Notes A E F. $11,960; $242,000 G. $14,337; $117,800 H % I. 12% K. a = 3, b = 2 or a = -3, b = -2

Transcription

1 1.01 A B C. D E F. $11,960; $4,000 G. $14,337; $117,800 H % I. 1% J log or e ln or K. a = 3, b = or a = -3, b = - L i 0 Algebra II Indicators p. 33 Answer Key

2 1.0 A. 43x 5-810x 4 y x 3 y - 70x y xy 4-3y 5 B. x 6 + 1x 5 y + 60x 4 y + 160x 3 y x y xy y 6 81 C. 4x + 11x+ 33+ x D. x 8x+ 36 x + 5 E. 4(x + 4y ) F. (x + 3)(4x - 6x + 9) G. (3x - 1)(x + 5) H. (x + 1)(x + 1)(x - 1) I. (3y + 4)(x + ) J. K. x x + 1 L. x x M. a a N. x - y O. 6.6 ohms P. 3.4 seconds Algebra II Indicators p. 34 Answer Key

3 .01 A. (x + 4) + (y - 5.5) = 6.5 (x + 4) + (y - 0.5) = 6.5 (x - 1) + (y - 5.5) = 6.5 (x - 1) + (y - 0.5) = 6.5 B. vertex: (3.5, -3.5) y-intercept: (0, 9) x-intercepts: 7+ 13, 0, 7 13, 0 C. vertex: (-10, ) x-intercept: (, 0) y-intercepts: 0 6 +, 30 3, 0 6, 30 3 ( ) ( ) ( ) D. x-intercepts: 3+ 6, 0, 3 6, 0 y-intercepts: ( 04, + 13), 04, 13 Algebra II Indicators p. 35 Answer Key

4 .0 A. Center: (, -) ( ) ( ) Foci: + 13,, 13, major axis: 8 Minor axis: 3 x-intercepts: none y-intercepts: (0, -0.5), (0, -3.5) B. Center: (-, 3) ( ) Vertices: ± 6,3 Asymptotes: y y x-intercepts: ± y-intercepts: none 7 = 3 x = 3 x ,0 Algebra II Indicators p. 36 Answer Key

5 C. x-intercepts: (-7.83, 0), (1.83, 0) y-intercepts: (0, 8.96). (0, -0.96) D. x-intercepts: (0.76, 0), (5.4, 0) y-intercepts: (0, 0.38), (0, -8.38) Algebra II Indicators p. 37 Answer Key

6 3.01 A. The data is widely scattered between $0.71 and $1.36. There is not an algebraic expression that will model year and gasoline prices very well because of the data s scattered nature. Students should research the petroleum industry and identify variables that are likely to affect the prices of petroleum-based products. Some variables they are likely to discover are: supplies available from petroleum producing nations; time of year; weather; wages and benefits for industry employees; refinery capacity; transportation costs; taxes; number of automobiles. Students should correlate these variables with prices and determine if algebraic models are appropriate. Gasoline: January Price per Gallon Algebra II Indicators p. 38 Answer Key

7 B. The best model available (linear, exponential, quadratic) is y = (0.8675) X ; this does not fit very well. A better idea is for a student to contact the PGA and find out how the prize money is distributed. Winnings are dependent upon the place an individual finishes. According to the model, the prize awarded depreciates in value about 13% for each place from first. Other variables that may affect prize money would include category of play (major championship event); sponsorships; ticket prices; television contracts. PGA Championship Place Algebra II Indicators p. 39 Answer Key

8 C. The data is increasing for the period shown. A linear model, y = x , r = , appears to fit reasonably well for estimation purposes. Transportation costs, weather, federal subsidies, and labor costs are a few variables that affect the price of food. Have students investigate variables specific to NC agriculture. Apples: Price per Pound D. B-40 L-Tower Area: 1n - 6 Volume: 3n - Cut Block Area: 6n Volume: 3n - 3n + 1 B-41 Blocks Area: good luck!! Volume: n-1 Steps Area: n + 5n Volume: 0.5n + 0.5n Algebra II Indicators p. 40 Answer Key

9 E. Circulation increases until 1985 and then begins to decline. Competition (TV, radio, cable, internet) is the major variable that affects circulation. Since the relation is increasing and then decreasing, a quadratic model is probably best, y = -0.01x + 1.6x According to the model, newspaper circulation will drop below 50 million about Newspaper Circulation Algebra II Indicators p. 41 Answer Key

10 3.0 A. The data is decreasing and should provide a good opportunity for a discussion of data and its outliers. Gate receipts, sponsorships, and TV/radio contracts are variables that can affect prize money. Ask a student to contact NASCAR and find out how prize money is determined. None of the three (linear, quadratic, exponential) provide a good model. If first place is excluded, exponential becomes the better model, y = 79901(0.93) X. Brickyard 400 1, Place Algebra II Indicators p. 4 Answer Key

11 B. The data is increasing over the period shown. A linear model (y = 0.43x ) fits well (r = 0.98). According to the linear model, the portion of food expenditures that will be spent outside the home will increase 0.43% annually. Increase in the number of parents working (two income families, working single parent) is probably the biggest contributor. Students may say more after school, extracurricular activities contribute. According to the model, 03 will be when we spend half of the money we spend on food outside the home. Food Expenditures Away from Home Algebra II Indicators p. 43 Answer Key

12 3.03 A. Two: -6 c < 6.15 One: c = 6.15 None: c > 6.15 B. Three: -45 c < -3 Two: c = -3 One: c > -3 None: there is always at least one real zero C. Four: e < 78 Three: 78 e < 79 Two: 79 < e < 446 One: 446 < e < 447 None: e > A. 33-5x B. -5x - 30x - 3 C. 13 D E. 33-5x F. x 3 5 G. 3 5x Algebra II Indicators p. 44 Answer Key

13 3.05 A. As b increases, the curves relocate, curling right to left. B. Same shape and y-intercept; reflections of each other across the y-axis. C. The curve opens up and the vertex moves right and down (+x, -y); the y-intercept remains at (0, 5). Algebra II Indicators p. 45 Answer Key

14 D. Open in opposite directions and intersect at (0, c). The relationship between the two curves could also be expressed as a combination of two reflections, one vertical and one horizontal or rotation about (0, c). E. The shape of the curve remains the same; as c increases, curve moves up the coordinate plane (+y). Algebra II Indicators p. 46 Answer Key

15 F. 197 surplus: y 1 > y deficit: y 1 < y G increasing circulation; decreasing circulation. Peaked in According to the equation, circulation approximated 45 million in 1950 and expects the same in 011. H. y-intercept: (0, 6) x-intercepts: (, 0), (6.5, 0) 17 vertex: 4 81, 8 Algebra II Indicators p. 47 Answer Key

16 3.06 A. 355 seconds; 90 seconds B. 1997; 1994; Yes, Algebra II Indicators p. 48 Answer Key

17 3.07 A. As d increases, the function moves up the coordinate plane (+y). B. The functions have the same y-intercept and are 180 o rotations of the other. The curves open in opposite directions. C. As a increases, the turns in the curve become less pronounced; straighter. Algebra II Indicators p. 49 Answer Key

18 D. As b increases, the turns in the curve become more pronounced. E. Low in early 199 (1.3, 0.696); high in late 1996 (4.9, 1.368); low in late 1998 (7.8, 0.975); increasing , after 1998; decreasing Prices fluctuate over the domain given. Since the function increases after 1998, you expect gas prices to increase the next three years. According to the function, gas prices will reach $ in late 000 (9.7,.015). Some variables that affect the price of gasoline are labor costs, technology, infrastucture, refining capacity, availability of crude petroleum, and political unrest in petroleum producing countries. Algebra II Indicators p. 50 Answer Key

19 F. 3x 3 + x = 9x + 6 given x (3x + ) = 3(3x + ) distributive property x (3x + ) - 3(3x + ) = 0 add/sub property of equality (x - 3)(3x + ) = 0 distributive property x - 3 = 0 or 3x + = 0 multiplicative property of zero (x + 3)(x - 3) = 0 3x = - add/sub property of equality difference of squares (distributive) x + 3 = 0 or x - 3 = 0 x = 3 mult/div property of equality mult property of zero x = 3 or x = 3 add/sub property of equality Therefore x = 3, 3, or 3 Algebra II Indicators p. 51 Answer Key

20 G. As c increases, the curve closes and looks more parabolic. H. As e increases, the curve moves vertcally (+y) I. As a increases positively, the curve closes and the turns are less pronounced. Algebra II Indicators p. 5 Answer Key

21 3.08 A. In the 7th month, $ ; in the 0th month, $17.577; decreasing, then increasing; by month 9, the metal reaches a value of $ B. 1956; the incidence of measles in 1940; decreasing; increasing; decreasing; 1963; increasing population, vaccinations. Algebra II Indicators p. 53 Answer Key

22 3.09 A. N-shaped curve intersecting the x-axis once at -. B. G(x) will intersect x-axis one unit to the left of F(x). C. H(x) will intersect the x-axis at, -, and -3. D. Generally increasing functions; N-shaped; H(x) shares an x-intercept with F(x) and G(x). Algebra II Indicators p. 54 Answer Key

23 3.10 A. The function is decreasing except at x = where it is undefined. Asymptotes at y = 0 and x = ; intercept at (0, -0.5) 1 The zero for y = x - is where y = x is undefined. B. The verical asymptote moves to the left (-x direction) as b increases.. Algebra II Indicators p. 55 Answer Key

24 C. The horizontal asymptote moves up (+y direction) as c increases. D. For the domain x < 0.5, function is increasing; for domain x > 0.5, function is decreasing; undefined at x = - and x = 3. The zeros for y = x - x - 6 are are 1 where y = is undefined. x x 6 Algebra II Indicators p. 56 Answer Key

25 E. The vertical asymptotte moves to the left (-x direction) as b increases. F. The horizontal asymptote moves up (+y direction) as c increases. Algebra II Indicators p. 57 Answer Key

26 G. The function is decreasing except at x = - and x = 3 where it is undefined. Asymptotes are x = -, x = 3 and y = 0 for the domain x < - and x > 3. x y = x and y = share an x x 6 intercept at the origin; the zeros for y = x - x - 6 are the vertical asymptotes x for y = x x 6 ; x the zero for y = is the vertical x x 6 x x 6 asymptote for y = x Algebra II Indicators p. 58 Answer Key

27 H. The function is increasing for x < 0 and decreasing for x > 0 except for x = - and x = 3 where the function is undefined. The origin is the only intercept. The asymptotes are x = -, x = 3, and y = 0. y = x and x y = share an intercept at the x x 6 origin; the zeros for y = x - x - 6 are the x asymptotes for y = ; the zero x x 6 x for y = is an asymptote for x x 6 x x 6 x y = ; y = and x x x 6 x x 6 y = share an asymptote, x y = 1, for domain x > 3. Algebra II Indicators p. 59 Answer Key

28 I. Decreasing for x < -.1, increasing for -.1 < x < 0.8, and decreasing for x > 0.8 except at x = -3, -1, and where the function is undefined. Asymptotes at x = -3, x = -1, x =, and y = 0. The 3 zeros for y=x + x 5x 6are 1 where y= 3 x + x 5x 6 is undefined. Algebra II Indicators p. 60 Answer Key

29 J. Increasing for x < 0.36 and undefined at x = -3, -1, and. Intercepts at (-, 0) and (0, 1 ); asymptotes at x = -3, 3 x = -1, x =, and y = 0 for x > and x < -3. y = x + shares an x-intercept with y= y= y= x+ 3 x + x 5x 6 ; x+ 3 x + x 5x 6 and 3 x + x 5x 6 are undefined for x+ x+ the same x-values; y= 3 x + x 5x 6 is undefined where y = x 3 + x - 5x - 6 has zeros. Algebra II Indicators p. 61 Answer Key

30 K. Decreasing for x < -7.4, increasing for -7.4 < x < -., decreasing for -. < x < 0.8, increasing for 0.8 < x < 8.8, and decreasing for x > 8.8; undefined at x =-3, -1, and. Intercepts at (-5, 0) and (5, 0); asymptotes are x = -3, x = -1, and x =. y = x - 5 and x 5 y= share x-intercepts; 3 x + x 5x 6 x 5 y= 3 x + x 5x 6 and 3 x + x 5x 6 y= are undefined at x 5 3 x = -3, -1, and. y=x + x 5x 6 has intercepts where x 5 y= is undefined. 3 x + x 5x 6 Algebra II Indicators p. 6 Answer Key

31 L. 1 = x 4 Given x ( 4)= 1 Multiplication property of equality 1 x 4= Division property of equality 7 x = Addition property of equality x= ± 7 Square root of equivalent expressions M. F(0) = $1 million and G(0) = $11.8 million; F(x) is increasing for x > 1, G(x) is increasing for x > 1; profits are increasing when the functions are increasing. F is making money beginning in the third year (x 3) and G is making money beginning in the fifth year (x 5). F is earning more than G for the period -14 years. Algebra II Indicators p. 63 Answer Key

32 3.11 A. The x-intercept moves closer to the origin and the curve gets steeper. B. The curve moves to the left (-x direction). C. The curve moves up the coordinate plane (+y direction) as c increases. Algebra II Indicators p. 64 Answer Key

33 D. As b increases, the curve moves to the left (-x direction). E. As b increases, the y-intercept moves up (+y direction) and the curve flattens out. F. Decreasing for x and increasing for x 3. Intercepts and minimums at (-, 0) and (3, 0); range only defined for y 0; undefined for - < x < 3. y = x x 6 and 1 y = are undefined for the x x 6 same domain. Algebra II Indicators p. 65 Answer Key

34 G. x + 5x+ 4 = x Given x + 5x+ 4= 4 4x+ x Square equivalent expressions 9x = 0 Add/sub properties of equality x = 0 Division property of equality H. 6x + 4 = x + 1 Given 6x + 4 = x + x + 1 Square equivalent expressions 0 = x - 4x - 3 Add/sub properties of equality x = ± 7 Apply quadratic formula; simplify I. x = 5.3 (approx.) J. x = -7.1 (approx.) Algebra II Indicators p. 66 Answer Key

35 K. No solution Zoom view, first quadrant L. Min = $17.80, max = $40.1; 14th month; in the 7th month (6., 50.07) M. Min = $605.39, max = $104.05; in the 18th month (17.7, ); in the 8th month (7.75, 50.9) Algebra II Indicators p. 67 Answer Key

36 3.1 A. 6.1 years; 7 years; 8.4 years B a b = c d C. Answers vary. An example is: y = 11 x 10 7 y 3 x 64 = y 5 x 3 = 5 D. (3.09, 0.6, 0.7) E. (1.71, 1.75, 3.08, -0.38) Algebra II Indicators p. 68 Answer Key

37 3.13 A. An example: y -3x - 0 y -3x - 5 B. x 6 10 y 40 x + y 35 I(x,y) = 10x + 6.5y I(6, 9) = $ y 4 x y 4 x Algebra II Indicators p. 69 Answer Key

38 3.14 A. The graph closes and moves closer to the y-axis. The vertex approaches the origin along the x-axis. B. The graph moves away from the y-axis (-x direction). C. rhombus Algebra II Indicators p. 70 Answer Key

39 D. Answers vary. An example is: y 9-4x - 6 E. 3x Given 3x and 3x Definition of inequality with respect to absolute value -17 3x Re-express inequalities as a compound inequalitiy -13 3x 1 Addition property of inequalities 13 x 7 3 Mult/div property of inequalities F. Solve by graphing y = 16-3x and y = x + 3; x = {3.5, 9.5} Algebra II Indicators p. 71 Answer Key

40 3.15 A. As b increases, the curve becomes steeper. B. As b approaches zero, the curve becomes steeper and approaches y = 0 more quickly. C. As a increases, the curve becomes steeper. Algebra II Indicators p. 7 Answer Key

41 D. () ( ) log 5 log 3.1,0 and (0, -8). E. (006, 754,468); 1 years; y = 49316( ) x (x = 0 for 1998) F. $18.31; 7 months Algebra II Indicators p. 73 Answer Key

42 G. Let x = 0 for The data generates y = 10498(1.178) x as the exponential curve of best fit. Assuming that we are in the 001 model year, a new vehicle costs $63,67. The depreciation function, based on the age of the vehicle in 001, is y = 6367(0.849) x A. c = B. d 6 = e 3.5 a C. x = e b D. x = log c a ( ) log b Algebra II Indicators p. 74 Answer Key

43 3.17 A. Solve = 465.6R 8 for R to determine the annual growth rate. R = R 7 = 78.9 million passengers in 005. B. y = 465.6(1.035) x is the model for airline passengers in millions since 1990 (x = 0 for 1990). C. $500; $4.39 D e r = given 350 = e r 00 division property of equality ln 350 = r 00 law of logarithms (equivalent expressions); definition of natural log ln = r division property of equality r simplify expression Algebra II Indicators p. 75 Answer Key

44 E. 960 = A given A = division property of equality A simplify expression F x = given 663 =.165 x 49 division property of equality log 663 = log(.165 x ) 49 law of logarithms (equivalent expressions) log 663 = xlog(.165) 49 law of logarithms (exponential expressions) log log(.165) = x division property of equality 3.37 x simplify expression Algebra II Indicators p. 76 Answer Key

45 4.01 A. 170; y = 0.5x - 1.5x B. Both sets of data are decreasing over time, flattening out for the last several Olympics. The women s performance have improved more over the domain shown. Training, health and nutrition, and increased number of swimmers competing are a few independent variables affecting the swimmers performances. Men: y = 337.1(0.9956) x (x = 4 for 194) Women: y = 399.5(0.9946) x (x = 4 for 194) Women showed the greater improvement, seconds compared to the men s seconds, within the domain shown. Accoring to the models: men (004, 16.58), women (004, 3.0) The actual results at the 004 Athens Olympics: men, 3.10; women, According to the models, the winning woman at the 068 or 07 Olympics (depends how much you round off constants in the algebraic model) will outpreform her male counterpart. Reality may be a different matter. It certainly would be an appropriate topic to address in conjuction with Health/PE, Anatomy/Physiology, Allied Health Sciences, Biomedical Technology, or Sports Medicine classes. Algebra II Indicators p. 77 Answer Key

46 C. y = 0.008x (x = 58 for 1958); (005, 0.38) According to the model, first class postage increases $0.008 annually. Technology, fuel, and labor costs are independent variables that affect postage rates. Algebra II Indicators p. 78 Answer Key

47 4.0 A. For ther domain given, the quadratic best-fit for Imports appears to fit the best and the correlation coefficient compares favorably to the exponential ( compared to ). Imports: y = 0.883x x (x = 70 for 1970) Imports: Exponential Best Fit Imports: Quadratic Best Fit Algebra II Indicators p. 79 Answer Key

48 For ther domain given, the quadratic best-fit for Exports appears to fit the best and the correlation coefficient compares favorably to the exponential ( compared to ). Exports: y = 0.691x x (x = 70 for 1970) Exports: Exponential Best Fit Exports: Quadratic Best Fit Trade was balanced, according to the best models, in 197. Algebra II Indicators p. 80 Answer Key

49 B. The exponential and quadratic curves of best fit both appear to fit the data well. The coorelation coefficients are virtually the same, and respectively. The graph of the residuals shows both curves fitting well, although the quadratic has one distinctly bigger bump. The exponential model, y = 0.394(1.0144) x (x = 0 for 1790), predicts the 000 population most accurately (7.93 million estimated versus million actual). Exponential Best Fit Quadratic Best Fit Exponential Residuals Quadratic Residuals Algebra II Indicators p. 81 Answer Key

50 C. Since the function is always decreasing, an exponential best-fit is probably the best model. y = 300(0.619) x (x = 6 for September 6). According to the model, 38.1% of the customers have their power restored daily. Power restored to all customers (<1000) on September 1. D. Within the domain provided, the data increases and then decreases, hence a quadratic curve as the choice for the best fit curve. y = x x (x = 0 for 1900) According to the model, US petroleum reserves will be exhausted by Technology, price, and political/economic situations in other petroleum producing countries are some variables that affect production. For 1859, x = -61, the model provides irrelevant data. The data can be adjusted to include 1859 (x= 0 for 1859) with the petroleum production for that year. It appears there will be no simple algebraic model for the data Algebra II Indicators p. 8 Answer Key

51 4.03 A. Solve 144 = 106.3R 1 for R to find the growth rate for the domain identified. (R = 1.056) Solve 144R x = 81.4 for x to determine how long until all Americans are connected. x = 6.5 months (October 00). B. Solve = 010R 50 for R to find the annual family income growth. (R = 1.016) Solve 44568R x = for x to determine how for family median income to reach $50,000. x = 7. years (004). C. 3454(1.0448) 3 = $ Solve 4500 = 3454(1.0448) x for x to determine how long it will take tuition to reach $4500 at the growth rate indicated. x = 6.0 years (010-11) Algebra II Indicators p. 83 Answer Key

52 4.04 A. Total Value of Goods = Balance of Trade = B. 1+a 3+a +a ; +b 4+b 5+b 1+a 3+a +a ; +b 4+b 5+b ; Algebra II Indicators p. 84 Answer Key

53 C = D or E ; F Algebra II Indicators p. 85 Answer Key