Nelson Functions 11 Errata

Transcription

1 Nelson Functions 11 Errata 1: Introduction to Functions Location Question Correct Answer Getting Started 6a Graph is correct but vertex and axis of symmetry are not labelled. Add blue point labelled (in black) (0, 6) and add broken black line on top of y-axis, labelled (in black) x = 0. Getting Started 6b Graph is correct but vertex and axis of symmetry are not labelled. Add blue point labelled (in black) (2, 1) and add broken red vertical line, labelled (in black) x = 2. Getting Started 6c Graph is correct but vertex and axis of symmetry are not labelled. Add blue point labelled (in black) ( 4, 2) and add broken black vertical line, labelled (in black) x = 4. Getting Started 6d Graph is correct but vertex and axis of symmetry are not labelled. Add blue point labelled (in black) (3, 9) and add broken black vertical line, labelled (in black) x = 3. Getting Started 7c 1 Vertical compression, scale factor, then translate right 2 1 unit and down 4 units Getting Started 9 In the table, right column, Number of quadrants should be: 2, 3, or 4 Last line of non-table text should be changed to: linear relations may only enter 2 or 3 quadrants, quadratic relations must enter at least 2 quadrants Insert at start of answer: y = 6; y = 2 or 3; 1.1 7c The text answer is correct. Change graph to a graph with x-scale 10 to 2, y-scale 8 to 4, standard labels + graph of y = 3 ( x + 3) 1 4, labelled with this equation Nelson Education Ltd. 1

2 1.1 13b Use graph with x-scale 1 to 11, y-scale 6 to 6, standard labels + rays as shown: d In line 2, change more to: less 1.2 6a domain = { 2, 2, 3, 5, 7}, f(x) =& x(281 x) Add between 1st and 2nd sentences: Advantages: function notation connects input with output; can write expressions involving more than one function. Change student examples will vary to: Function notation makes relations clearer, for example, T(d) = d helps show that the temperature depends on the depth and T(3585) = d shows that the temperature at a depth of 3585 m is being determined f Change last part to: range = {y R y = 6, 2 y < 2, y 4} Above graph, insert table: Speed (km/h) Time (h) d Change last part to: range = {P R 5 2 P 40} 1.5 2de Switch the entire answers for parts d and e; that is, graphs and text y = 0.38x a 1 1 Change label on graph from y = x to: y = x Nelson Education Ltd. 2

3 1.7 7b Use graph with x-scale 12 to 12, y-scale 2 to 6, standard labels + graphs as shown; label right hand graph y = x, 1 1 upper left graph y = x, lower left graph y = x ] a g(x) = 3x a Insert before g(x) : f(x): domain = {x R}, range = {y R y 0} b Insert before g(x) : f(x): domain = {x R x 0}, range = {y R y 0} c Insert before g(x) : f(x): domain = {x R x 0}, range = {y R y 0} Replace C with: C: horizontal compression, factor ; b Replace with graph with x-scale 14 to 10, y-scale 6 to 4, standard labels + graphs as shown; label curve 1 y = f ( x + 1) ] Change range to: range = {y R y 2} 1.8 7c Change range to: range = {y R y 1} 1.8 9c Change range to: range = {y R y 4} 2009 Nelson Education Ltd. 3

4 Graph the following with unchanged scale, labels: Change the radical sign to: y = 3 ( x 5) b a = 2 1, k = 1, c = 4, d = c 6, In 2nd line, before apply vertical stretch, insert: C a Change translation 3 units left and 1 unit up to: translation 6 units left and 2 units up b 2 y = ( x + 6) or y = ( x + 6) a Yes; vertical stretch must be done before vertical translation 19a Change inequality sign in domain to: 19b Change inequality sign in domain to: 19c Change inequality sign in domain to: 19d domain = {x R x 9}, range = {y R y > 3} Self- 6a Test Replace with the following graph with x-scale 5000 to , y-scale 500 to 2500, x, y labels + ray with point as shown, label ray f(x) = 0.04(x 2500) scale exactly as shown: 2009 Nelson Education Ltd. 4

5 Self- Test 6c Replace graph with the following graph with x-scale 500 to 2500, y-scale 5000 to , x, y labels + ray with point as shown, label ray f 1 (x) = 25(x 1500) scale exactly as shown: Self- Test 6e Replace the final value with: = $8500 Self- 8a 1 Test Insert after d = 2 : ; y = 2 ( x 2) 1 Change function label on graph to: y = 2 ( x 2) Self- 8b 4 Test Insert after d = 2 : ; y = 3 x Change function label on graph to: y = 3 x + 2 2: Equivalent Algebraic Expressions Location Question Correct Answer Add: Answers may vary. For example, c Replace answer with: 5x + 24, where x is the number in the top left corner; 5x + 18, where x is the number in the top right corner; 5x 24, where x is the number in the bottom right corner a Add: Answers may vary. For example, b Add: Answers may vary. For example, 2.2 9a a Add: Answers may vary. For example, b Add: Answers may vary. For example, will have terms a Answers may vary. For example, 45 i) 4 ii) = 20 iii) 2025 = b The area of the region between the outside of the inner ring and the outside of the outer ring c d 2009 Nelson Education Ltd. 5

6 Mid- 2a Mid- 2b Mid- 2c Mid- 2d 2.4 4d The answer is just: no The answer is just: yes The answer is just: yes The answer is just: no 2.4 5a 2.4 5b, 2.4 6a Add: denominator cannot equal zero 2.4 6b Add: the denominator cannot equal zero 2.4 6c Add: the denominator cannot equal zero 2.4 6d Add: the denominator cannot equal zero 2.4 6e Add: the denominator never equals zero 2.4 6f Add: a iii b yes; a and b and c and b Add 2.7 2d 2.7 9d b Factor the quadratic denominators and determine the common denominator by taking the product of the unique factors of each denominator. Answers will vary. For example, and factor to; b The LCD is and. Answers may vary. For example, or 2009 Nelson Education Ltd. 6

7 17a or 17bi 17bii, so he cannot win 3: Quadratic Functions Location Question Correct Answer c between $ and $ It is possible, because maximum rectangular area occurs when rectangle is 125 m by m 3.3 6a b The graph is correct but replace with and replace with c a iv c iv Mid 10 Mid 12a Mid 12b 3.5 5a about (2.59, 0), about ( 0.26, 0) 3.5 6c 836 or d 900 or c Add "about": about 2.1 m a Add "about": about 1.68 s and about s Add to the answer of $2.75: (It is unreasonable to raise the fare to $14.25.) A: break-even at x = 4.8 B: break-even at about x = 0.93 or 5.22 C: break-even at about x = 2.24 or 6.19 Buy Machine B. It has the earliest break-even point c Change equation to: f(x) = 443(x 1.35) b about (1.91, 8.91), about ( 1.57, 5.43) 2009 Nelson Education Ltd. 7

8 3.8 4d about ( 1.59, 3.97), about ( 0.16, 3.22) either $3.00 or $ Before the answer, add: Answers may vary. For example, Add "about": about 7.20 s Add "about": Yes, at about 0.18 s after kick at (0.18, 4.0) 16 Use k instead of x: k < 0.5 or k > b Add "about": about 8.8 m s 1 3 Cumulative Location Cumulative s 1 3 Cumulative s 1 3 Question Correct Answer 24a. d. 35a The answer in the book is correct, but in each part of the equation, change 1 in the inequality to: 0 8 or 30 students 4: Exponential Functions Location Question Correct Answer 4.1 2c They are similar in that both first difference tables show a multiplicative pattern. They are different in that in the first case the values decrease sharply and then level off while in the second case the values start level and then increase sharply e 4.2 1f c (m + 4n) b 4.3 3d c 4.3 6d 4.3 6f 8 2 = a Answers may vary. For example, m = 1, n = b n = b x 2 y 2 = d y < 0 Mid- 1b 4.5 2a exponential; the values decrease at a fast rate 4.6 2c The base function is. Vertical stretch factor 7 and translation of 1 unit down and 4 units to the right Replace graph, graph should not show x < 0: 2009 Nelson Education Ltd. 8

9 Replace graph with the following c 16 years ago b Add the word "about": 49.3% b Answers may vary. For example, y = 4.25 x c There are too few pieces of data to make a model, and the exponential growth cannot continue indefinitely b Add the word "about": 32.2% 5a a 0 5b b 1 5f 6 7b for a, b > 0 ; therefore 11b The graph is correct, but type should be: y = 2 x ; vertical compression by a factor of, reflection in the y-axis, and translation of 1 unit up 11c The graph is correct, but type should be: y = 3 x ; reflection in the x-axis, vertical stretch by a factor of 2, horizontal compression by a factor of, and a translation of 2 units left 2009 Nelson Education Ltd. 9

10 11d The graph is correct, but: y = 5 x ; reflection in the x-axis, vertical compression by a factor of, horizontal compression by a factor of, and a translation of 3 units right and 10 units up 12 Change equation to: y = 0.5 -x c Graph should not show x < 0. Change y to: f(t) 14f There would be a horizontal compression of the graph; that is, the values would decrease more quickly. 17d about 7.2% Self- Test Self- Test 1b 1c If the second differences are 0 then the relationship is linear. If the second differences are equal but non-zero then the relationship is quadratic. If the second differences show a multiplicative pattern then the relationship is exponential. reflection in the x-axis, vertical compression of, horizontal compression of, and a translation of 2 units left and 5 units up. 5: Trigonometric Ratios Location Question Correct Answer 5.1 5b ii b iii c Add: Answers may vary. For example, 5.3 3b 5.4 2a For r, sin θ, cos θ, and θ, replace = by =& 5.4 2b For r, sin θ, cos θ, and θ, replace = by =& 5.4 2c For r, sin θ, cos θ, and θ, replace = by =& 5.4 2d For θ, replace = by =& 5.4 5abcd For sin, cos, and tan, replace = by =& 5.4 6abcdef For θ, β, replace = by =& 5.4 7b c e f ab For θ, replace = by =& 2009 Nelson Education Ltd. 10

11 Mid- 3 tan 54 or 234, csc 46 or 134, sec 44 or 316, cot 36 or b add: except 90 and d add: except 90 and = R.S d L.S. and R.S. are reversed, but the identify is proven not an identity; for example, csc sec b Last line should read: where sin α ± b iv sin β cos β b vi sin x 0, cos x b 68 or m about m m Carol on same side as 66 only. Distance to 66 is 11 m: a) 28 m b) 36 m c) 63 m Carol on same side as 66 only. Distance to 35 is 11 m: a) 5 m b) 6 m c) 2 m Carol on same side as 35 only. Distance to 66 is 11 m: a) 9 m b) 11 m c) 5 m Carol on same side as 35 only. Distance to 35 is 11 m: a) 19 m b) 24 m c) 24 m All on same side. Distance to 35 is 11 m: a) 19 m b) 24 m c) 7m All on same side. Distance to 66 is 11 m: a) 28 m b) 36 m c) 16 m Carol on same side as neither. Distance to 35 is 11 m: a) 5 m b) 6 m c) 6 m Carol on same side as neither. Distance to 66 is 11 m: a) 9 m b) 11 m c) 19 m 2009 Nelson Education Ltd. 11

12 a) N L M sin L sin M l > m, = l m height = msin L b) L N M or L N M h = msin L msin L < l < m c) N L h = msin L l < h M b 3 km The art in the question should have 8.8 m, 6.5 m, not cm a Add: (Answer: 85 ) 5.7 9b Add: (Answer: 273 m) In art for the answer, change: 85.2 to m b A is higher by 210 m 5.8 3b 38 cm 5.8 3d a m Determine angle D using the sum of angles rule. Then, determine b using the sine law. Finally, determine h using the sine trigonometric ratio The question needs to change 13.5 m to: 8.8 m Add to current answer: First, calculate the distance from Tara to the boat. Since the angle of elevation for both girls is the same, the distance between each girl and the boat is the same. Then, the cosine law can be applied to determine the angle between Tara and 2009 Nelson Education Ltd. 12

13 the boat b You need the altitude of the balloon and the angle formed by the horizontals of the friends sight lines m 3cii Graphic should show β = 55, not θ = 55 (keep in fourth quadrant); show θ = 55 (in first quadrant) on diagram too 6 Add: β 0, 180, 360 7a Add: α 90 or 270 7b Change last line to: φ 0, 90, 180, 270, or 360 7c Change last line to: x 0, 90, 180, 270, or 360 7d Change last line to: θ 90 or km or 30.5 km m Self-Test Self-Test 4ai 4bi θ should be replaced by: φ Add: φ 90, 270, α 0, 180, 360 6: Sinusoidal Functions Location Question Correct Answer Getting 3a 31 Started Getting 3b 153 Started Getting 7 Replace by: or Started 6.1 4a period: about b period: about e period: Graph should show more than one cycle, and add: Answers may vary. For example, Graph should show more than one cycle, and add: Answers may 2009 Nelson Education Ltd. 13

14 vary. For example, c period: 7 min; axis: d = 25; amplitude: 15 cm c Change domain to: domain: 6.2 3a The graph x-axis should be adjusted to show max and min at +1 and Add: Answers may vary. For example, Replace circle with: function b Dots should be removed from the graph d The amplitude is the maximum positive or negative displacement from rest e Answers may vary. For example, yes, because she is never closer than 2 m from the motion detector a Graph should be horizontally translated so that there is a minimum at 0 s, and should be extended to 120 s b Graph should be horizontally translated so that there is a minimum at 0 s, and should be extended to 45 s d Replace graph with the following, with x-label t and y-label h (to match other graphs in this question): 6.3 8a Graph should be shifted up so that the range is from 0 to 52 cm b approximately 50 cm 6.3 8c approximately 337 cm In the question, the in the equation should be after 2sin(12t) c Table for displacement of small gear should be extended to 24 s d 0.52 m e 0.87 m Mid- 2e Add units: psi Mid- 5e Change last part to: 2: 503 cm/s 2009 Nelson Education Ltd. 14

15 Mid- 6a Graph should be replaced by a screen (and window settings): Mid- 6c axis: P = 0; the average position is 0 with respect to due west 6.4 2d Remove: horizontal translation of 30 ; 6.5 1abcd Add the words: in any order 6.5 1e Add the words: in any order, as long as the horizontal translation is after the horizontal stretch Add the words: Order may vary, as long as any horizontal translations are after any horizontal stretches or compressions and any vertical translations are after any vertical stretches or compressions d The 335 on the x-axis should be replaced by f Graph should extend to ( 12.4, 720). Replace these: 6 has been dropped from 360 on x-axis and 16 on y-axis 6.5 8c Change X max to: d Switch Y min and Y max values Add: Answers may vary. For example, Reflection in x-axis, vertical compression of, vertical translation of 30 upward, horizontal compression of Add: Answers may vary. For example, Add: Answers may vary. For example, Add: Answers may vary. For example, 6.6 5b y = 4cos(0.5x) c y = 3sin(1.5x) Add: Answers may vary. For example, 6.6 8a Graph should be horizontally shifted so that it runs from 0 to 11 months. Also, graph should show data points listed so it represents a scatter plot with a line of good fit drawn b A sinusoidal model can be used because the data is waveshaped d Answers may vary. For example, 8.1 or 10.3 using the chart c The fit is somewhat close e Answers may vary. For example, 0.8 s and 2.2 s from model or 0.6 s and 2.4 s interpolating from the chart Nelson Education Ltd. 15

16 6.6 10a Graph should show sinusoidal curves drawn to represent these points c Latitude affects average temperature as well as maximum and minimum temperatures a y = 3 sin(9000t) cm h = 7 cos(22.74t) + 8, t in seconds, h in metres 6.7 1d The range is correct. For the equation, change to: d = 0.5 sin(180t) e Add: Answers may vary. For example, Add: Answers may vary. For example, 6.7 4c Add: Answers may vary. For example, 6.7 5a Add: Answers may vary. For example, d = 30 cos[18(t 12)] 6.7 5b Add: Answers may vary. For example, d = 9 cos[18(t 12)] 6.7 6ae Add: Answers may vary. For example, Add: Answers may vary. For example, 6.7 8a Add: Answers may vary. For example, 6.7 9a Add: Answers may vary. For example, h = 30 cos(1.43d) The periods are the same. The rabbit population has a higher average value and amplitude. The fox population increases when the rabbit population is above average and decreases when the rabbit population is below average The period, amplitude, location of the axis, and horizontal shift Add: Answers may vary. For example, assuming the paint drop started at the lowest point a Add: Answers may vary. For example, 9a Graph should be extended to Nelson Education Ltd. 16

17 9b Graph should be replaced with the graph below but with the degree symbol for 90, 180, 270, and 360, x on the horizontal axis, and f(x) on the vertical axis. 11a 11g 11h 13f Graph should show data points listed so it represents a scatter plot with a line of good fit drawn. Add: Answers may vary. For example, Change February to: March Add: Answers may vary. For example, 14 To determine the equation of a sinusoidal function, calculate the period, amplitude, equation of the axis, and horizontal translation. This information will help you to determine the values of k, a, c, and d, respectively, in the equations g(x) = a sin(k(x d)) + c and h(x) = a cos(k(x d)) + c. 1f No. Since the period is 40 s, at 300 s the stair will be at the same level as it is at 20 s, which is 4 m. ; the positive root 1+ 5 r = approximates the ratio 2 Self- Test 7: Discrete Functions: Sequences and Series Location Question Correct Answer Add: Answers may vary. For example, Add: Answers may vary. For example, Add: about Change Yes, to: Yes, for n > 2, 7.4 2a of as n increases 7.4 3b Add: about Mid- 4f-ii change recursive formula (leave general term and t 6 = ):, where n > Nelson Education Ltd. 17

18 Mid- 5b-i Change geometric to: neither Mid- 10b t 1 = 3, t 2 = 2, t n = t n-2 + t n-1, where n > a Add: where n > e For both, change = to: =& b Using,. As n approaches infinity, 7.7 4e ( 2 + 3) = 8x x + 180x x + 270x x e Trailing + should be (i.e. + to ) 22e Add: about Self-Test 8b p 6 + 6q, p 7 7q, p 8 + 8q 8: Discrete Functions: Financial Applications Location Question Correct Answer Getting Started 8 Add: Answers may vary. For example, c 24 years and 158 days Plan B: Plan A = $ , Plan B = $ $ Answers may vary slightly depending on rounding. For example, $ or $ Answers may vary slightly depending on rounding. For example, $ or $ Mid- 5 about 11 years and 5 months 8.4 5d Answers may vary slightly depending on rounding. For example, $ or $ about 5 years and 9 months Add: Answers may vary. For example; b Answers may vary slightly depending on rounding. For example, $ or $ Answers may vary slightly depending on rounding. For example, $ or $ Nelson Education Ltd. 18

19 $ a Add "about": about 12 years 8.6 1b Add "about": about 7 years 8.6 1c Add "about": about 19 years 8.6 1d Add "about": about 8 years 8.6 4b about 5 years and 5 months 8.6 4c $ c Add "about": about 18 years and 6 months 12 about 12 years and 3 months 19 Add "about": about 4 years Self %/a compounded monthly. This is equivalent to 6.04%/a Test compounded annually Self- Test s 7 8 Cumulative Location Question Correct Answer Cumulative 12 (d) s 7 8 Cumulative s 7 8 Cumulative s Answers may vary slightly depending on rounding. For example, $ or $ c Change right-hand column head from C to: D 17 Add "about": about 5.4 h Appendix A of Essential Skills Location Question Correct Answer A 5 2a Replace graph, exactly the same scales, grid, and labels as before, but redrawing blue line (which is y = 3x 1), slope of 3, y-intercept 1: A 6 3e 9 28, Nelson Education Ltd. 19

20 A 6 3f 1 5, 2 2 A 9 3c (5a 3)(a + 2) A 14 2c ( 2, 1) A 14 2d ( 5, 4) A 14 3d y = x 2 2 A 16 5d 24.3 A 17 2a 16.0 A 17 2c Add unit cm: 23.4 cm A 17 2d Add unit cm: 13.2 cm A 17 2f Add unit cm: 30.3 cm A 17 3a t =& 6.1 cm, A =& 74, C =& 47 A 17 3b A =& 34, B =& 42, C =& 104 A 17 3c F =& 32, E =& 109, DF =& 25.8 cm 2009 Nelson Education Ltd. 20