercepts. 45. 47. 49. y 3 1 2x y x2 4x 3 y 2x x 1 51. y 3x 46. 48. 50. 52. y 2 3x 1 y x2 x 2 y 4 x2 1 y 3x 1 Section 1.2 Graphs of Equations 23 53. 55. y xx 6 y x 3 54. 56. y 6 xx y 2 x In Exercises 57–64, write the standard form of the equation of the circle with the given characteristics. radius: 4 58. Center: 0, 0; radius: 5 57. Center: 59. Center: 60. Center: 61. Center: 62. Center: 0, 0; 2, 1; 7, 4; 1, 2; 3, 2; radius: 4 radius: 7 solution point: solution point: 0, 0 1, 1 63. Endpoints of a diameter: 64. Endpoints of a diameter: 0, 0, 6, 8 4, 1, 4, 1 67. 65. In Exercises 65–70, find the center and radius of the circle, and sketch its graph. x2 y 2 25 x 12 y 32 9 x2 y 12 22 y 32 16 9 x2 y 2 16 70. 69. 66. 68. 2 2 71. Depreciation A manufacturing plant purchases a new y by Sketch the graph of the molding machine for $225,000. The depreciated value (reduced y 225,000 20,000t, equation. t 0 ≤ t ≤ 8. value) given years after is 72. Consumerism You purchase a jet ski for $8100. The is given by Sketch the graph of the 0 ≤ t ≤ 6. years after y t depreciated value y 8100 929t, equation.
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 A x520 3 and its area is x. y 520 3 x (c) Use a graphing utility to graph the area equation. Be sure to adjust your window settings. (d) From the graph in part (c), estimate the dimensions of the rectangle that yield a maximum area. (e) Use your school’s library, the Internet, or some other reference source to find the actual dimensions and area of a regulation NFL playing field and compare your findings with the results of part (d). The symbol indicates an exercise or a part of an exercise in which you are instructed to use a graphing utility. 333202_0102.qxd 12/7/05 8:31 AM Page 24 24 Chapter 1 Function and Their Graphs 74. Geometry A soccer playing field of length and width x y has a perimeter of 360 meters. (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 and its area is A x180 x. w 180 x (c) Use a graphing utility to graph the area equation. Be sure to adjust your window settings. (d) From the graph in part (c), estimate the dimensions of the rectangle that yield a maximum area. (e) Use your school’s library, the Internet, or some other reference source to find the actual dimensions and area of a regulation Major League Soccer field and compare your findings with the results of part(d). Model It 75. Population Statistics The table shows the life expectancies of a child (at birth) in the United States for selected years from 1920 to 2000. (Source: U.S. National Center for Health Statistics) 76. Electronics The resistance y (in ohms) of 1000 feet of solid copper wire at 68 degrees Fahrenheit can be approxiy 10,770 5 ≤ x ≤ 100 is the diameter of the wire in mils (0.001 inch). mated by the model 0.37, x2 x where (Source: American Wire Gage) (a) Complete the table. 5 10 20
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 diameter of the copper wire and the resistance? Year Life expectancy, y Synthesis 1920 1930 1940 1950 1960 1970 1980 1990 2000 54.1 59.7 62.9 68.2 69.7 70.8 73.7 75.4 77.0 A model for the life expectancy during this period is y 0.0025t2 0.574t 44.25, 20 ≤ t ≤ 100 y where in years, with represents the life expectancy and t corresponding to 1920. t 20 (a) Sketch a scatter plot of the data. is the time (b) Graph the model for the data and compare the scatter plot and the graph. (c) Determine the life expectancy in 1948 both graph- ically and algebraically. (d) Use the graph of the model to estimate the life expectancies of a child for the years 2005 and 2010. (e) Do you think this model can be used to predict the life expectancy of a child 50 years from now? Explain. True or False? the statement is true or false. Justify your answer. In Exercises 77 and 78, determine whether 77. A graph is symmetric with respect to the -axis if, when- x ever x, y is on the graph, x, y is also on the graph. 78. A graph of an equation can have more than one -intercept. y y 79. Think About It Suppose you correctly enter an expression for the variable on a graphing utility. However, no graph appears on the display when you graph the equation. Give a possible explanation and the steps you could take to remedy the problem. Illustrate your explanation with an example. 80. Think About It Find b if the graph of y ax 2 bx3 is symmetric with respect to (a) the y -axis and (b) the origin. (There are many correct answers.) and a Skills Review 81. Identify the terms: 9x5 4x3 7. 82. Rewrite the expression using exponential notation. (7 7 7 7) In Exercises 83–88, simplify the expression. 83
. 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 • Use slope to graph linear equations in two variables. • Find slopes of lines. • Write linear equations in two variables. • Use slope to identify parallel and perpendicular lines. • Use slope and linear equations in two variables to model and solve real-life problems. Why you should learn it Linear equations in two variables can be used to model and solve real-life problems. For instance, in Exercise 109 on page 37, you will use a linear equation to model student enrollment at the Pennsylvania State University. Courtesy of Pennsylvania State University Using Slope y mx b. The simplest mathematical model for relating two variables is the linear equation The equation is called linear because its graph is a in two variables line. (In mathematics, the term line means straight line.) By letting you can see that the line crosses the -axis at as shown in Figure 1.28. In other y words, the -intercept is y mx b The steepness or slope of the line is y 0, b. y b, x 0, m. Slope y-Intercept The slope of a nonvertical line is the number of units the line rises (or falls) vertically for each unit of horizontal change from left to right, as shown in Figure 1.28 and Figure 1.29. y y-intercept y = mx + b (0, b) 1 unit m units, m > 0 x y (0, b) 1 unit y-intercept m units, m < 0 y = mx + b x Positive slope, line rises. FIGURE 1.28 Negative slope, line falls. FIGURE 1.29 A linear equation that is written in the form y mx b is said to be written in slope-intercept form. The Slope-Intercept Form of the Equation of a Line The graph of the equation y mx b is a line whose slope is m and whose y- intercept is 0, b. Exploration Use a graphing utility to compare the slopes of the lines m 0.5, 1, 2, 1, 2, and obtain a true geometric perspective. What can you conclude about the
slope and the “rate” 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 is undefined. Once you have determined the slope and the -intercept of a line, it is a relatively simple matter to sketch its graph. In the next example, note that none of the lines is vertical. A vertical line has an equation of the form y x a. Vertical line The equation of a vertical line cannot be written in the form the slope of a vertical line is undefined, as indicated in Figure 1.30. y mx b because Example 1 Graphing a Linear Equation Sketch the graph of each linear equation. a. b. c. y 2x 1 y 2 x y 2 Solution a. Because m 2, shown in Figure 1.31. the b 1, Moreover, because the slope is the line rises two units for each unit the line moves to the right, as -intercept is 0, 1. y b. By writing this equation in the form y 0x 2, you can see that the and the slope is zero. A zero slope implies that the line is y -intercept is horizontal—that is, it doesn’t rise or fall, as shown in Figure 1.32. 0, 2 c. By writing this equation in slope-intercept form x y 2 y x 2 y 1x 2 Write original equation. Subtract x from each side. Write in slope-intercept form. Moreover, because the slope is the line falls one unit for each unit the line moves to the right, as -intercept is y 0, 2. you can see that the m 1, shown in Figure 1.33. y 5 4 3 2 y = 2x + 1 m = 2 (0, 10, 2− (0, 2 When m is positive, the line rises. FIGURE 1.31 When m is 0, the line is horizontal. FIGURE 1.32 When m is negative, the line falls. FIGURE 1.33 Now try Exercise 9. 333202_0103
.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 − y1 x 2 − 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 still find the slope of a line. For instance, suppose you want to find the slope of the line passing x1, y1, as shown in Figure 1.34. As you move through the points from left to right along this line, a change of units in the vertical direction corresponds to a change of units in the horizontal direction. x2, y2 y1 y2 and x2 x1 y2 y1 the change in y rise and the change in x run y2 x2 and x1 x2, y2 x2 x1 The ratio of through the points y1 to x1, y1 Slope change in y change in x represents the slope of the line that passes. rise run y2 x2 y1 x1 The Slope of a Line Passing Through Two Points x2, y2 The slope x1, y1 and m is of the nonvertical line through y1 x1 m where x1 y2 x2 x2.. x2, y2 When this formula is used for slope, the order of subtraction is important. and However, once you have done this, you must form the numer- Given two points on a line, you are free to label either one of them as the other as ator and denominator using the same order of subtraction. y1 x2 y1 x1 y2 x2 x1, y1 m m m y1 x1 y2 x2 y2 x1 Correct Correct Incorrect For instance, the slope of the line passing through the points be calculated as m 7 4 5 3 3 2 3, 4 and 5, 7 can or, reversing the subtraction order in both the numerator and denominator, as m 4 7 3 5 3 2 3 2. 333202_0103.qxd 12/7/05 8:33 AM Page 28 28 Chapter 1 Functions and Their Graphs Example 2 Finding the Slope of a Line Through Two Points Find the
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 Figure 1.36. c. The slope of the line passing through 0, 4 and 1, 1 is m 1 4 1 0 5 1 5. See Figure 1.37. d. The slope of the line passing through 3, 4 and 3, 1 is m 1 4 3 3 3 0. See Figure 1.38. Because division by 0 is undefined, the slope is undefined and the line is vertical. In Figures 1.35 to 1.38, note the relationships between slope and the orientation of the line. a. Positive slope: line rises from left to right b. Zero slope: line is horizontal c. Negative slope: line falls from left to right d. Undefined slope: line is vertical y 4 3 2 1 −1 (−2, 0) −2 −1 FIGURE 1.35 m = 1 5 (3, 1) (−1, 22, 2) 1 2 3 −2 −1 −1 FIGURE 1.36 x x y (0, 4) 4 3 2 1 m = −5 −1 −1 3 2 (1, −1) 4 x x y 4 3 2 1 Slope is undefined. (3, 4) (3, 1) −1 −1 1 2 4 FIGURE 1.37 FIGURE 1.38 Now try Exercise 21. 333202_0103.qxd 12/7/05 8:33 AM Page 29 Section 1.3 Linear Equations in Two Variables 29 Writing Linear Equations in Two Variables x1, y1 If then is a point on a line of slope m and x, y is any other point on the line, y y1 x x1 m. This equation, involving the variables and x y, can be rewritten in the form y y1 mx x1 which is the point-slope form of the equation of a line. Point-Slope
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. −2 −1 y 1 −1 −2 −3 −4 −5 FIGURE 1.39 y = 3x − 5 Example 3 Using the Point-Slope Form Find the slope-intercept form of the equation of the line that has a slope of 3 and passes through the point 1, 2. 1 3 4 3 x Solution Use the point-slope form with m 3 and x1, y1 1, 2. 1 (1, −2) y y1 mx x1 y 2 3x 1 y 2 3x 3 y 3x 5 Point-slope form Substitute for m, x1, and y1. Simplify. Write in slope-intercept form. The slope-intercept form of the equation of the line is this line is shown in Figure 1.39. y 3x 5. The graph of Now try Exercise 39. When you find an equation of the line that passes through two given points, you only need to substitute the coordinates of one of the points into the point-slope form. It does not matter which point you choose because both points will yield the same result. The point-slope form can be used to find an equation of the line passing To do this, first find the slope of the line through two points and. x2, y2 x1, y1 m y2 x2 y1 x1, x1 x2 and then use the point-slope form to obtain the equation y y1 y2 x2 y1 x1 x x1. Two-point form This is sometimes called the two-point form of the equation of a line. 333202_0103.qxd 12/7/05 8:33 AM Page 30 30 Chapter 1 Functions and Their Graphs in terms of Exploration m1 d1 d2 Find and m2, and respectively (see figure). Then use the Pythagorean Theorem to find a relationship and between m2. m1 y d1 (0, 0) (1, m1) x Parallel and Perpendicular Lines Slope can be used to decide whether two nonvertical lines in a plane are parallel, perpendicular, or neither. Parallel and Per
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 slope-intercept forms of the equations of the lines that pass through the 2x 3y 5. and are (a) parallel to and (b) perpendicular to the line point 2, 1 Solution By writing the equation of the given line in slope-intercept form 3 y = − x + 2 2 2x − 3y = 5 2x 3y 5 3y 2x 5 y 2 3x 5 3 Write original equation. Subtract 2x from each side. Write in slope-intercept form. x you can see that it has a slope of m 2 3, as shown in Figure 1.40. a. Any line parallel to the given line must also have a slope of So, the line that is parallel to the given line has the following equation. through 2 3. y 3 2 1 −1 1 4 5 (2, −1) 2 y = x − 3 7 3 FIGURE 1.40 Te c h n o l o g y On a graphing utility, lines will not appear to have the correct slope unless you use a viewing window that has a square setting. For instance, try graphing the lines in Example 4 using the standard 10 ≤ x ≤ 10 setting 10 ≤ y ≤ 10. viewing window with the square setting 6 ≤ y ≤ 6. the lines y 3 perpendicular? and On which setting do 3 x 5 and 3 appear to be Then reset the and 2, 1 y 1 2 x 2 3 3y 1 2x 2 3y 3 2x 4 y 2 3x 7 3 Write in point-slope form. Multiply each side by 3. Distributive Property Write in slope-intercept form. 3 3 because 2 2 that is perpendi- b. Any line perpendicular to the given line must have a slope of. 2, 1 2 3 So, the line through is the negative reciprocal of cular to the given line has the following equation. x 2 y 1 3 2 2y 1 3x 2 2y 2 3x 6 y 3 2x 2 Now try Exercise 69. Write in point-slope form.
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 Equations in Two Variables 31 Applications In real-life problems, the slope of a line can be interpreted as either a ratio or a rate. If the -axis and -axis have the same unit of measure, then the slope has no units and is a ratio. If the -axis and -axis have different units of measure, then the slope is a rate or rate of change. x y x y Example 5 Using Slope as a Ratio The maximum recommended slope of a wheelchair ramp is A business is installing a wheelchair ramp that rises 22 inches over a horizontal length of 24 feet. (Source: Americans with Disabilities Is the ramp steeper than recommended? Act Handbook) 1 12. Solution The horizontal length of the ramp is 24 feet or Figure 1.41. So, the slope of the ramp is 1224 288 inches, as shown in Slope vertical change horizontal change 22 in. 288 in. 0.076. Because 1 12 0.083, the slope of the ramp is not steeper than recommended. y 22 in. 24 ft x FIGURE 1.41 Now try Exercise 97. Example 6 Using Slope as a Rate of Change A kitchen appliance manufacturing company determines that the total cost in x dollars of producing units of a blender is C 25x 3500. Cost equation Describe the practical significance of the -intercept and slope of this line. y 0, 3500 Solution y The -intercept tells you that the cost of producing zero units is $3500. This is the fixed cost of production—it includes costs that must be paid regardless of the number of units produced. The slope of tells you that the cost of producing each unit is $25, as shown in Figure 1.42. Economists call the cost per unit the marginal cost. If the production increases by one unit, then the “margin,” or extra amount of cost, is $25. So, the cost increases at a rate of $25 per unit. m 25 Manufacturing C C = 25 + 3500 x Marginal cost: m = $25 Fixed cost: $3500 )
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 property that has a useful life of more than 1 year. Such costs must be depreciated (decreased in value) over the useful life of the property. If the same amount is depreciated each year, the procedure is called linear or straight-line depreciation. The book value is the difference between the original value and the total amount of depreciation accumulated to date. Example 7 Straight-Line Depreciation A college purchased exercise equipment worth $12,000 for the new campus fitness center. The equipment has a useful life of 8 years. The salvage value at the end of 8 years is $2000. Write a linear equation that describes the book value of the equipment each year. Solution V Let the initial value of the equipment by the data point value of the equipment by the data point represent the value of the equipment at the end of year You can represent and the salvage 0, 12,000 The slope of the line is 8, 2000. t. m 2000 12,000 8 0 $1250 which represents the annual depreciation in dollars per year. Using the pointslope form, you can write the equation of the line as follows. V 12,000 1250t 0 Write in point-slope form. V 1250t 12,000 Write in slope-intercept form. Useful Life of Equipment V The table shows the book value at the end of each year, and the graph of the equation is shown in Figure 1.43. 12,000 (0, 12,000) V − t = 1250 +12,000 Year, t Value 10,000 8,000 6,000 4,000 2,000 (8, 2000) 4 6 2 Number of years 8 t 10 FIGURE 1.43 Straight-line depreciation 0 1 2 3 4 5 6 7 8 12,000 10,750 9,500 8,250 7,000 5,750 4,500 3,250 2,000 Now try Exercise 107. In many real-life applications, the two data points that determine the line are often given in
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 linear equation that gives the sales per (Source: share in terms of the year. Then predict the sales per share for 2003. Starbucks Corporation) Starbucks Corporation (3, 9.97) (2, 8.47) (1, 6.97) y = 1.5t + 5.47 1 2 3 4 Year (1 ↔ 2001) t y 10 FIGURE 1.44 y Given points Estimated point Linear extrapolation FIGURE 1.45 y Given points Estimated point Linear interpolation FIGURE 1.46 x x represent 2001. Then the two given values are represented by the data 2, 8.47. The slope of the line through these points is Solution t 1 Let points 1, 6.97 and m 8.47 6.97 2 1 1.5. Using the point-slope form, you can find the equation that relates the sales per share and the year t y to be y 6.97 1.5t 1 Write in point-slope form. y 1.5t 5.47. Write in slope-intercept form. to According y 1.53 5.47 $9.97, tion is quite good—the actual sales per share in 2003 was $10.35.) in 2003 was as shown in Figure 1.44. (In this case, the predic- the sales per share this equation, Now try Exercise 109. The prediction method illustrated in Example 8 is called linear extrapolation. Note in Figure 1.45 that an extrapolated point does not lie between the given points. When the estimated point lies between two given points, as shown in Figure 1.46, the procedure is called linear interpolation. Because the slope of a vertical line is not defined, its equation cannot be written in slope-intercept form. However, every line has an equation that can be written in the general form Ax By C 0 General form A where can be represented by the general form x a 0. are not both zero. For instance, the vertical line given by and B x a Summary of Equations of Lines 1. General form: 2.
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 1–6, fill in the blanks. 1. The simplest mathematical model for relating two variables is the ________ equation in two variables y mx b. 2. For a line, the ratio of the change in y to the change in x is called the ________ of the line. 3. Two lines are ________ if and only if their slopes are equal. 4. Two lines are ________ if and only if their slopes are negative reciprocals of each other. 5. When the -axis and -axis have different units of measure, the slope can be interpreted as a ________. y x 6. The prediction method ________ ________ is the method used to estimate a point on a line that does not lie between the given points. 7. Match each equation of a line with its form. (a) (b) (c) (d) (e) Ax By C 0 x a y b y mx b y y1 mx x1 (i) Vertical line (ii) Slope-intercept form (iii) General form (iv) Point-slope form (v) Horizontal line PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1 and 2, identify the line that has each slope. 7. y 8. y 1. (a) (b) (c) is undefined. m 2 3 m m 2 y L1 L3 2. (a) (b) (c) m 0 m 3 4 m 1 y L1 L2 x L3 x L2 In Exercises 3 and 4, sketch the lines through the point with the indicated slopes on the same set of coordinate axes. Point 3. 4. 2, 3 4, 1 (a) 0 (b) 1 (a) 3 (b) 3 Slopes (c) 2 (d) 1
2 3 (d) Undefined (c) In Exercises 5–8, estimate the slope of the line. 5. y 6 In Exercises 9–20, 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 x 2 0 In Exercises 21–28, plot the points and find the slope of the line passing through the pair of points. 21. 23. 25. 27. 28. 3, 2, 1, 6 6, 1, 6, 4 11, 3 2, 1 2, 4 4.8, 3.1, 5.2, 1.6 1.75, 8.3, 3 3 2.25, 2.6 22. 24. 26. 2, 4, 4, 4 0, 10, 4, 0 7 4,1 8, 3, 5 4 4 333202_0103.qxd 12/7/05 8:33 AM Page 35 In Exercises 29–38, use the point on the line and the slope of the line to find three additional points through which the line passes. (There are many correct answers.) Section 1.3 Linear Equations in Two Variables 35 and passing through the pairs of points are parallel, perpen- In Exercises 65–68, determine whether the lines L2 dicular, or neither. L1 Point 2, 1 4, 1 5, 6 10, 6 8, 1 3, 1 5, 4 0, 9 7, 2 1, 6 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. is undefined. is undefined. Slope In Exercises 39–50, find the slope-intercept form of the equation of the line that passes through the given point and has the indicated slope. Sketch the line. Point 0, 2 0, 10 3, 6 0, 0 4, 0 2, 5 6, 1 10, 4 4, 5 2 1 2, 3 2 5.1, 1.8 2.3, 8.5 39. 40. 41. 42.
43. 44. 45. 46. 47. 48. 49. 50. Slope is undefined. is undefined In Exercises 51– 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 8, 1, 8, 7 2, 1, 1 2, 5, 9 1 10, 3 10, 9 5 1, 0.6, 2, 0.6 8, 0.6, 2, 2.4 2, 1, 1 5, 2, 6, 2 1 7 3, 1 3, 8, 7 1.5, 2, 1.5, 0.2 3, 1 65. 67. L1: 0, 1, 5, 9 L2: 0, 3, 4, 1 L1: 3, 6, 6, 0 L2: 0, 1, 5, 7 3 66. 68. L1: 2, 1, 1, 5 L2: 1, 3, 5, 5 L1: (4, 8), (4, 2) L2: 3, 5, 1, 1 3 In Exercises 69–78, write the slope-intercept forms of the equations of the lines through the given point (a) parallel to the given line and (b) perpendicular to the given line. Point 8 2, 1 3, 2 2 3, 7 7 8, 3 4 1, 0 4, 2 2, 5 5, 1 2.5, 6.8 3.9, 1.4 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. Line 4x 2y 3 x y 7 3x 4y 7 5x 3y 6x 2y 9 In Exercises 79–84, use the intercept form to find the equation of the line with the given intercepts. The intercept form of the equation of a line with intercepts and 0, b a, 0 is x a y b 1, a 0, b 0. 79. x -intercept: y -intercept: 81. x -intercept: y
-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 85–88, identify any relationships that exist among the lines, and then use a graphing utility to graph the three equations in the same viewing window. Adjust the viewing window so that the slope appears visually correct—that is, so that parallel lines appear parallel and perpendicular lines appear to intersect at right angles. 85. (a) 86. (a) y 2x y 2 3x (b) (b) y 2x y 3 2x y 1 2x (c) (c) y 2 3x 2 333202_0103.qxd 12/7/05 8:33 AM Page 36 36 Chapter 1 Functions and Their Graphs 87. (a) 88. (a) y 1 2x y x 8 (b) (b) 2x 3 y 1 y x 1 (c) (c) y 2x 4 y x 3 96. Net Profit The graph shows the net profits (in millions) for Applebee’s International, Inc. for the years 1994 through 2003. (Source: Applebee’s International, Inc.) In Exercises 89–92, find a relationship between such that two points. y is equidistant (the same distance) from the x, y and x 89. 90. 91. 92. 4, 1, 2, 3 6, 5, 1, 8, 7, 1 3, 5 2 2, 4, 7 1 2, 5 4 93. Sales The following are the slopes of lines representing in years. Use the slopes to annual sales interpret any change in annual sales for a one-year increase in time. in terms of time x y (a) The line has a slope of (b) The line has a slope of (c) The line has a slope of m 135. m 0. m 40. 94. Revenue The following are lines y representing daily revenues in days. Use the slopes to
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. (Source: Educational Research Service) 85,000 80,000 75,000 70,000 65,000 60,000 55,000 ) 12, 83,944) (10, 79,839) (8, 74,380) (6, 69,277) (4, 64,993) (2, 61,768) (0, 55,722) 2 6 4 Year (0 ↔ 1990) 8 10 12 (a) Use the slopes to determine the time periods in which the average salary increased the greatest and the least. (b) Find the slope of the line segment connecting the years 1990 and 2002. (c) Interpret the meaning of the slope in part (b) in the con- text of the problem ( 100 90 80 70 60 50 40 30 20 10 (13, 99.2) (12, 83.0) (10, 63.2) (11, 68.6) (8, 50.7) (9, 57.2) (6, 38.0) (7, 45.1) (5, 29.2) (4, 16.6) 4 5 6 8 7 Year (4 9 10 11 12 13 14 ↔ 1994) (a) Use the slopes to determine the years in which the net profit showed the greatest increase and the least increase. (b) Find the slope of the line segment connecting the years 1994 and 2003. (c) Interpret the meaning of the slope in part (b) in the con- text of the problem. 97. Road Grade You are driving on a road that has a 6% uphill grade (see figure). This means that the slope of the road is Approximate the amount of vertical change in your position if you drive 200 feet. 6 100. 98. Road Grade From the top of a mountain road, a and x as shown in the table ( surveyor takes several horizontal measurements several vertical measurements and are measured in feet). y, y x x y 300 600 900 1200 1500 1800 2100 25 50 75 100 125 150 175 (a) Sketch a scatter plot of
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 the road. For instance, a surveyor would put up a sign that states “8% grade” on a road 100. What with a downhill grade that has a slope of should the sign state for the road in this problem? 8 333202_0103.qxd 12/7/05 8:33 AM Page 37 Rate of Change In Exercises 99 and 100, you are given the dollar value of a product in 2005 and the rate at which the value of the product is expected to change during the next 5 years. Use this information to write a linear equation that gives the dollar value of the product in terms of the year t. represent 2005.) (Let V t 5 2005 Value Rate 99. $2540 100. $156 $125 decrease per year $4.50 increase per year Graphical Interpretation In Exercises 101–104, match the description of the situation with its graph. Also determine the slope and y-intercept of each graph and interpret the slope and y-intercept in the context of the situation. [The graphs are labeled (a), (b), (c), and (d).] (a) y (b) y 2 4 6 8 40 30 20 10 (c) y 24 18 12 6 200 150 100 50 −2 (d) y 800 600 400 200 x x 2 4 6 8 10 101. A person is paying $20 per week to a friend to repay a $200 loan. 102. An employee is paid $8.50 per hour plus $2 for each unit produced per hour. 103. A sales representative receives $30 per day for food plus $0.32 for each mile traveled. 104. A computer that was purchased for $750 depreciates $100 per year. 105. Cash Flow per Share The cash flow per share for the Timberland Co. was $0.18 in 1995 and $4.04 in 2003. Write a linear equation that gives the cash flow per share t 5 in terms of the year. Let represent 1995. Then predict the cash flows for the years 2008 and 2010.
(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? Explain. (Source: J.C. Penney Co.) t 9 Section 1.3 Linear Equations in Two Variables 37 107. Depreciation A sub shop purchases a used pizza oven for $875. After 5 years, the oven will have to be replaced. Write a linear equation giving the value of the equipment during the 5 years it will be in use. V 108. Depreciation A school district purchases a high-volume printer, copier, and scanner for $25,000. After 10 years, the equipment will have to be replaced. Its value at that time is expected to be $2000. Write a linear equation giving the value of the equipment during the 10 years it will be in use. V 109. College Enrollment The Pennsylvania State University had enrollments of 40,571 students in 2000 and 41,289 students in 2004 at its main campus in University Park, Pennsylvania. (Source: Penn State Fact Book) (a) Assuming the enrollment growth is linear, find a linear model that gives the enrollment in terms of the t, year where corresponds to 2000. t 0 (b) Use your model from part (a) to predict the enroll- ments in 2008 and 2010. (c) What is the slope of your model? Explain its meaning in the context of the situation. 110. College Enrollment The University of Florida had enrollments of 36,531 students in 1990 and 48,673 students in 2003. (Source: University of Florida) (a) What was the average annual change in enrollment from 1990 to 2003? (b) Use the average annual change in enrollment to estimate the enrollments in 1994, 1998, and 2002. (c) Write the equation of a line that represents the given data. What is its slope? Interpret the slope in the context of the problem. (d) Using the results of parts (a)–(c), write a short paragraph discussing the concepts of slope and average rate of change. 111. Sales A discount outlet is offering a 15% discount on S all items. Write a linear equation giving the sale price L. for an item with a list price 112. Hourly
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 shingle delivery truck with a shingle elevator for $36,500. The vehicle requires an average expenditure of $5.25 per hour for fuel and maintenance, and the operator is paid $11.50 per hour. (a) Write a linear equation giving the total cost of opert ating this equipment for hours. (Include the purchase cost of the equipment.) C 333202_0103.qxd 12/7/05 8:33 AM Page 38 38 Chapter 1 Functions and Their Graphs (b) Assuming that customers are charged $27 per hour of R machine use, write an equation for the revenue t derived from hours of use. (c) Use the formula for profit P R C to write an equation for the profit derived from hours of use. t (d) Use the result of part (c) to find the break-even point—that is, the number of hours this equipment must be used to yield a profit of 0 dollars. 114. Rental Demand A real estate office handles an apartment complex with 50 units. When the rent per unit is $580 per month, all 50 units are occupied. However, when the rent is $625 per month, the average number of occupied units drops to 47. Assume that the relationship between the monthly rent and the demand is linear. p x (a) Write the equation of the line giving the demand x in terms of the rent p. (b) Use this equation to predict the number of units occu- pied when the rent is $655. (c) Predict the number of units occupied when the rent is $595. 115. Geometry The length and width of a rectangular garden are 15 meters and 10 meters, respectively. A walkway of width surrounds the garden. x (a) Draw a diagram that gives a visual representation of the problem. (b) Write the equation for the perimeter of the walkway y in terms of x. (c) Use a graphing utility to graph the equation for the perimeter. (d) Determine the slope of the graph in part (c). For each additional one-meter increase in the width of the walkway,
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 salesperson’s monthly wage W S. in terms of monthly sales 117. Business Costs A sales representative of a company using a personal car receives $120 per day for lodging and meals plus $0.38 per mile driven. Write a linear equation giving the daily cost the number of miles driven. to the company in terms of x, C 118. Sports The median salaries (in thousands of dollars) for players on the Los Angeles Dodgers from 1996 to 2003 are shown in the scatter plot. Find the equation of the line represent the that you think best fits these data. (Let t 6 t median salary and let corresponding to 1996.) represent the year, with (Source: USA TODAY) y y 2500 2000 1500 1000 500 FIGURE FOR 118 9 10 11 12 13 8 7 Year (6 ↔ 1996) t Model It 119. Data Analysis: Cell Phone Suscribers The num(in millions) in the bers of cellular phone suscribers is the United States from 1990 through 2002, where year, are shown as data points (Source: Cellular Telecommunications & Internet Association) x, y. x y (1990, (1991, (1992, (1993, (1994, (1995, (1996, (1997, (1998, (1999, (2000, (2001, (2002, 5.3) 7.6) 11.0) 16.0) 24.1) 33.8) 44.0) 55.3) 69.2) 86.0) 109.5) 128.4) 140.8) (a) Sketch a scatter plot of the data. Let x 0 corre- spond to 1990. (b) Use a straightedge to sketch the line that you think best fits the data. (c) Find the equation of the line from part (b). Explain the procedure you used. (d) Write a short paragraph explaining the meanings of the slope and -intercept of the line in terms of the data. y (e) Compare the values obtained using your model with the actual values. (f) Use your model to estimate the number of cellular phone suscribers in 2008. 333202_0103.qxd 12/7/05 8:33 AM Page 39 120. Data Analysis: Average Scores An instructor gives regular 20
-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)–(d).] is the average quiz score and 15, 82. 18, 87, 10, 55, where x, y and x (a) Sketch a scatter plot of 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) Use the equation in part (c) to estimate the average test score for a person with an average quiz score of 17. (e) The instructor adds 4 points to the average test score of each student in the class. Describe the changes in the positions of the plotted points and the change in the equation of the line. Synthesis True or False? In Exercises 121 and 122, determine whether the statement is true or false. Justify your answer. 121. A line with a slope of 5 7 is steeper than a line with a slope of 6 7. 122. The line through 0, 4 and 7, 7 8, 2 and are parallel. 1, 4 and the line through 123. Explain how you could show that the points B 2, 9, and C 4, 3 are the vertices of a right triangle. A 2, 3, 124. Explain why the slope of a vertical line is said to be undefined. 125. With the information shown in the graphs, is it possible to determine the slope of each line? Is it possible that the lines could have the same slope? Explain. (a) y (b) y x 2 4 x 2 4 126. The slopes of two lines are 4 and Which is steeper? 5 2. Explain. 127. The value V purchased is of a molding machine years after it is t V 4000t 58,500, 0 ≤ t ≤ 5. Explain what the V -intercept and slope measure. Section 1.3 Linear Equations in Two Variables 39 128. Think About It Is it possible for two lines with positive slopes to be perpendicular? Explain. Skills Review In Exercises 129–132, match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).] (a) y
(b) y 6 4 2 −2 −6 −4 6 4 2 −2 x 2 x 2 −6 −4 (c) y (d) y 12 8 4 −4 −4 x 4 8 12 8 4 −4 −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 133–138, 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 2x2 21x 49 0 x2 8x 3 0 x 9 15 0 3x 16x 5 0 139. Make a Decision To work an extended application analyzing the numbers of bachelor’s degrees earned by women in the United States from 1985 to 2002, visit this text’s website at college.hmco.com. (Data Source: U.S. Census Bureau) 333202_0104.qxd 12/7/05 8:35 AM Page 40 40 Chapter 1 Functions and Their Graphs 1.4 Functions What you should learn • Determine whether relations between two variables are functions. • Use function notation and evaluate functions. • Find the domains of functions. • Use functions to model and solve real-life problems. • Evaluate difference quotients. Why you should learn it Functions can be used to model and solve real-life problems. For instance, in Exercise 100 on page 52, you will use a function to model the force of water against the face of a dam. Introduction to Functions Many everyday phenomena involve two quantities that are related to each other by some rule of correspondence. The mathematical term for such a rule of correspondence is a relation. In mathematics, relations are often represented by I mathematical equations and formulas. For instance, the simple interest earned on I 1000r. $1000 for 1 year is related to the annual interest rate by the formula represents a special kind of relation that matches each item from one set with exactly one item from a different set. Such a relation is called a function. The formula I 1000r r Definition of Function f A function from a set element domain (or set of inputs) of the function and the set (or set of outputs). exactly one element f, in the set to a set A A B x y is a relation that assigns to each B. in the set The set B
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 range. Set Outputs: 9, 10, 12, 13, 15 © Lester Lefkowitz/Corbis This function can be represented by the following ordered pairs, in which the first coordinate ( -value) is the input and the second coordinate ( -value) is the output. x y 1, 9, 2, 13, 3, 15, 4, 15, 5, 12, 6, 10 Characteristics of a Function from Set A to Set B 1. Each element in must be matched with an element in A B. 2. Some elements in may not be matched with any element in B A. 3. Two or more elements in may be matched with the same element in A B. 4. An element in elements in B. A (the domain) cannot be matched with two different 333202_0104.qxd 12/7/05 8:35 AM Page 41 Functions are commonly represented in four ways. Section 1.4 Functions 41 Four Ways to Represent a Function 1. Verbally by a sentence that describes how the input variable is related to the output variable 2. Numerically by a table or a list of ordered pairs that matches input values with output values 3. Graphically by points on a graph in a coordinate plane in which the input values are represented by the horizontal axis and the output values are represented by the vertical axis 4. Algebraically by an equation in two variables To determine whether or not a relation is a function, you must decide whether each input value is matched with exactly one output value. If any input value is matched with two or more output values, the relation is not a function. Example 1 Testing for Functions Determine whether the relation represents as a function of y x. a. The input value y value is the number of senators. x is the number of representatives from a state, and the output b. Input, x Output, y c. 2 2 3 4 5 11 10 8 5 1 y 3 2 1 −3 −2 −1 1 2 3 x −2 −3
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 Figure 1.48 does describe as a function of Each input value x. y is matched with exactly one output value. Now try Exercise 5. Representing functions by sets of ordered pairs is common in discrete mathematics. In algebra, however, it is more common to represent functions by equations or formulas involving two variables. For instance, the equation y x 2 y is a function of x. represents the variable y as a function of the variable x. In this equation, isx 333202_0104.qxd 12/7/05 8:35 AM Page 42 42 Chapter 1 Functions and Their Graphs the independent variable and function is the set of all values taken on by the independent variable range of the function is the set of all values taken on by the dependent variable is the dependent variable. The domain of the and the y. x © Historical Note Leonhard Euler (1707–1783), a Swiss mathematician, is considered to have been the most prolific and productive mathematician in history. One of his greatest influences on mathematics was his use of symbols, or notation. The y fx function notation was introduced by Euler. Example 2 Testing for Functions Represented Algebraically Which of the equations represent(s) as a function of y x? a. x2 y 1 b. x y2 1 Solution To determine whether y is a function of x, try to solve for y in terms of x. a. Solving for yields y x2 y 1 Write original equation. y 1 x2. Solve for y. x To each value of of x. there corresponds exactly one value of So, y. y is a function b. Solving for yields y x y2 1 y2 1 x y ± 1 x. Write original equation. Add x to each side. Solve for y. indicates that to a given value of x there correspond two values of y. ± The y So, is not a function of x. Now try Exercise 15. Function Notation When an equation is used to represent a function, it is convenient to name the function so that it can be referenced easily. For example, you know that the
Suppose you give this equation function the name “ ” 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 the name of the function, whereas f x. f x f x at or simply of x. The symbol y f x. So, you can write Keep in f x is the value of the function f x 3 2x f 2, f1, f0, For has function values denoted by substitute the specified input values into the given equation. f 1 3 21 3 2 5. f 0 3 20 3 0 3. f 2 3 22 3 4 1. x 1, x 0, x 2, For For and so on. To find these values, 333202_0104.qxd 12/7/05 8:35 AM Page 43 Section 1.4 Functions 43 Although x as the independent variable, you can use other letters. For instance, is often used as a convenient function name and f is often used f x x2 4x 7, f t t2 4t 7, and gs s2 4s 7 all define the same function. In fact, the role of the independent variable is that of a “placeholder.” Consequently, the function could be described by f 2 4 7. Example 3 Evaluating a Function gx 2 In Example 3, note that In is not equal to general, gu v gu gv. gx g2. Let a. gx x2 4x 1. g2 gt b. c. Find each function value. gx 2 Solution a. Replacing with 2 in x gx x2 4x 1 yields the following. g2 22 42 1 4 8 1 5 b. Replacing with yields the following. t x gt t2 4t 1 t 2 4t 1 c. Replacing with x yields the following. gx 2 x 22 4x 2 1 x 2 x 2 4x 4 4x 8 1 x 2 4x 4 4x 8 1 x 2 5 Now try Exercise 29. A function defined by two or more equations over a specified domain is called a piecewise-defined function. Example 4 A Piecewise-Defined Function Evaluate the function when x 1,
0, and 1. f x x2 1, x 1, x < 0 x ≥ 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 graphing utility to graph the functions given by y x2 4. and domain of each function? Do the domains of these two functions overlap? If so, for what values do the domains overlap? The Domain of a Function The domain of a function can be described explicitly or it can be implied by the expression used to define the function. The implied domain is the set of all real numbers for which the expression is defined. For instance, the function given by fx 1 x2 4 Domain excludes x-values that result in division by zero. These two has an implied domain that consists of all real other than values are excluded from the domain because division by zero is undefined. Another common type of implied domain is that used to avoid even roots of negative numbers. For example, the function given by x x ±2. fx x Domain excludes x-values that result in even roots of negative numbers. is defined only for In general, the domain of a function excludes values that would cause division by zero or that would result in the even root of a negative number. So, its implied domain is the interval x ≥ 0. 0,. Example 5 Finding the Domain of a Function Find the domain of each function. a. f : 3, 0, 1, 4, 0, 2, 2, 2, 4, 1 c. Volume of a sphere: V 4 3 r3 b. gx 1 x 5 d. hx 4 x2 Solution a. The domain of consists of all first coordinates in the set of ordered pairs. f Domain 3, 1, 0, 2, 4 b. Excluding -values that yield zero in the denominator, the domain of x g is the x set of all real numbers except x 5. c. Because this function represents the volume of a sphere, the values of the radius must be positive. So, the domain is the set of all real numbers such that r r > 0. r
d. This function is defined only for -values for which x 4 x 2 ≥ 0. By solving this inequality (see Section 2.7), you can conclude that 2 ≤ x ≤ 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 equation V 4 3 r3 you would have no reason to restrict implies that a sphere cannot have a negative or zero radius. r to positive values, but the physical context 333202_0104.qxd 12/7/05 8:35 AM Page 45 h r = 4 r Applications Section 1.4 Functions 45 Example 6 The Dimensions of a Container You work in the marketing department of a soft-drink company and are experimenting with a new can for iced tea that is slightly narrower and taller than a standard can. For your experimental can, the ratio of the height to the radius is 4, as shown in Figure 1.49. h a. Write the volume of the can as a function of the radius b. Write the volume of the can as a function of the height r. h. Solution a. Vr r 2h r 24r 4r3 2 Vh h 4 h3 16 h b. Write V as a function of r. Write V as a function of h. FIGURE 1.49 Now try Exercise 87. Example 7 The Path of a Baseball A baseball is hit at a point 3 feet above ground at a velocity of 100 feet per second and an angle of 45º. The path of the baseball is given by the function fx 0.0032x2 x 3 y where and are measured in feet, as shown in Figure 1.50. Will the baseball clear a 10-foot fence located 300 feet from home plate? x f(x) 80 60 40 20 ) 30 60 90 Baseball Path f(x) = − 0.0032x 2 + x + 3 150 120 Distance (in feet) 180 210 240 270 300 x FIGURE 1.50 Solution When x 300, the height of the baseball is f 300 0.00323002 300 3 15 feet. So, the baseball will clear the fence. Now try Exercise 93. In the equation in Example 7, the height of the baseball is a function of the distance from home plate. 333202_0104.qxd 12/7/05 8:35 AM Page 46 46 Chapter
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 ↔ 1995) 12 t V The number (in thousands) of alternative-fueled vehicles in the United States increased in a linear pattern from 1995 to 1999, as shown in Figure 1.51. Then, in 2000, the number of vehicles took a jump and, until 2002, increased in a different linear pattern. These two patterns can be approximated by the function Vt 18.08t 155.3 38.20t 10.2, 5 ≤ t ≤ 9 10 ≤ t ≤ 12 t 5 t where represents the year, with corresponding to 1995. Use this function to approximate the number of alternative-fueled vehicles for each year from 1995 to 2002. (Source: Science Applications International Corporation; Energy Information Administration) Solution From 1995 to 1999, use 245.7 263.8 Vt 18.08t 155.3. 281.9 299.9 318.0 1995 1996 1997 1998 1999 From 2000 to 2002, use 392.2 430.4 Vt 38.20t 10.2. 468.6 2000 2001 2002 Now try Exercise 95. Difference Quotients One of the basic definitions in calculus employs the ratio f x h f x h, h 0. This ratio is called a difference quotient, as illustrated in Example 9. Example 9 Evaluating a Difference Quotient For f x x2 4x 7, find f x h f x h. Solution f x h f x h x h2 4x h 7 x 2 4x 7 h x 2 2xh h2 4x 4h 7 x 2 4x 7 h h2x h 4 h 2xh h2 4h h 2x h 4, h 0 Now try Exercise 79. The symbol indicates an example or exercise that highlights algebraic techniques specifically used in calculus. 333202_0104.qxd 12/7/05 2:47 PM Page 47 Section 1.4 Functions 47 You may find it easier to calculate the difference quotient in Example 9 by and then substituting the resulting expression into the f x h, first finding difference quotient, as follows. f x h x h2 4x h 7 x2 2xh h2 4x 4h 7 f x h f x h x2 2xh h
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. is the dependent variable. is the independent variable. x f x is the value of the function at x. Domain: The domain of a function is the set of all values (inputs) of the is in the domain independent variable for which the function is defined. If ff, is said to be of undefined at x is not in the domain of is said to be defined at ff, If x. x. x Range: The range of a function is the set of all values (outputs) assumed by the dependent variable (that is, the set of all function values). Implied Domain: is not specified, the implied domain consists of all real numbers for which the expression is defined. is defined by an algebraic expression and the domain If f W RITING ABOUT MATHEMATICS Everyday Functions In groups of two or three, identify common real-life functions. Consider everyday activities, events, and expenses, such as long distance telephone calls and car insurance. Here are two examples. a. The statement,“Your happiness is a function of the grade you receive in this course” is not a correct mathematical use of the word “function.“ The word ”happiness” is ambiguous. b. The statement,“Your federal income tax is a function of your adjusted gross income” is a correct mathematical use of the word “function.” Once you have determined your adjusted gross income, your income tax can be determined. Describe your functions in words. Avoid using ambiguous words. Can you find an example of a piecewise-defined function? 333202_0104.qxd 12/7/05 8:35 AM Page 48 48 Chapter 1 Functions and Their Graphs 1.4 Exercises VOCABULARY CHECK: Fill in the blanks. 1. A relation that assigns to each element x from a set of inputs, or ________, exactly one element y in a set of outputs, or ________, is called a ________. 2. Functions are commonly represented in four different ways, ________,
________, ________, 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 ≥ 0 is an example of a ________ function. 5. If the domain of the function f is not given, then the set of values of the independent variable for which the expression is defined is called the ________ ________. 6. In calculus, one of the basic definitions is that of a ________ ________, given by f x h f x h, h 0. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 4, is the relationship a function? 1. Domain −2 −1 0 1 2 Range 2. 5 6 7 8 3. Domain Range 4. National League American League Cubs Pirates Dodgers Orioles Yankees Twins Range 3 4 5 Range (Number of North Atlantic tropical storms and hurricanes) 7 8 12 13 14 15 19 Domain − − 2 1 0 1 2 Domain (Year) 1994 1995 1996 1997 1998 1999 2000 2001 2002 In Exercises 5–8, does the table describe a function? Explain your reasoning. 5. Input value Output value. 7. 8. Input value 0 1 Output value 4 2 Input value 10 Output value Input value Output value 12 12 3 0 4 10 15 15 3 In Exercises 9 and 10, which sets of ordered pairs represent functions from B? A 9. 10. to A 0, 1, 2, 3 (a) Explain. B 2, 1, 0, 1, 2 and 0, 1, 1, 2, 2, 0, 3, 2 0, 1, 2, 2, 1, 2, 3, 0, 1, 1 0, 0, 1, 0, 2, 0, 3, 0 0, 2, 3, 0, 1, 1 (b) (c) B 0, 1, 2, 3 (d) A a, b, c (a) and a, 1, c, 2, c, 3, b, 3 a, 1, b, 2, c, 3 1, a, 0, a, 2, c, 3, b (c
) (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 11. Is the circulation of morning newspapers a function of the year? Is the circulation of evening newspapers a function of the year? Explain. 12. Let f x x. year Find f 1998. represent the circulation of evening newspapers in In Exercises 13–24, determine whether the equation represents as a function of x. y 13. 15. 17. 19. 21. 23. x2 y 2 4 x2 y 4 2x 3y 14 14. 16. 18. 20. 22. 24. x y 2 x y 2 4 x 22 75 In Exercises 25–38, evaluate the function at each specified value of the independent variable and simplify. 32. 33. 34. 35. 36. 37. 38. 31. qx 1 x2 9 q0 (a) qt 2t 2 3 t 2 x (a) q2 f x x f 2 (a) f x x 4 f 2 (a) Section 1.4 Functions 49 (b) q3 (c) qy 3 (b) q0 (c) qx (b) f 2 (c) f x 1 (b) f 2 (c) f x2 (a) (b) f 1 f x 2x 1, x < 0 2x 2, x ≥ 0 f 0 f x x2 2, x ≤ 1 2x2 2, x > 1 f 1 (b 3x 1, f 2 (a) 4, x2, f 2 (a) f x 4 5x, 0, x2 1, 2 f 1 (bc) f 2 (c) f 2 (c) f 3 (a) f 3 (b) f 4 (c) f 1 In Exercises 39 –44, complete the table. 39. f x x2 3 2 1 0 1 2 x f x 40. gx x 3 25. 26. 27. 28. 29. 30. r3 f x 2x 3 f 1 (a) g y 7 3y g

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