Answers Chapter 1 Sequences and Series 1.1 Arithmetic Sequences, pages 1 to 1 1. a) arithmetic sequence: t 1 = 1, d = 1; net three terms: 9, 11, 18 not arithmetic arithmetic sequence: t 1 = -, d = -; net three terms: -19, -, - d) arithmetic sequence: t 1 =, d = -; net three terms: -1, -1, -18. a), 8, 11, 1-1, -, -9, -1,,, d) 1., 1.,.7,.. a) t 1 = 11 t 7 = 9 t 1 =. a) 7, 11, 1, 19, ; t 1 = 7, d =, 9,, ; t =, d = - 1,,, 8, 1; t 1 =, d =. a) 8 d) 17. a) t = 1, t = t = 19, t = t = 7, t = 7. a), 8, 11, 1, 17 t n = n + t = 1, t = d) The general term is a linear equation of the form = m + b, where t n = and n =. Therefore, t n = n + has a slope of. e) The constant value of in the general term is the -intercept of. 8. A and C; both sequences have a natural-number value for n. 9. 1. t n = -n + 8; t 1 = -7 11. = -1; first three terms: -78, -11, -1 1. z = - 1. a) t n = n + 8 1 1. a), 8, 1, plaers t n = 8n - 8 d) 1:1 e) Eample: weather, all foursomes starting on time, etc. 1. 1 square inches 1. a) t n = n - 1 1st da Susan continues the program until she accomplishes her goal. 17. a) Carbon Atoms 1 Hdrogen Atoms 8 1 t n = n + or H = C + 1 carbon atoms 18. Multiples of 8 7 1 Between 1 and 1 and and First Term, t 1 8 Common Difference, d 8 7 1 nth Term, t n 98 9 9 General Term t n = 8n t n = 7n + 97 t n = 1n + Number of Terms 1 19. a) 1.7, 9.,.1, 8.8; t n = 1.7n, where n represents ever increment of ft in depth. 9 psi at 1 ft and 98 psi at ft Water Pressure 1 as Depth Changes Water Pressure (psi) 1 8 1 -ft Depth Changes d) 1.7 psi e) 1.7 f) The -intercept represents the first term of the sequence and the slope represents the common difference.. Other lengths are cm, 1 cm, and 18 cm. Add the four terms to find the perimeter. Replace t with t 1 + d, t with t 1 + d, and t with t 1 + d. Solve for d. 1. a), 8, 1, 1, t n = n min. -9 beekeepers..8 million carats. This value represents the increase of diamond carats mined each ear.. 19. m. a) 1:, 1:9, 1:, 1:9, 1:1; t 1 = 1:, d = : t n = :n + 1:9 Assume that the arithmetic sequence of times continues. d) 1:9 Answers MHR 1
. a) d > d < d = d) t 1 e) t n 7. Definition: An ordered list of terms in which the difference between consecutive terms is constant. Common Difference: The difference between successive terms, d = t n - t n - 1 Eample: 1, 19,, Formula: t n = 7n + 8. Step 1 The graph of an arithmetic sequence is alwas a straight line. The common difference is described b the slope of the graph. Since the common difference is alwas constant, the graph will be a straight line. Step a) Changing the value of the first term changes the -intercept of the graph. The -intercept increases as the value of the first term increases. The -intercept decreases as the value of the first term decreases. Yes, the graph keeps it shape. The slope stas the same. Step a) Changing the value of the common difference changes the slope of the graph. As the common difference increases, the slope increases. As the common difference decreases, the slope decreases. Step The common difference is the slope. Step The slope of the graph represents the common difference of the general term of the sequence. The slope is the coefficient of the variable n in the general term of the sequence. 1. Arithmetic Series, pages 7 to 1 1. a) 9 7 1-181 d) = 1.. a) t 1 = 1, d =, S 8 = t 1 =, d = -, S 11 = 1 t 1 =, d = 1, S =. 7 d) t 1 = -., d =., S = 1.7. a) 19 d) 9 e) 1. a) 1 8. d) 1. a) 1 1. a) t 1 =, S 1 = 7 t 1 = -17, S 1 = - t 1 = -, S 1 = -8 d) t 1 = 7, S 1 = 7. 7. a) 1 8 8. 1 times 9. a) 1. 8 11. + 1 + 17 + 1. a) S n = n [t + (n - 1)d] 1 S n = n [() + (n - 1)1] S n = n S n = n(1n) S n = 1n S n = n [1 + 1n - 1] S 1 = 1 [() + (1-1)1] S 1 = 1 [1 + 99] S 1 = 1 (1) S 1 = n (1 + n) d(1) = (1) d(1) = (1 ) d(1) = 1. 171 1. a) the number of handshakes between si people if the each shake hands once 1 + + + + + + 7 + 8 + 9 d) Eample: The number of games plaed in a home and awa series league for n teams. 1. a) t 1 =., d = 1. t = 9 S = 1. 17 cm 17. a) True. Eample: + + + 8 =, + 8 + 1 + 1 =, = False. Eample: + + + 8 =, + + + 8 + 1 + 1 + 1 + 1 = 7, 7 True. Eample: Given the sequence,,, 8, multipling each term b gives 1,,,. Both sequences are arithmetic sequences. 18. a) 7 + 11 + 1 d) S n = n [t + (n - 1)d] 1 S n = n [(7) + (n - 1)] S n = n [1 + n - ] [n + 1] S n = n(n + ) S n = n + n S n = n 1 MHR Answers
19. a) + + + + S n = n + n 189 d) Nathan will continue to remove an etra 1 bushels per hour.. (-7) + (-) + (-17) 1. Jeanette and Pierre have used two different forms of the same formula. Jeanette has replaced t n with t 1 + (n - 1)d.. a) 1 S green = 1 + + + + 1 S blue = + 1 + + + + 9 S total = S green + S blue S total = 1 (1 + 1) + 1 ( + 9) S total = (11) + (9) S total = + S total = 1. a) The nth triangular number is represented b S n. S n = n [t + (n - 1)d] 1 S n = n [(1) + (n - 1)(1)] S n = n [ + (n - 1)] S n = n (1 + n) 1. Geometric Sequences, pages 9 to 1. a) geometric; r = ; t n = n - 1. not geometric geometric; r = -; t n = (-) n - 1 d) not geometric e) geometric; r = 1.; t n = 1(1.) n - 1 f) geometric; r = ; t n = -1() n - 1 a) Geometric Sequence Common Ratio th Term 1th Term, 18,, 18 118 98 1.8,.,.,... 1,, 9, 19 8. a),, 18, -, 1, -8, 19, -1,, -18 d), 1,,. 18.9,.1, 1.9. a) t n = () n - 1 t n = 19 (- ) n - 1 t n = 9 ()n - 1 d) t n = () n - 1. a) 7 d) e) 9 f) 8 7. 7 8. 1, 1, 9; t n = 1 ( ) n - 1 9. a) t 1 = ; r =.7 t n = (.7) n - 1 approimatel.9 cm d) 7 1. a) 9% 1, 9, 9., 8.77.9 d) about 9.87% e) After 7 washings, % of the original colour would remain in the jeans. Eample: The geometric sequence continues for each washing. 11. 1.77 1. a) 1,,, 8, 1 t n = 1() n - 1 9 or 87 91 1. a) 1.1 1. cm jumps 1. a) 1,,, 8, 1, t n = 1() n - 1 or d) All cells continue to double and all cells live. 1..9% 1. 8 weeks 17.. m 18..9 19. a) 7. ml h. a) Time, d (das) Charge Level, C (%) 1 1 98 9. 9.1 t n = 1(.98) n - 1 The formula in part includes the first term at d = in the sequence. The formula C = 1(.98) n does not consider the first term of the sequence. d) 81.7% 1. a).1 mm 117.77 mm. Eample: If a, b, c are terms of an arithmetic sequence, then b - a = c - b. If a, b, c are terms of a geometric series, then b = c a and b b - a = c - b. Therefore, b - a = c - b. So, when a, b, c form a geometric sequence, then a, b, c form an arithmetic sequence.. ; 9, 1,. a).9 cm 19. cm.1 cm d).1 cm e).1, 1.9, 1.79; arithmetic; d = -.11 cm. Mala s solution is correct. Since the aquarium loses 8% of the water ever da, it maintains 9% of the water ever da. Answers MHR 1
. 1 1 1 1 1 1 1 1 1 1 1 1 18 1 1 9 8 1 1 1 1 1 1 7. a).8 cm 1.7 cm. cm d) 19.88 cm 1. Geometric Series, pages to 7 1. a) geometric; r = geometric; r = - not geometric d) geometric; r = 1.1. a) t 1 =, r = 1., S 1 = 17 7, S 79.98 1 t 1 = 18, r = -., S 1 = 1 8 1, S 1. 1 t 1 =.1, r =, S 9 = 1 71 1, S = 17.1 9 d) t 1 =., r =.1, S 1 = 1, S. 1. a) 1 7 8 81-9 71 8 d). a)..9 19 d) 9. a) 9.7. 7 7. a) 81 81 + 7 + 9 + + 1 8. t = - 81 1 ; S = 7.8 9. a) If the person in charge is included, the series is 1 + + 1 + +. If the person in charge is not included, the series is + 1 + +. If the person in charge is included, the sum is 9. If the person in charge is not included, the sum is 1 98 1. 1.. m m 11. 79. km 1. Stage Number Length of Each Line Segment Number of Line Segments Perimeter of Snowflake 1 1 1 1 1 9 1 7 1 81 8 1 19 9 78 7 length, t n = ( ) n - 1 ; number of line segments, t n = () n - 1 ; perimeter, t n = ( ) n - 1 d) 1 81 1. 1. 98 79 1. 91 mm 1. a).9 mg 7. mg 1. 8 8 17. 8 18. a =, b = 1, c = or a =, b = 1, c = 19. 1 1. π 1. General Term Formula Arithmetic Eample Sequences General Term Formula Geometric Eample t n = t 1 + (n - 1)d 1,,, 7, t n = t 1 r n - 1, 9, 7, 81, General Sum Formula n S n = (t 1 + t n ) Arithmetic or n S n = [t 1 + (n - 1)d ] Eample Series General Sum Formula Geometric Eample 1 + + + 7 + rt + 9 + 7 + n - t S 1 81 + n = r, r 1-1 or t 1 (r n - 1) S n =, r 1 r - 1. Eamples: a) All butterflies produce the same number of eggs and all eggs hatch. No. Tom determined the total number of butterflies from the first to fifth generations. He should have found the fifth term, which would determine the total number of butterflies in the fifth generation onl. 1 MHR Answers
This is a reasonable estimate, but it does include all butterflies up to the fifth generation, which is. 1 7 more butterflies than those produced in the fifth generation. d) Determine t = 1() or. 1 1. 1. Infinite Geometric Series, pages to 1. a) divergent convergent convergent d) divergent e) divergent. a) no sum no sum d) e).. a).87 +.87 +. 87 + ; S = 87 or 9 99.7 +. 7 + ; S = 7 999. Yes. The sum of the infinite series representing.999 is equal to 1.. a) 1 or.8 1. t 1 = 7; 7 + 18 + 1 + 7. r = ; -8-1 - - 1-8. a) barrels of oil Determining the lifetime production assumes the oil well continues to produce at the same rate for man months. This is an unreasonable assumption because 9% is a high rate to maintain. 9. = ; 1 + + 9 1 + 7 + 1. r = 11. a) -1 < < 1 - < < - < < 1. cm 1. cm 1. No sum, since r = 1.1 > 1. Therefore, the series is divergent. 1. 8 m 1. a) approimatel 17.8 cm cm 17. a) Rita ; therefore, r < -1, and the series is divergent. 18. 1 m 19. 7 cm. a) Eample: + ( ) + ( ) + + ( ) n r = - and + ( ) + ( ) + + ( ) n S = t 1 1 - r = 1 - = = and 1 - t S = 1 1 - r = = = 1. Geometric series converge onl when -1 < r < 1.. a) S n = - 8 n + 11 8 n S n = ( ) n - 1 - S n = - ( ) n + S = 1 1 - S =. Step n 1 Fraction of Paper 1 Step + 1 + +, Eample: S = Chapter 1 Review, pages to 8 1. a) arithmetic, d = arithmetic, d = - not arithmetic d) not arithmetic. a) C D E d) B e) A. a) term, n = 1 not a term term, n = d) not a term. a) A Compare Two Sequences Term Value 1 1 8 Sequence 1 Sequence 1 1 Term Number In the graph, sequence 1 has a larger positive slope than sequence. The value of term 17 is greater in sequence 1 than in sequence. Answers MHR 17
. t 1 = 1. cm 7. a) S 1 = 19 S 1 = 8 S 1 = -7 d) S = 1 8. S = 9. a) 9 das 1. a) 1 9 11. 117 1. a) not geometric geometric, r = -, t 1 = 1, t n = (-) n - 1 geometric, r =, t = 1, t = 1 n ( d) not geometric 1. a) 7 bacteria t n = (1.8) n 1. π cm or approimatel.8 cm 1. Arithmetic Sequence Definition A sequence in which the difference between consecutive terms is constant Formula Formula t n = t 1 + (n - 1)d t n = t 1 r n - 1 ) n - 1 Geometric Sequence Definition A sequence in which the ratio between consecutive terms is constant Eample Eample,, 9, 1,, 1,, 18, 1. a) arithmetic geometric geometric d) arithmetic e) arithmetic f) geometric 17. a) S 1 = 17 7, S 79.98 1 S 1 = 8 1, S.99 1 S =, S.7 d) S 9 = 9 9, S 1.7 9 18. a) 19.1 mm 1.7 m 19. a) S = 1 S =. a) convergent, S = 1 divergent convergent, S = -8 d) convergent, S = 1. a) r = -. S 1 = 7, S =., S =., S =.87, S =.1 d) S =. 1,. Yes. The areas form a geometric sequence. The common ratio is. a) 1,, 1 or 1.8 1 square units square units. a) A series is geometric if there is a common ratio r such that r 1. An infinite geometric series converges if -1 < r < 1. An infinite geometric series diverges if r < -1 or r > 1. Eample: + + 1 +. + ; S = 8 1-1. +. -. + ; S = 1 Chapter 1 Practice Test, pages 9 to 7 1. D. B. B. B. C. 11. cm 7. Arithmetic sequences form straight-line graphs, where the slope is the common difference of the sequence. Geometric sequences form curved graphs. 8. A = 1, B = 9 9..7 km 1. a),, 7, 98, 19, 1 t n = 1n -, 1,,, 8, 1 d) t n = () n - 1 11. a) 17,, 1, 8, 8 t n = 17n million ears d) Assume that the continents continue to separate at the same rate ever ear. 1. a) s, s, 9 s, 1 s, 1 s arithmetic das d) 91 min 18 MHR Answers
Chapter Trigonometr.1 Angles in Standard Position, pages 8 to 87 1. a) No; the verte is not at the origin. Yes; the verte is at the origin and the initial arm is on the -ais. No; the initial arm is not on the -ais. d) Yes; the verte is at the origin and the initial arm is on the -ais.. a) F C A d) D e) B f) E. a) I IV III d) I e) III f) II. a) 7 9. 19. 1. a) dogwood (-., ), white pine (., -), river birch (-., -) red maple, flowering dogwood 1, river birch 1, white pine m 11. cm 1. a) A (, -), A (-, ), A (-, -) A OC = - θ, A OC = 18 - θ, A OB = 18 + θ 1. ( - ) m or ( - 1 ) m 1. 1. Cu (copper), Ag (silver), Au (gold), Uuu (unununium) 1. a) 1 8 das 18 das 17. a) 7 7 1 17 d) 8 17 8 d) 1. a) 1 1 7 d). a) 1,, 1 1,, 1, 1, d) 1,, 8 7. a) 88 1 198 d) 8. θ sin θ cos θ tan θ º or or or 1 18. a) Angle Height (cm) 1. 1...8 1. 7. 9 7. A constant increase in the angle does not produce a constant increase in the height. There is no common difference between heights for each pair of angles; for eample,. cm - 1 cm = 11. cm,. cm -. cm = 1.9 cm. When θ etends beond 9, the heights decrease, with the height for 1 equal to the height for 7 and so on. 19. and 1. a) 19. m i) 19 ii) 9.1 m 1. a) B D Answers MHR 19
. + = r. a) θ 8 sin θ..8.8.988 sin (18 - θ)..8.8.988 sin (18 + θ) -. -.8 -.8 -.988 sin ( - θ) -. -.8 -.8 -.988 Each angle in standard position has the same reference angle, but the sine ratio differs in sign based on the quadrant location. The sine ratio is positive in quadrants I and II and negative in quadrants III and IV. The ratios would be the same as those for the reference angle for cos θ and tan θ in quadrant I but ma have different signs than sin θ in each of the other quadrants.. a) ft 1 As the angle increases to the distance increases and then decreases after. The greatest distance occurs with an angle of. The product of cos θ and sin θ has a maimum value when θ =.. Trigonometric Ratios of An Angle, pages 9 to 99 1. a) (, ) - - - O θ d) (-1, ) θ - - - O -. a) sin =, cos =, tan = sin = - or -, cos = - or -, tan = 1 sin 1 =, cos 1 = -, tan 1 = - or - d) sin 9 = 1, cos 9 =, tan 9 is undefined, tan θ =. a) sin θ = sin θ = -, cos θ = 1, cos θ = - 1, tan θ = 1 1 sin θ = - 1, cos θ = 8, tan θ = - 1 17 17 8 d) sin θ = - or -, cos θ = or, tan θ = -1. a) II I III d) IV. a) sin θ = 1, cos θ = -, tan θ = - 1 1 sin θ = - or -, cos θ = or, tan θ = - sin θ = or, cos θ = or, tan θ = d) sin θ = - 1, cos θ = - 1 1. a) positive positive negative d) negative 7. a), tan θ = 1 (-, ) θ (-1, ) (1, ) θ O θ - - - O θ - - - O - (-, -) or 17 8. a) sin θ =, tan θ = - cos θ =, tan θ = sin θ = - or - 1 1 1, cos θ = or 1 1 1 MHR Answers
d) cos θ = -, tan θ = e) sin θ = - or -, cos θ = - or - 9. a) and 1 and 1 and d) and e) and f) 1 and 1 1. θ sin θ cos θ tan θ º 1 9º 1 undefined 18º -1 7º -1 undefined º 1 11. a) = -8, =, r = 1, sin θ =, cos θ = - =, = -1, r = 1, sin θ = - 1 cos θ = 1 1. a) (-9, ) θ R, tan θ = -, tan θ = - 1 1 1. a) θ θ θ R (7, -) 1, 7 8 1. a) sin θ = or sin θ = or sin θ = or d) The all have the same sine ratio. This happens because the points P, Q, and R are collinear. The are on the same terminal arm. 1. a) 7 and 1 sin θ =, cos θ = ± 7, tan θ = ± 7 1. sin θ = 17. sin =, cos = 1, tan =, sin 9 = 1, cos 9 =, tan 9 is undefined 18. a) True. θ R for 11 is 9 and is in quadrant II. The sine ratio is positive in quadrants I and II. True; both sin and cos 1 have a reference angle of and sin = cos =. False; tan 1 is in quadrant II, where tan θ <, and tan is in quadrant III, where tan θ >. d) True; from the reference angles in a - -9 triangle, sin = cos =. e) True; the terminal arms lie on the aes, passing through P(, -1) and P(-1, ), respectivel, so sin 7 = cos 18 = -1. 19. θ sin θ cos θ tan θ 1 or or or 1 9 1 undefined 1-1 - 1 or - 1 or - -1 1 - - 1 or - 18-1 1-1 - or - 1 or - - 1 or - 1 - - 1 7-1 undefined - - 1-1 or - or -1-1 - 1 or - 1. a) A =, B = 1, C =, D = 1 A (, ), B ( -, ), C ( -,- ), D (,- ) Answers MHR 1
1. a) Angle Sine Cosine Tangent º 1 1º.88.99.79 º..8.77 º.771.771 1 º.8. 1.71 7.99.88.71 9 1 undefined 1.99 -.88 -.71 1.8 -. -1.71 1.771 -.771-1 1. -.8 -.77 1.88 -.99 -.79 18-1 As θ increases from to 18, sin θ increases from a minimum of to a maimum of 1 at 9 and then decreases to again at 18. sin θ = sin (18 - θ). Cos θ decreases from a maimum of 1 at and continues to decrease to a minimum value of -1 at 18. cos θ = -cos (18 - θ). Tan θ increases from to being undefined at 9 then back to again at 18. For θ 9, cos θ = sin (9 - θ). For 9 θ 18, cos θ = -sin (θ - 9 ). d) Sine ratios are positive in quadrants I and II, and both the cosine and tangent ratios are positive in quadrant I and negative in quadrant II. e) In quadrant III, the sine and cosine ratios are negative and the tangent ratios are positive. In quadrant IV, the cosine ratios are positive and the sine and tangent ratios are negative.. a) sin θ = or 7 7 7, cos θ = or 7 7 7, tan θ =. As θ increases from to 9, decreases from 1 to, increases from to 1, sin θ increases from to 1, cos θ decreases from 1 to, and tan θ increases from to undefined.. tan θ = 1 - a a. Since BOA is, the coordinates of point A are (, ). The coordinates of point B are (1, ) and of point C are (-1, ). Using the Pthagorean theorem d = ( - 1 ) + ( - 1 ), d AB = 1, d BC =, and d AC =. Then, AB = 1, AC =, and BC =. So, AB + AC = BC. MHR Answers The measures satisf the Pthagorean Theorem, so ABC is a right triangle and CAB = 9. Alternativel, CAB is inscribed in a semicircle and must be a right angle. Hence, CAB is a right triangle and the Pthagorean Theorem must hold true.. Reference angles can determine the trigonometric ratio of an angle in quadrant I. Adjust the signs of the trigonometric ratios for quadrants II, III, and IV, considering that the sine ratio is positive in quadrant II and negative in quadrants III and IV, the cosine ratio is positive in the quadrant IV but negative in quadrants II and III, and the tangent ratio is positive in quadrant III but negative in quadrants II and IV. 7. Use the reference triangle to identif the measure of the reference angle, and then adjust for the fact that P is in quadrant III. Since tan θ R = 9, ou can find the reference angle to be 1. Since the angle is in quadrant III, the angle is 18 + 1 or 1. 8. Sine is the ratio of the opposite side to the hpotenuse. The hpotenuse is the same value, r, in all four quadrants. The opposite side,, is positive in quadrants I and II and negative in quadrants III and IV. So, there will be eactl two sine ratios with the same positive values in quadrants I and II and two sine ratios with the same negative values in quadrants III and IV. 9. θ =. Both the sine ratio and the cosine ratio are negative, so the terminal arm must be in quadrant III. The value of the reference angle when sin θ R = is. The angle in quadrant III is 18 + or.. Step a) As point A moves around the circle, the sine ratio increases from to 1 in quadrant I, decreases from 1 to in quadrant II, decreases from to -1 in quadrant III, and increases from -1 to in quadrant IV. The cosine ratio decreases from 1 to in quadrant I, decreases from to -1 in quadrant II, increases from -1 to in quadrant III, and increases from to 1 in quadrant IV. The tangent ratio increases from to infinit in quadrant I, is undefined for an angle of 9, increases from negative infinit to in the second quadrant, increases from to positive infinit in the third quadrant, is undefined for an angle of 7, and increases from negative infinit to in quadrant IV.
The sine and cosine ratios are the same when A is at approimatel (.,.) and (-., -.). This corresponds to and. The sine ratio is positive in quadrants I and II and negative in quadrants III and IV. The cosine ratio is positive in quadrant I, negative in quadrants II and III, and positive in quadrant IV. The tangent ratio is positive in quadrant I, negative in quadrant II, positive in quadrant III, and negative in quadrant IV. d) When the sine ratio is divided b the cosine ratio, the result is the tangent ratio. This is true for all angles as A moves around the circle.. The Sine Law, pages 18 to 11 1. a) 8.9. 8 d). a).9 mm. m. a) 8. a) C = 8, A = 7, a =. m or C = 9, A = 19, a =. m C =, c =.7 m, a =. m B = 119, c =.9 mm, a = 1. mm d) B = 71, c = 19. cm, a = 1. cm. a) AC =. cm B cm 7 A 7 AB =. cm B 8 cm A AB =.7 m C C B d) BC =. cm B 1 cm A. a) two solutions one solution one solution d) no solutions 7. a) a > b sin A, a > h, b > h a > b sin A, a > h, a < b a = b sin A, a = h d) a > b sin A, a > h, a b 8. a) A = 8, B = 11, b = 7. cm or A = 1, B = 17, b =. cm P =, R = 7, r =.9 cm or P = 11, R =, r = 8. cm no solutions 9. a) a 1 cm a =. cm. cm < a < 1 cm d) a <. cm 1. a) C 78 9 Ro m 9.9 m 11..7 m 1. 1 1.. m 1. a) M 1 Maria h m.1 m 7. m 1. a) 1.1 Å.11 mm 1. least wingspan 9.1 m, greatest wingspan 9. m 17. a) Since a < b ( < ) and a > b sin A ( > sin ), there are two possible solutions for the triangle. second stop B cairn C second m.8 m 17.8 stop B cairn 17. m C 9. A 7 m C m A first stop A first stop Answers MHR
Armand s second stop could be either 191.9 m or 7. m from his first stop. 18. 911. m 19. Statements Reasons. sin C = h b h = b sin C h = c sin B sin B = h c b sin C = c sin B sin C c b = sin B b C a sin B ratio in ABD sin C ratio in ACD Solve each ratio for h. Equivalence propert or substitution Divide both sides b bc. A B c Given A = B, prove that side AC = BC, or a = b. Using the sine law, a sin A = b sin B But A = B, so sin A = sin B. Then, a sin A = b sin A. So, a = b. 1. 1.1 km. a).1 cm < a <. cm C. cm A B a < 1. cm C 1.7 cm A B a = 1.8 cm C. B a c C A b In ABC, sin A = a c and sin B = b c Thus, c = a sin A and c = b sin B. Then, a sin A = b sin B. This is onl true for a right triangle and does not show a proof for oblique triangles.. a) 1.9 cm ( + ) cm or ( + 1) cm.9 cm d).1 cm e) The spiral is created b connecting the angle vertices for the reducing golden triangles. 7. Concept maps will var. 8. Step 1 B A C Step a) No. There are no triangles formed when BD is less than the distance from B to the line AC. Step D 7.7 cm 7 A B. 1.7 m. a) There is no known side opposite a known angle. There is no known angle opposite a known side. There is no known side opposite a known angle. d) There is no known angle and onl one known side. B A D C a) Yes. One triangle can be formed when BD equals the distance from B to the line AC. MHR Answers
Step e) B 18. m 9. m B f) A 1.8 m A = C B A D D C a) Yes. Two triangles can be formed when BD is greater than the distance from B to the line AC. Step a) Yes. One triangle is formed when BD is greater than the length AB. Step The conjectures will work so long as A is an acute angle. The relationship changes when A > 9.. The Cosine Law, pages 119 to 1 1. a). cm 1. mm. m. a) J = L = P = 17 d) C = 19. a) Q =, R =, p =. km S = 1, R =, T = 7. a) B cm 7 A cm C BC =.1 cm B 1 cm 8 cm. m. m A. m C C = 17. a) Use the cosine law because three sides are given (SSS). There is no given angle and opposite side to be able to use the sine law. Use the sine law because two angles and an opposite side are given. Use the cosine law to find the missing side length. Then, use the sine law to find the indicated angle.. a). cm 7. m 7.. cm 8. 9 m 9. The angles between the buos are, 88, and 7. 1.. 11.. km 1.. km 1. 8. cm 1. a) 17 8 km km d) A AC = 8. m B C 9 cm 8 A 1 cm AB = 7.8 cm B 9 cm 1 cm A 1 cm B = C C Julia and Isaac base camp 9.1 km 1. 9.7 m 1. Use the cosine law in each oblique triangle to find the measure of each obtuse angle. These three angles meet at a point and should sum to. The three angles are 118, 1, and 99. Since 118 + 1 + 99 =, the side measures are accurate. 17. The interior angles of the bike frame are 7,, and. 18. 98.8 m 19. 1 km Answers MHR
. 8.1 m 1. The interior angles of the building are,, and 8.. Statement Reason Use the Pthagorean c = (a - ) + h Theorem in ABD. Epand the square of a c = a - a + + h binomial. Use the Pthagorean b = + h Theorem in ACD. c = a - a + b Substitute b for + h. cos C = b Use the cosine ratio in ACD. = b cos C Multipl both sides b b. Substitute b cos C for in c = a - ab cos C + b step. c = a + b - ab cos C Rearrange... km. No. The three given lengths cannot be arranged to form a triangle ( + b < c ). When using the cosine law, the cosines of the angles are either greater than 1 or less than -1, which is impossible.. 1. cm. ABC =, ACD = 97 7. 9 km 8..1 m 9. B(-, ) b θ R C(, ) c θ a A(a, ) cos θ R = -cos θ = - + b = + c = (a + ) + Prove that c = a + b - ab cos C: Left Side = ( (a + ) + ) = (a + ) + = a + a + + Right Side = a + ( + ) - a ( + ) (- + ) = a + + + a = a + a + + Left Side = Right Side Therefore, the cosine law is true.. 11. m 1. a) 8. cm 8. cm These methods give the same measure when C = 9. d) Since cos 9 =, ab cos 9 =, so a + b - ab cos 9 = a + b. Therefore, c can be found using the cosine law or the Pthagorean Theorem when there is a right triangle.. Concept Summar for Solving a Triangle Given Right triangle Two angles and an side Three sides Three angles Two sides and the included angle Two sides and the angle opposite one of them Begin b Using the Method of. Step a) A = 9, B = 1, C = 7 The angles at each verte of a square are 9. Therefore, = ABC + 9 + GBF + 9 18 = ABC + GBF GBF = 7, HCI = 1, DAE = 11 GF =. cm, ED = 1. cm, HI = 11.1 cm Step a) For HCI, the altitude from C to HI is.1 cm. For AED, the altitude from A to DE is 1. cm. For BGF, the altitude from B to GF is. cm. For ABC, the altitude from B to AC is.9 cm. area of ABC is 11.7 cm, area of BGF is 11.7 cm, area of AED is 11.7 cm, area of HCI is 11.7 cm A B C D C B MHR Answers
Step All four triangles have the same area. Since ou use reference angles to determine the altitudes, the product of bh will determine the same area for all triangles. This works for an triangle. Chapter Review, pages 1 to 18 1. a) E D B d) A e) F f) C g) G. a) θ R sin 1 =, cos 1 = -, tan 1 = - sin = -, cos =, tan = - or - d) sin 1 = or, cos 1 = - or -, tan 1 = -1. a) Q(-, ) θ Quadrant III, θ R = θ R 1 Quadrant II, θ R = Quadrant I, θ R = d) θ R Quadrant IV, θ R =. No. Reference angles are measured from the -ais. The reference angle is.. quadrant I: θ =, quadrant II: θ = 18 - or 1, quadrant III: θ = 18 + or 1, quadrant IV: θ = - or. a) sin = - or -, cos = - or -, tan = 1 or sin θ = or, cos θ = - or -, tan θ = - d) 117 7. (, ), (-, ), (-, -) 8. a) sin 9 = 1, cos 9 =, tan 9 is undefined sin 18 =, cos 18 = -1, tan 18 = 9. a) cos θ = -, tan θ = sin θ = - 8 or -, tan θ = - 8 or - sin θ = 1, cos θ = 1 1 1. a) 1 or 1 or 7 or 9 11. a) Yes; there is a known angle (18-18 - 11 = 8 ) and a known opposite side ( cm), plus another known angle. Yes; there is a known angle (9 ) and opposite side ( cm), plus one other known side. No; there is no known angle or opposite side. 1. a) C = 7, c =.9 mm A = 78, B = 1. R q p. 1. P. cm Q R =., q =. cm, p =. cm Answers MHR 7
1..8 km 1. a) Ship B,. km S h 7 9 A B 8 km Use tan 9 = h and tan 7 = h 8 -. Solve tan 9 = (8 - ) tan 7. =.8 km Then, use cos 9 =.8 and cos 7 =. BS AS to find BS and AS. AS = 1. km, BS =. km 1. no solutions if a < b sin A, one solution if a = b sin A or if a b, and two solutions if b > a > b sin A 17. a) 7 km 7 km 7 E of S 99. km 18. a) The three sides do not meet to form a triangle since + < 7. A + C > 18 Sides a and c lie on top of side b, so no triangle is formed. d) A + B + C < 18 19. a) sine law; there is a known angle and a known opposite side plus another known angle cosine law; there is a known SAS (side-angle-side). a) a = 9.1 cm B = 7 1. 17. d. a) C 1.8 m 9. m A 18. m A = C 8 1 cm 9 cm A AB = 7.8 cm B B. a) C A 8 m 1 m B A =, C = 1, AC = 8. m. km/h 8 km/h 18. km. a) cm 1 cm 8 8.8 cm and. cm Chapter Practice Test, pages 19 to 1 1. A. A. C. B. C. - 7. a) Oak Ba 1.1 km 79 7 Ross Ba 1.9 km. km 8. a) two B =, C = 97, c = 19.9 or B = 17, C =, c = 7.8 9. R = 17 1. a) Q P 1 cm 1 cm R R =, Q = 8, r = 7.8 cm or R = 8, Q = 9, r =.7 cm 11.. cm 1. a) 1.7 m 1. quadrant I: θ = θ R, quadrant II: θ = 18 - θ R, quadrant III: θ = 18 + θ R, quadrant IV: θ = - θ R 8 MHR Answers
1. a) second base pitcher s mound home plate 7 ft ft 7 ft first base a + b = c 7 + 7 = c c = 99 Second base to pitcher s mound is 99 - or 9 ft. Distance from first base to pitcher s mound is = + 7 - ()(7) cos or 9. ft. 1. Use the sine law when the given information includes a known angle and a known opposite side, plus one other known side or angle. Use the cosine law when given oblique triangles with known SSS or SAS. 1. patio triangle: 8,,. m; shrubs triangle:,.7 m,. m 17..1 km Cumulative Review, Chapters 1, pages 1 to 1 1. a) A D E d) C e) B. a) geometric, r = ; 1, 9, 7 arithmetic, d = -;,, -1 arithmetic, d = ; -1,, 9 d) geometric, r = -; 8, -9, 19. a) t n = -n + 1 t n = n -. t n = (-) n - 1 t = - or -1 8 7. a) S 1 = 17 S = 8. a) Phtoplankton Production 1 1 Phtoplankton (t) 1 Number of 11-Da Ccles t n = 1n The general term is a linear equation with a slope of 1. 7. 1 m 8. a) r =.1, S = 1 Answers will var. 9. 1. sin θ = 8 17, cos θ = 1, tan θ = 8 17 1 11. a) 1 d) 1. a) 9 1 (, ) - -1 1 sin θ = 1, cos θ =, tan θ is undefined 1. a) sin = or cos = tan = 1 d) cos 18 = -1 e) tan 1 = = - or - f) sin 7 = -1 1. The bear is 8.9 km from station A and 7. km from station B. 1. 9. 1. a) woodpecker.8 m Chelsea 1 m 17. 1. 7 Unit 1 Test, pages 1 to 17 1. B. C. D Answers MHR 9
. C. D. $.1 per cup 7. 8. 9. 77 1. a) - t n = n - 11 d) S 1 = 1 11. $1 88. 1. km 1. a),, 1, 8, t n = ( ) n - 1 games 1. a) The shapes of the graphs are the same with the parabola of = ( - ) being two units to the right. verte: (, ), ais of smmetr: =, domain: { R}, range: = ( - ) - {, R}, -intercept occurs at (, ), -intercept occurs at (, ) The shapes of the graphs are the same with the parabola of = - being four units lower., 1, 18,,, t n = n 1. a) 8. m 1. 8 Chapter Quadratic Functions.1 Investigating Quadratic Functions in Verte Form, pages 17 to 1 1. a) Since a > in f() = 7, the graph opens upward, has a minimum value, and has a range of {, R}. Since a > in f() =, the graph opens upward, has a minimum value, and has a range of {, R}. Since a < in f() = -, the graph opens downward, has a maimum value, and has a range of {, R}. d) Since a < in f() = -., the graph opens downward, has a maimum value, and has a range of {, R}.. a) The shapes of the graphs are the same with the parabola of = + 1 being one unit higher. verte: (, 1), ais of smmetr: =, domain: { R}, = + 1 range: { 1, R}, no -intercepts, -intercept occurs at (, 1) - - - - - = - verte: (, -), ais of smmetr: =, domain: { R}, range: { -, R}, -intercepts occur at (-, ) and (, ), -intercept occurs at (, -) d) The shapes of the graphs are the same with the parabola of = ( + ) being three units to the left. = ( + ) 1 - - - 8 verte: (-, ), ais of smmetr: = -, domain: { R}, range: {, R}, -intercept occurs at (-, ), -intercept occurs at (, 9) MHR Answers
. a) Given the graph of =, move the entire graph units to the left and 11 units up. Given the graph of =, appl the change in width, which is a multiplication of the -values b a factor of, making it narrower, reflect it in the -ais so it opens downward, and move the entire new graph down 1 units. Given the graph of =, appl the change in width, which is a multiplication of the -values b a factor of, making it narrower. Move the entire new graph units to the left and 1 units down. d) Given the graph of =, appl the change in width, which is a multiplication of the -values b a factor of, making it wider, 8 reflect it in the -ais so it opens downward, and move the entire new graph. units to the right and 1.8 units up.. a) = -( - ) + 9 8 d) = -( - 1) + 1 1 1 8 - - verte: (1, 1), ais of smmetr: = 1, opens downward, maimum value of 1, domain: { R}, range: { 1, R}, -intercepts occur at (-1, ) and (, ), -intercept occurs at (, 9) - - verte: (, 9), ais of smmetr: =, opens downward, maimum value of 9, domain: { R}, range: { 9, R}, -intercepts occur at (, ) and (, ), -intercept occurs at (, ) =.( + ) + 1-8 - - - verte: (-, 1), ais of smmetr: = -, opens upward, minimum value of 1, domain: { R}, range: { 1, R}, no -intercepts, -intercept occurs at (, ) - - = ( - ) - - verte: (, -), ais of smmetr: =, opens upward, minimum value of -, domain: { R}, range: { -, R}, -intercepts occur at (, ) and (, ), -intercept occurs at (, ). a) 1 =, = +, = -, = - 1 = -, = - +, = - -, = - - 1 = ( + ), = ( + ) +, = ( + ) -, = ( + ) - d) 1 = -, =, = -, = -. For the function f() = ( - 1) - 1, a =, p = 1, and q = -1. a) The verte is located at (p, q), or (1, -1). The equation of the ais of smmetr is = p, or = 1. Since a >, the graph opens upward. Answers MHR 1
d) Since a >, the graph has a minimum value of q, or -1. e) The domain is { R}. Since the function has a minimum value of -1, the range is { -1, R}. f) Since the graph has a minimum value of -1 and opens upward, there are two -intercepts. 7. a) verte: (, 1), ais of smmetr: =, opens downward, maimum value of 1, domain: { R}, range: { 1, R}, two -intercepts verte: (-18, -8), ais of smmetr: = -18, opens upward, minimum value of -8, domain: { R}, range: { -8, R}, two -intercepts verte: (7, ), ais of smmetr: = 7, opens upward, minimum value of, domain: { R}, range: {, R}, one -intercept d) verte: (-, -), ais of smmetr: = -, opens downward, maimum value of -, domain: { R}, range: { -, R}, no -intercepts 8. a) = ( + ) - = -( - 1) + 1 = ( - ) + 1 d) = - ( + ) + 9. a) = - = - = -( - ) + d) = ( + ) - 1 1. a) (, 1) (-1, 1) (-1, ) (, 1) (, ) (, -) (, 1) (, -1) (1, -1) d) (, 1) (, 8) (, ) 11. Starting with the graph of =, appl the change in width, which is a multiplication of the -values b a factor of, reflect the graph in the -ais, and then move the entire graph up units. 1. Eample: Quadratic functions will alwas have one -intercept. Since the graphs alwas open upward or downward and have a domain of { R}, the parabola will alwas cross the -ais. The graphs must alwas have a value at = and therefore have one -intercept. 1. a) = The new function could be = ( - ) - or = ( + ) -. Both graphs have the same size and shape, but the new function has been transformed b a horizontal translation of units to the right or to the left and a vertical translation of units down to represent a point on the edge as the origin. 1. a) The verte is located at (, ), it opens downward, and it has a change in width b a multiplication of the -values b a factor of. of the graph =. The equation of the ais of smmetr is =, and the graph has a maimum value of. times people 1. Eamples: If the verte is at the origin, the quadratic function will be =.. If the edge of the rim is at the origin, the quadratic function will be =.( - ) - 1. 1. a) Eample: Placing the verte at the origin, 9 or the quadratic function is =.. Eample: If the origin is at the top of the left tower, the quadratic function is = 9 ( - 8) - or.( - 8) -. If the origin is at the top of the right tower, the quadratic function is = 9 ( + 8) - or.( + 8) -. 8.17 m; this is the same no matter which function is used. 17. = - 9 11 ( - 11) + 9 18. = - ( - ) + 9 19. Eample: Adding q is done after squaring the -value, so the transformation applies directl to the parabola =. The value of p is added or subtracted before squaring, so the shift is opposite to the sign in the bracket to get back to the original -value for the graph of =.. a) = - 7 1 ( - 8) + 1 domain: { 1, R}, range: { 7 1, R} 1. a) Since the verte is located at (, ), p = and q =. Substituting these values into the verte form of a quadratic function and using the coordinates of the given point, the function is = -1.( - ) +. Knowing that the -intercepts are -1 and -, the equation of the ais of smmetr must be = -1. Then, the verte is located at (-1, -). Substituting the coordinates of the verte and one of the -intercepts into the verte form, the quadratic function is =.7( + 1) -. MHR Answers
. a) Eamples: I chose = 8 as the ais of smmetr, I choose the position of the hoop to be (1, 1), and I allowed the basketball to be released at various heights ( ft, 7 ft, and 8 ft) from a distance of 1 ft from the hoop. For each scenario, substitute the coordinates of the release point into the function = a( - 8) + q to get an epression for q. Then, substitute the epression for q and the coordinates of the hoop into the function. M three functions are = - 1 ( - 8) + 1, = - 1 ( - 8) + 97 1, and = - 1 ( - 8) + 8 1. Eample: = - 1 ( - 8) + 1 ensures that the ball passes easil through the hoop. domain: { 1, R}, range: { 1, R }. (m + p, an + q). Eamples: a) f() = -( - 1) + Plot the verte (1, ). Determine a point on the curve, sa the -intercept, which occurs at (, 1). Determine that the corresponding point of (, 1) is (, 1). Plot these two additional points and complete the sketch of the parabola.. Eample: You can determine the number of -intercepts if ou know the location of the verte and the direction of opening. Visualize the general position and shape of the graph based on the values of a and q. Consider f() =.( + 1) -, g() = ( - ), and h() = -( + ) -. For f(), the parabola opens upward and the verte is below the -ais, so the graph has two -intercepts. For g(), the parabola opens upward and the verte is on the -ais, so the graph has one -intercept. For h(), the parabola opens downward and the verte is below the -ais, so the graph has no -intercepts.. Answers ma var.. Investigating Quadratic Functions in Standard Form, pages 17 to 179 1. a) This is a quadratic function, since it is a polnomial of degree two. This is not a quadratic function, since it is a polnomial of degree one. This is not a quadratic function. Once the epression is epanded, it is a polnomial of degree three. d) This is a quadratic function. Once the epression is epanded, it is a polnomial of degree two.. a) The coordinates of the verte are (-, ). The equation of the ais of smmetr is = -. The -intercepts occur at (-, ) and (-1, ), and the -intercept occurs at (, -). The graph opens downward, so the graph has a maimum of of when = -. The domain is { R} and the range is {, R}. The coordinates of the verte are (, -). The equation of the ais of smmetr is =. The -intercepts occur at (, ) and (1, ), and the -intercept occurs at (, ). The graph opens upward, so the graph has a minimum of - when =. The domain is { R} and the range is { -, R}. The coordinates of the verte are (, ). The equation of the ais of smmetr is =. The -intercept occurs at (, ), and the -intercept occurs at (, 8). The graph opens upward, so the graph has a minimum of when =. The domain is { R} and the range is {, R}.. a) f() = -1 + f() = 1 - +. a) f() - - - - f() = - - verte is (1, -); ais of smmetr is = 1; opens upward; minimum value of - when = 1; domain is { R}, range is { -, R}; -intercepts occur at (-1, ) and (, ), -intercept occurs at (, -) f() 1 f() = - + 1 1 8 - - Answers MHR
verte is (, 1); ais of smmetr is = ; opens downward; maimum value of 1 when = ; domain is { R}, range is { 1, R}; -intercepts occur at (-, ) and (, ), -intercept occurs at (, 1) p() - - - - verte is (1.,.1); ais of smmetr is = 1.; opens downward; maimum value of.1 when = 1.; domain is { R}, range is {.1, R}; -intercepts occur at (-., ) and (, ), -intercept occurs at (, ) - - d) p() = + -8 verte is (-, -9); ais of smmetr is = -; opens upward; minimum value of -9 when = -; domain is { R}, range is { -9, R}; -intercepts occur at (-, ) and (, ), -intercept occurs at (, ) g() d) verte is (., 1.); ais of smmetr is =.; opens downward; maimum value of 1. when =.; domain is { R}, range is { 1., R}; -intercepts occur at (, ) and (1., ), -intercept occurs at (, ). a) - - g() = - + 8-1 - - -8-1 verte is (, -); ais of smmetr is = ; opens downward; maimum value of - when = ; domain is { R}, range is { -, R}; no -intercepts, -intercept occurs at (, -1) verte is (-1., -1.1); ais of smmetr is = -1.; opens upward; minimum value of -1.1 when = -1.; domain is { R}, range is { -1.1, R}; -intercepts occur at (-, ) and (.7, ), -intercept occurs at (, -) verte is (-., 11.9); ais of smmetr is = -.; opens upward; minimum value of 11.9 when = -.; domain is { R}, range is { 11.9, R}; no -intercepts, -intercept occurs at (,.). a) (-, -7) (, -7) (, ) 7. a) 1 cm, h-intercept of the graph cm after s, verte of the parabola approimatel. s, t-intercept of the graph d) domain: {t t., t R}, range: {h h, h R} e) Eample: No, siksik cannot sta in the air for. s in real life. 8. Eamples: a) Two; since the graph has a maimum value, it opens downward and would cross the -ais at two different points. One -intercept is negative and the other is positive. Two; since the verte is at (, 1) and the graph passes through the point (1, -), it opens downward and crosses the -ais at two different points. Both -intercepts are positive. MHR Answers
Zero; since the graph has a minimum of 1 and opens upward, it will not cross the -ais. d) Two; since the graph has an ais of smmetr of = -1 and passes through the - and -aes at (, ), the graph could open upward or downward and has another -intercept at (-, ). One -intercept is zero and the other is negative. 9. a) domain: { R}, range: { 8, R} domain: {., R}, range: { 8, R} Eample: The domain and range of algebraic functions ma include all real values. For given real-world situations, the domain and range are determined b phsical constraints such as time must be greater than or equal to zero and the height must be above ground, or greater than or equal to zero. 1. Eamples: a) = 1 (-1, ) (, ) - - - d) (, 1) - - - (, ) = 11. a) { 8, R} (, 1) The maimum depth of the dish is cm, which is the -coordinate of the verte (, -). This is not the maimum value of the function. Since the parabola opens upward, this the minimum value of the function. d) {d - d, d R} e) The depth is approimatel 17.19 cm, cm from the edge of the dish. 1. a) - (1, -) = - (-, ) (-1, ) - - - - (-, -) - 8 (-1, ) (, ) (1, ) - - = 1 The h-intercept represents the height of the log..1 s; 1.9 cm d). s e) domain: {t t., t R}, range: {h h 1.9, h R} f) 1. cm 1. Eamples: a) {v v 1, v R} v f 1. 7 11. 1 1 1. 1 Answers MHR
f(v) 1 f(v) =.v 7 1 1 1 The graph is a smooth curve instead of a straight line. The table of values shows that the values of f are not increasing at a constant rate for equal increments in the value of v. d) The values of the drag force increase b a value other than. When the speed of the vehicle doubles, the drag force quadruples. e) The driver can use this information to improve gas consumption and fuel econom. 1. a) v d) The verte indicates the maimum area of the rectangle. e) domain: { - 1, R}, range: {A A 7, A R}; the domain represents the values for that will produce dimensions of a rectangle. The range represents the possible values of the area of the rectangle. f) The function has both a maimum value and a minimum value for the area of the rectangle. g) Eample: No; the function will open downward and therefore will not have a minimum value for a domain of real numbers. 1. Eample: No; the simplified version of the function is f() = + 1. Since this is not a polnomial of degree two, it does not represent a quadratic function. The graph of the function f() = - + ( - ) + 1 is a straight line. 17. a) A = - + 1; this is a quadratic function since it is a polnomial of degree two. The coordinates of the verte are (81, 11 ). The equation of the ais of smmetr is = 81. There are no -intercepts. The -intercept occurs at (, 1 ). The graph opens upward, so the graph has a minimum value of 11 when = 81. The domain is {n n, n R}. The range is {C C 11, C R}. Eample: The verte represents the minimum cost of $11 to produce 81 units. Since the verte is above the n-ais, there are no n-intercepts, which means the cost of production is alwas greater than zero. The C-intercept represents the base production cost. The domain represents thousands of units produced, and the range represents the cost to produce those units. 1. a) A = - + 1 + (, ); The verte represents the maimum area of m when the width is m. d) domain: { 7, R}, range: {A A, A R} The domain represents the possible values of the width, and the range represents the possible values of the area. e) The function has a maimum area (value) of m and a minimum value of m. Areas cannot have negative values. f) Eample: The quadratic function assumes that Maria will use all of the fencing to make the enclosure. It also assumes that an width from m to 7 m is possible. 18. a) Diagram Diagram Diagram The values between the -intercepts will produce a rectangle. The rectangle will have a width that is greater than the value of and a length that is less times the value of. Diagram : square units Diagram : square units Diagram : 8 square units A = n + n Quadratic; the function is a polnomial of degree two. MHR Answers
d) {n n 1, n N}; The values of n are natural numbers. So, the function is discrete. Since the numbers of both diagrams and small squares are countable, the function is discrete. e) A 1 A = n + n, n N d) Eample: Using the graph or table, notice that as the speed increases the stopping distances increase b a factor greater than the increase in speed. Therefore, it is important for drivers to maintain greater distances between vehicles as the speed increases to allow for increasing stopping distances. 1. a) f() = + +, f () = + 8 +, and f() = + 1 + 9 f() f() = + 1 + 9 1 f() = + 8 + 8 19. a) A = πr 1 domain: {r r, r R}, range: {A A, A R} n 1 f() = + + d) The -intercept and the -intercept occur at (, ). The represent the minimum values of the radius and the area. e) Eample: There is no ais of smmetr within the given domain and range.. a) d(v) = 1.v. + v 1 v d 1 7 7 1 119 1 17 1 17 8 91 d(v) 1 1.v v d(v) = +. 1 1 1 No; when v doubles from km/h to km/h, the stopping distance increases b a factor of =.7, and when the 1 velocit doubles from km/h to 1 km/h, the stopping distance increases b a factor of 119 =.98. Therefore, the stopping distance increases b a factor greater than two. v Eample: The graphs have similar shapes, curving upward at a rate that is a multiple of the first graph. The values of for each value of are multiples of each other. d) Eample: If k =, the graph would start with a -intercept times as great as the first graph and increase with values of that are times as great as the values of of the first function. If k =., the graph would start with a -intercept of the original -intercept and increase with values of that are of the original values of for each value of. f() 1 1 1 8 f() = + 1 + 1 f() =. + + 1. 1 Answers MHR 7
e) Eample: For negative values of k, the graph would be reflected in the -ais, with a smooth decreasing curve. Each value of would be a negative multiple of the original value of for each value of. f() f() = - - - 1 - - - -8 f() = - - 8 - f) The graph is a line on the -ais. g) Eample: Each member of the famil of functions for f () = k( + + ) has values of that are multiples of the original function for each value of.. Eample: The value of a in the function f() = a + b + c indicates the steepness of the curved section of a function in that when a >, the curve will move up more steepl as a increases and when -1 < a < 1, the curve will move up more slowl the closer a is to. The sign of a is also similar in that if a >, then the graph curves up and when a <, the graph will curve down from the verte. The value of a in the function f () = a + b indicates the eact steepness or slope of the line determined b the function, whereas the slope of the function f() = a + b + c changes as the value of changes and is not a direct relationship for the entire graph.. a) b = b = - and c = 1. a) Earth Moon h(t) = -.9t + t + h(t) = -.81t + t + h(t) = -1t + 8t h(t) = -.9t + 8t h(t) = -.9t + 1 h(t) = -.81t + 1 Eample: The first two graphs have the same -intercept at (, ). The second two graphs pass through the origin (, ). The last two graphs share the same -intercept at (, 1). Each pair of graphs share the same -intercept and share the same constant term. d) Eample: Ever projectile on the moon had a higher trajector and staed in the air for a longer period of time.. Eamples: a) (m, r); appl the definition of the ais of smmetr. The horizontal distance from the -intercept to the -coordinate of the verte is m -, or m. So, one other point on the graph is (m + m, r), or (m, r). (-j, k); appl the definition of the ais of smmetr. The horizontal distance from the given point to the ais of smmetr is j - j, or j. So, one other point on the graph is (j - j, k), or (-j, k). ( s + t, d ); appl the definitions of the ais of smmetr and the minimum value of a function. The -coordinate of the verte is halfwa between the -intercepts, or s + t. The -coordinate of the verte is the least value of the range, or d.. Eample: The range and direction of opening are connected and help determine the location of the verte. If q, then the graph will open upward. If q, then the graph will open downward. The range also determines the maimum or minimum value of the function and the -coordinate of the verte. The equation of the ais of smmetr determines the -coordinate of the verte. If the verte is above the -ais and the graph opens upward, there will be no -intercepts. However, if it opens downward, there will be two -intercepts. If the verte is on the -ais, there will be onl one -intercept. 7. Step The -intercept is determined b the value of c. The values of a and b do not affect its location. Step The ais of smmetr is affected b the values of a and b. As the value of a increases, the value of the ais of smmetr decreases. As the value of b increases, the value of the ais of smmetr increases. Step Increasing the value of a increases the steepness of the graph. 8 MHR Answers
Step Changing the values of a, b, and c affects the position of the verte, the steepness of the graph, and whether the graph opens upward (a > ) or downward (a < ). a affects the steepness and determines the direction of opening. b and a affect the value of the ais of smmetr, with b having a greater effect. c determines the value of the -intercept.. Completing the Square, pages 19 to 197 1. a) + + 9; ( + ) - + ; ( - ) + 1 + 9; ( + 7) d) - + 1; ( - 1). a) = ( + ) - 1; (-, -1) = ( - 9) - 1; (9, -1) = ( - ) + ; (, ) d) = ( + 1) - 7; (-1, -7). a) = ( - ) - 18; working backward, = ( - ) - 18 results in the original function, = - 1. = ( + ) - 7; working backward, = ( + ) - 7 results in the original function, = + + 17. = 1( - 8) - ; working backward, = 1( - 8) - results in the original function, = 1-1 + 8. d) = ( + 7) - ; working backward, = ( + 7) - results in the original function, = + - 9.. a) f() = -( - ) + 1; working backward, f() = -( - ) + 1 results in the original function, f() = - + 1. f() = -( + 1) + 177; working backward, f() = -( + 1) + 177 results in the original function, f() = - - -. f() = -( + 1) + 91; working backward, f() = -( + 1) + 91 results in the original function, f () = - - +. d) f() = -7( - 1) + 111; working backward, f() = -7( - 1) + 111 results in the original function, f () = -7 + 18-7.. Verif each part b epanding the verte form of the function and comparing with the standard form and b graphing both forms of the function.. a) minimum value of -11 when = - minimum value of -11 when = maimum value of when = - d) maimum value of when = 7. a) minimum value of - 1 minimum value of maimum value of 7 d) minimum value of -1.9 e) maimum value of 18.9 f) maimum value of 1. ) - 11 8. a) = ( + 1 9 ) + 88 = -( + 1 ) + = ( - 9. a) f() = -( - ) + 8 Eample: The verte of the graph is (, 8). From the function f() = -( - ) + 8, p = and q = 8. So, the verte is (, 8). 1. a) maimum value of ; domain: { R}, range: {, R} Eample: B changing the function to verte form, it is possible to find the maimum value since the function opens down and p =. This also helps to determine the range of the function. The domain is all real numbers for non-restricted quadratic functions. 11. Eample: B changing the function to verte form, the verte is ( 1, - ) or (., -.7). 1. a) There is an error in the second line of the solution. You need to add and subtract the square of half the coefficient of the -term. = + 8 + = ( + 8 + 1-1) + = ( + ) + 1 There is an error in the second line of the solution. You need to add and subtract the square of half the coefficient of the -term. There is also an error in the last line. The factor of disappeared. f() = - 9 - f() = [ -. +. -.] - f() = [( -. +.) -.] - f() = [( -.) -.] - f() = ( -.) - 1.1 - f() = ( -.) -.1 There is an error in the third line of the solution. You need to add and subtract the square of half the coefficient of the -term. = 8 + 1-1 = 8[ + ] - 1 = 8[ + + 1-1] - 1 = 8[( + + 1) - 1] - 1 = 8[( + 1) - 1] - 1 = 8( + 1) - 8-1 = 8( + 1) - 1 Answers MHR 9