Review of Essential Skills- Part Operations with Rational Numbers, page. (e) 8 Exponent Laws, page 6. (a) 0 + 5 0, (d) (), (e) +, 8 + (h) 5, 9. (h) x 5. (d) v 5 Expanding, Simplifying, and Factoring Algebraic Expressions, page 8. (f) x 9y. (b) h 6. (f) question may be wrong 7. (i) (5x )(x + ) Solving Linear and Quadratic Equations Algebraically, page 0. (f) y = 6 (h) n = 5. (d) y = Practise, Apply, Solve., page 9. (a). 56,.89 07 5,.090 507 7,.0 7 78,.0 897 9,.00 889 86,.005 9 90,.00 7 75,.00 5 70,.000 677 0. (c) Example: t n = n; t n =.5 +.5( ) n ; t n = () n ; t n = n + n +. (b) Example: The number of toothpicks is h(h + ), where h is the number of toothpicks per side. Practise., page 6 (d),. 56,.7 050 808,,.6 067 977,.9 89 7,.65 75,.88 7 5,,.6 77 660 Practice., page 8 Part (c) t 0 = 0., t 5 = 0.067, t 0 = 0.05 (d) t 0 =.6, t 5 =.87, t 0 =.7 (g) t 0 = 5.97, t 5 =.77, t 0 = 67.75 (h) t 0 = 0.009 766, t 5 = 0.000 05, t 0 = 0.000 009 57 Practise, Apply, Solve.7, page 57 8. (a) dots in first quadrant curving up to right through (, ) (b) dots in fourth quadrant curving down to right through (, ) (c) dots in first quadrant curving up to left through (, 5) (d) dots alternating above and below x-axis in increasing magnitude to right through (, ) (e) dots in first quadrant curving up to right through (, 500) (f) dots in first quadrant curving up to right through (, ) (g) dots in first quadrant curving up to right through (, 0.5) (h) dots in fourth quadrant curving down to right through (, 00) Practise, Apply, Solve.0, page 85 8. (e) 5m n Chapter, Review and Practice, page 98 5. (a) dots in first quadrant in straight line through (, 5) and (, 9) (b) dots in first quadrant curving up to right through (, 0) (c) dots in first quadrant curving up to right through (, ) (d) dots in fourth quadrant in straight line through (, ) and (, ) (e) dots in first quadrant curving up to right through (, ) (f) dots in first quadrant curving up to left through (, 0) Chapter Review Test, page 0. d = 0 Practise, Apply, Solve., page 7. (a) S 7 S 7 = ( + 8 + + 8 + 5 + 08 + 89) ( + 8 + + 8 + 5 + 08 + 89) S 7 = 768 766 S 7 = S 7 = 0 9
(b) S 7 7 ar ( ) = r 7 ( ) = ( 6 8) = = 0 9 Practise., page 86 (c) $80. (e) Practise., page 8 (b) nonlinear with points (quarter, balance) at {(, 860), (, 8.0), (, 889.66),..., (0, 7 66.)} A B C D Period Prev.Int. Quarterly Bal.8.0%/a Balance 8000 = B * 0.0 = B + C = A + = D = B * 0.0 = B + C Total Interest after 8 years = $7076. Balance after 8 years = $5 076. (d) A B C D Period Prev.Int. Monthly Bal.6.0%/a Balance 0 000 = B * 0.005 = B + C = A + = D = B * 0.005 = B + C Balance after 0 years = $5 58.90 Practise, Apply, Solve.7, page 5 7. years, each divided into equal segments. Amount of each payment, proceeding from most recent to first: 600, 600(.05), 600(.05),..., 600(.05) 5 8. (d) 7 years, each divided into equal segments. Amount of each payment, proceeding from most recent to first: 00, 00(.075), 00(.075),..., 00(.075) ; A = 00 + 00(.075) +... + 00(.075) ; $8 86.66 9. (d) A = 00(.075) + 00(.075) +... + 00(.075) = $86 760.9 Practise, Apply, Solve.8, page 6 6. (c) years, each divided into 5 segments. Present value of each payment, proceeding from most recent to first: 60(.5) 56, 60(.5) 55, 650(.5) 5,..., 60(.5) P = 60(.5) 56 + 60(.5) 55 +... + 60(.5), $0.00 8. (b) P = R(.05) 5 + R(.05) +... + R(.05) 0. (b).99% compounded annually Chapter, Review and Practice, page 97. (a) 00 7. t n = ar n 5. (c) about months; $0.7 (d) $8.7. (a) n years each divided into segments. Amount of each payment, proceeding from most recent to first: 75, 75(.005), 75(.005),..., 75(.005) n, 75(.005) n, 75(.005) n 0. (d) $6.9. (a) $6.9 6. (a) $59.7 (b) year : $79 78.7 year : $77 68.55 year : $7 7.9 year : $7 887.6 year 5: $68 780.96 (c) $09.0, $5.00 (d) $95.5 (e) $9 57.9 Cumulative Review Test, page 05. (b) t n = 50 000 + 000n. (a) t 5 = 86 09 0. (c) 6 months: final payment: $0.06 (d) $560.06 (e) $.. (b) $9 09.9 (c) $5 06.88
5U. (a) t n = 50 + 0.(t n ) (c) an infinite length of time Review of Essential Skills- Part Using Properties of Relations to Sketch Their Graphs, page. (d) straight line through (0, ) and (8, 0); x = 8, y =. (d) straight line through, 6 0 and (0,.5); m = 9, b =. (a) parabola opening up, vertex (.5, 0.5) through (0, 5) and (, 5) (c) parabola opening up, vertex ( 0.5,.5) through (, ) and (0, ). (b) parabola opening up, vertex (.5,.5) through (, 0) and (5, 0) 5. (a) parabola opening up, vertex (, ) through (0, 7) and (, 7) Completing the Square to Convert to the Vertex Form of a Parabola, page. (a) (x + ) + 9. (d) y = ( x 5) + Solving Quadratic Equations: The Quadratic Formula, page i. (b) m = 7 ± 7 0 Practise, Apply, Solve., page 6. (d) One curve in the first quadrant going down with a negative slope through (, ); second curve in second quadrant going up with a positive slope through (, ) D = {x x 0, x R}, R = {y y > 0, y R}; function passes vertical line test 8. (b) D = {x 0 x 00, x R} 6. (c) 0.9 559 70 5,... Practise, Apply, Solve., page 5. (i) solid line between closed dots at and 8. x x x 9. D = {x < x <, x R}, R = {y < y < 8, y R} Practise, Apply, Solve., page 55. (h) connected points: (, ) to (, 0.5) to (,.5) to (.8, ) to (, 0.5) to (, ) to (, ) 5. (e) (, ), (, ) and (5, 5) (g) intersect at (, ), (, ) and, 0 Practise.5, page 6 x +. ii. f (x) = ± + iv. (c) D = {x 0. x., x R}; 7. (c) g (x) = (x ) 8. (c) f (x) = x + Practise, Apply, Solve.7, page 85 9. curve in first quadrant down to right through (, ), curve in third quadrant down to right through (, ). (b) y = ( x 5) Chapter, Review and Practice, page 9 6. (a) solid line between closed dot at and open dot at 7. (b) x = y, x (c) f is y = x, y. (a) x g + y Chapter Review Test, page 97 6. (c) g (x) = ± x + 5+ 8. (b) point from (0.5, ) to ( 0.5, ) to (, 5) to (, ) to (, 7) Chapter Getting Ready, page 0. (f) 57 5 (i) 85 6 (l) 7 0 7. (b) 6m n (m + n) (c) p q r(5pr + 7pqr) (d) (x 5)(x ) (e) (y + )(y 8) (f) (x + )(x + 5) (g) (x + )(x )
(h) (x 7)(x + 7) (i) (5x 6)(5x + 6) (j) (x ) (k) (x )(x + ) (l) (a )(5a + ) 8. (f) line, x-intercept:, y-intercept: 5 Practice., page 06. (a) f(x) = (x ) (c) f(x) = (x ) 9 9. (d), 98 0. vertical stretch with factor 0.; horizontal shift.5 units to the right; vertical shift unit up Practise, Apply, Solve., page. (a) R(x) = x + 7x; $ 50 (b) R(x) = x + x; $0 08.50 (c) R(x) = 0.x + x, $ 500 0. Complete the square to find minimum possible value for f(x). f(x) has a minimum value since a > 0; min possible value is.. opens downward; the x-coordinate of the vertex is always between the x-coordinates of the vertices for f(x) and g(x). The y-coordinate of the vertex for f(x) + g(x) may be above, between, or below f(x) and g(x), depending on its positions.. (a) 7. (a) A line going through the first quadrant with the equation: y =.98 79 67x + 9.6 985 7 (b) R(x) = x + 9x (c) P(x) = x + 0x or P(x) = (x 5) 6; when x = 5, then 5000 pizzas should be sold to maximize profit.. (b) y(x) = x + 965x 7. y = x Practise, Apply, Solve., page 5. 68 items 6. graphically: Create the profit function, P(x) = R(x) C(x), where R(x) is the quadratic revenue and C(x) is the linear cost function. P(x) will be a quadratic function. Graph the profit function and find the zeros. 7. The function will have two zeros regardless of k s value. Practise, Apply, Solve., page 5. (e) 5.66i (f) 7.07i 6. (e) 0.5 ± 0.66i (f) ±.i 9. (c) ± 6 (d) ± i. (a) (x + i)(x + +i) 7. sum of two complex numbers may be real or complex, when two complex conjugates are added, the result is always a real number; product of two complex numbers may be real or complex, when two complex numbers multiplied, result is always a real number; answers will vary Practise, Apply, Solve.6, page 5. (a) x-intercept: 8. (c) W = VI, where I and W are linear 9. (b) f (x) = x x 5. (a) =, f ( x) = f ( x) x + D of f = x R, R of f = y R; D of = {x x f, x R}, R of = {y y R}; f D of f = x R, R of f = y R (b) =, f ( x) =± x + f ( x) x D of f = x R, R of f = {y y, y R}; D of = {x x ±, x R}, f R of = { y y > 0 or y, y R}; f D of f = {x x, x R}, R of f = y R} (c) =, f ( x) = ( x + ) + f ( x) x D of f = {x x, x R}, R of f = {y y, y R}; D of f = {x x, x R}, R of = {y y 0, y R}; f D of f = {x x, x R}, R of f = {y y, y R} 8. (x, A(x)) = (, 0), (, 0.69), (,.), (,.9), (5,.6), (6,.79), (7,.95), (8,.08), (9,.0), (0,.0), (,.0), (,.8)
Practise, Apply, Solve.8, page 5. (a), y x (b), y x (c), y x (d), x y (e), x y (f), x y x 5y 9. (d), x + y x ± 5 5 y. (a) 0 mg (b) 7.6 mg Surface Area ( r + h) 5. =, r 0, h 0 Volume rh 6. No, he made his error by incorrectly factoring the numerator. Practise, Apply, Solve.9, page 59 xy. (a) (b) 0 a, a 0, b 0 b (c) 9x, x 0, y 0 (d), p 0, q 0 (e) y, x 0, y 0. (a) a b, b 0 m (b), m 0, n 0 (c) 8ab, a 0, b 0 9r (d) pq, p, q, r 0 a (e) b, a 0, b 0. (a), x 0, 5( y 5) (b), y 0, (c), a 0,, (d) x, x bb ( + ) (e), b, ± ( b ) (f) s + ss ( + ). (a) 8 y, y 0, (b) 5, p, (c) x + y x +, x ±, y 5. (a), k 0, ± k (b) (x )(x + ), x ±, q (c), q ± + q, ± 6. When you divide two rational expressions, take reciprocal of second expression so numerator of second expression is now the denominator. This new denominator cannot be zero so there may be additional restrictions to state. ( x + )( x + 5) 7. (d) ( x )( x ), x, ± 5 5 ( x + 5)( x ) 8. (b), x ±,, 0 xx ( + ) x + 9. (a), x, x + (e), a 5,, 9+ p (f), p,, 0,, 7 8p p 0. (a) x, x,, 5 (b), k 5,, ±. (a) 0.5 billion dollars (b) 5 billion dollars (c) 50 billion dollars a+ b. (a), a 0, ±b a b (b) x, x 0, ±,,, 6. f(x) = g(x) ( x )( x + ) 7. (a) ( x )( x ), x ± +, ±, ± ( a + b)( a ) (b) ( 5 )( ), b, 5,, a b ± b ± a + b a + mm ( + n) (c) ( m n)( m n), m n, + + n, ± n, ± n, n Practise, Apply, Solve.0U, page 6. (a) i x y (f) x y x y i, + + x, y. (b) + i 0 Practise, Apply, Solve., page 69 b+ 7a. (a), ab, 0 ab 9q q 9. (d) q 6 ( q+ )( q )( q ),,, 0. (a) x = + i or x = i. Arshia s speed: 8. km/h; Sarah s speed: 8 km/h; Arshia s time: 5 h; Sarah s time: 5.5 h 5
. 5 km/h. 6 km/h 5. 8 shirts Practise, Apply, Solve., page 7 5. (a) x + x x (c) x x + 6x x + 5 7. (f) 6x + x 6x + 5 8. (a) x + x + Chapter, Review and Practice, page 78 8. (b) min, x =, 5 (c) max, t =, 0 (d) max, x = 0.55, 0.70 5 9. (b) P(x) = 5x + 0x 5 (c) x = (d) 000 and 000 (e) P(x) = 5(x ) + 5; parabola, opens down; vertex (, 5), zeros:,, y-int: 5 9. (d) Q, R, C (g) Q, R, C (h) Q, R, C (i) W, R, C, I, Q. (f) 5 i (i) 50 i 5. (a) 6; i; 0 (b) 6; 8i; 5 (c) 0; i; 9. (b) For f(x), D = {x x R}, R = {y y, y R} For, D = {x x ±.7, x R}, f ( x) R = {y y or y > 0, y R} For f (x), D = {x x, x R}, R = y R (c) For f(x), D = {x x, x R}, R = {y y, y R} For, D = {x x, x, x R}, f ( x) R = {y y 0, y R} For f (x), D = {x x, x R}, R = {y y, y R} ( x + ). (g) 0 ( ), x x,, 6. (c) + i,, + i i (d) i,, i + i 6b 5. (j) 0 ± 0 ( a + b)( a b), a, b, b 5. (c) x =±, 0 Chapter Review Test, page 87. (a) + i, 5 + i,, + 5 0 i b. (c) 0 a, a, ± b, b 5. (b) + i; x x + 5 = 0 8. (a) 8. mg (b) 5.6 mg Cumulative Review Test, page 89 5. (a) 8 6 y 8 6 0 6 8 6 8 7. (b) g (x) = x 8, g(x): upper branch of parabola opens right, vertex (, 0), y-int:.87; g (x): parabola opens up; vertex (0, ); zero at.87, D of g(x) = {x x, x R}, R of g(x) = {y y > 0, y R}; D of g (x) = x R, R of g (x) = {y y, y R} 9. (a) x f y 0. (e) parabola opens down, vertex (, ), zeros at 0.6 and 5.5, y-int:. (b) D of f(x) = x R, R of f(x) = {y y 9, y R} (c) D of f(x) = {x x, x R}, R of f(x) = {y y 5, y R}; D of = {x x, x R}, f ( x) R of = {y y 0, y R}; f ( x) D of f (x) = x R, R of f (x) = {y y, y R} x 6
5. (c) 7. (a), x ± x (b) p,q 0 q (c) 5 ( x ), x,, ( x + ) 5 (d) ( a 5)( a+ )( a+ ) ( a )( a a ), a,., 7., 5 5 8. (f) 5 + 5i Review of Essential Skills Part The Trigonometry of Right Triangles, page 99 5. (a).5 cm y Chapter 5, Getting Ready, page 06 0. (d) maximum: (5, 0.5), minimum: none, zeros:, 7 0 Practise, Apply, Solve 5., page. (d) The sun rises in the same spot as the earth rotates, forming a self-replicating function. 9. (b) 8 min 0. (b) min Practise, Apply, Solve 5., page 8. (h) 69. The point ( 5, 9) forms a right triangle in quadrant III with horizontal side 5 and vertical side 9. The angle x, which is the related acute angle for θ, is linked to the sides 5 and 9 by the tangent ratios. Once the acute value is known, then θ = (80 + related acute angle) for a quadrant III angle θ. x Practise, Apply, Solve 5., page. (a) i (b) ii (c) iii (d) iv 6. (b) 8 5. (e) y = csc θ: D = {θ θ = m, m R, θ 80 n, n I}, R = {y y or y, y R}; y = sec θ: D = {θ θ = m, θ 90 + 80 n, m R, n I}; R = {y y or y >, y R}; y = cot θ: D = {θ θ = m, θ 80 n, m R, n I}; R = y R Practise, Apply, Solve 5., page 8. (a) (θ, f(θ)) = ( ) π 7 07 π 6 0 5π π,,,.,,,, 07., π (, ),,., π,, π π π 07,.,(, ),,., 0 07 0 07 π 0 π 5π π 07 07 6 0 7π,,,.,( π, ),,.,,,, 07.,( π, ) 5. (d) The graph is periodic and sinusoidal. y = sin θ, 0 θ π 9. (c) (.5, ) Practise, Apply, Solve 5.6, page 55 7. (c), 080, 5, 9. (c) zeros: π, min. at (0, ), max at (π, ), axis of symmetry at y = 0 8. (a) axis of symmetry y =, min. ( 95, ), ( 5, ), (65, ), (5, ), max. ( 85, ), ( 05, ), (75, ), (55, ), passing through points (x, ) where x = 0, 0, 50, 60, 0, 0, 0, 00 (b) axis of symmetry y =, min. ( 5, 7), ( 95, 7), ( 75, 7), (5, 7), (65, 7), (85, 7), max.( 55, ), ( 5, ), ( 5, ), (05, ), (5, ), (5, ), passing through points (x, ) where x = 5, 85, 5, 65, 05, 5, 5, 75, 5, 95, 55, 5 (d) axis of symmetry y =, min.( 5,.5), max. (5,.5), passing through (80, ) and ( 80, ) 0. cos θ: stretched vertically by factor of, reflected in θ- axis, moved up one unit on the y-axis, horizontally shifted π units right, period: 80 ; sin θ: vertically 8 stretched by factor of, reflected in θ-axis, moved up one unit on y-axis, period: 80, horizontal phase shift of π units right 7
. (e) SHOULD BE. (f). (f) Let x-axis represent the day of the year and y-axis represent the number of hours of daylight axis of symmetry D(t) =, min. (0, 8) and (55, 8) max. (8.5, 6). (b) max.(7.6, 8 ) 5. a: vertical stretch by factor of a, b: horizontal phase shift by b units; shift right for b < 0, shift left for b > 0 Practise, Apply, Solve 5.7, page 6. (a) min. (90, ) and ( 90, ) 6. d = 5 7. d = 9 maximum minimum 0. (a) a = Practise, Apply, Solve 5.8, page 7 6. (e) 6.9, 6.9 9. (b).8,.8 Chapter 5, Review and Practice, page 85. (b) The first cycle starts at t = 0 s and ends at t = 50 s. The minimum of the graph is h =.5 m and its maximum is h = 5 m. Repeat the given relation starting at t = 50 s. π π 8. (c),, 0 maximum minimum. (b) amplitude = Chapter 5 Review Test, page 9 9. (a) 95, 55 0. (e) T(t) = 8.9cos π 6 t + 5.8 (f) phase shift 6 units right Practise, Apply, Solve 6., page 509 8. (c) ( l, f, F) = (50, 6. cm, 95 ), (0,. cm, 5 ) (e) ( X, Z, z) = (56, 86, 8. cm), (, 8, 5.7 cm) 9. ( E, F, f) = (50, 89, 5. cm), (0, 9, 5.5 cm) Practise, Apply, Solve 6., page 5. Plane Able will land first. Cumulative Review Test, page 55 5. (b) period: π (f) zero at.59 and 0.069 6. (c) 8 s 7. (c) θ = 7.86,. (d) x =.0,.0 Review of Essential Skills- Part Analytic Geometry: Lines and Line Segments, page 560. (a) m = 8 Practise, Apply, Solve 7.U, page 57. (d) rectangle with a diagonal line leading from bottom left corner to a point just left of the opposite corner. 0. The locus of B is another circle with radius r also inside the first circle. Practise, Apply, Solve 7.U, page 509 6. (f) D = {x 6 x 6, x R}, R = {y y, y R} Practise, Apply, Solve 7.6, page 60. (f) focus 0,. (c) vertex: (0, ), opens right. (d) (y + ) = (x ), vertex: (, ) 7. (b) x = 8y 5. (y ) = (x ) 7. 8 m Practise, Apply, Solve 7.9U, page 65. (d) y-int: ±, ±. (f) asymptotes: y = x + and y = x Practise, Apply, Solve 7.0U, page 60 5. (b) ( x ) ( y ) = 8. (a) minor axis: Practise, Apply, Solve 6., page 5 5. (a) AB = 6, AD =, BC = 6, CD = 6, AC = 6 + 6 8
Practise, Apply, Solve 7., page 66. radius (7, ) 0. vertex: (0, 0) Chapter 7, Review and Practice, page 69 6. y-int: (0, 8), (0, 8) x y 7. (b) = a b. (d) ellipse, x ( y ) + 6 8 =, vertices (, ) and (, ) Chapter 7U Review Test, page 6. (c) ellipse, vertices: (, 0), (, 0), (0, ), (0, ), foci: (0, 0) (d) parabola opening to the right, vertices: (, 0), foci: (, 0), (, 0). In the equation: x y =, the transverse axis is b a vertical (along the y-axis). Cumulative Review Test, page 65 0. (a) D = {x x, x R}, R = {y y R} 5. (b) k >, k = ±, k < 7. Insert (b) before no zeros 7. (a) x = π π,, 0.,.. (h) 5 9 i 9