Selected Answers for Core Connections Algebra

Transcription

1 Selected Answers for Core Connections Algebra

2 Lesson a: = 2! 6 and then =! 5 b: Yes, reverse the order of the machines ( =! 5 and then = 2! 6 ) and use an input of = a: 54 b:!7 3 5 c: 2 d: a: b: It grows b two tiles each time. c: 1 Figure 4 Figure a: 59 b: 17 c: 72 d: 6 e: 24 f: 25 g: 25 h: 25 i: a: 5 b: 3 c: 4 Lesson (Da 1) a: 24 1 = 24 minutes; 24 2 = 12 minutes b: Speed (in blocks per minute) c: The time decreases Time to Get to Friend s House (in minutes) a: 11 b: 5 2 c: 22 d: a: b: c: d: e: a: = 0 b: = an number c: = 14 d: no solution a: 5 18 b:! c:! 3 5 d: Core Connections Algebra

3 Lesson (Da 2) a: $18 b: 8.4 gallons c: The line would get steeper a: = 3 b: = 5 c: = a: 8 b: 56 c: 3 d: a: b: c: d: a: Function A = 84, Function B = no solution b: He cannot get an output of 0 with Function A. He can get an output of 0 b putting a 4 in Function B. 7 Lesson See graph at right. The graph is a parabola opening up. There is a vertical line of smmetr through (0, 3). (0, 3) is the verte and a minimum. There are no -intercepts. The -intercept is (0, 3) a: b: c: d: There is onl one line of smmetr: horizontal through the middle a: = 3 b: = 1 c: = 1.5 d: = Either 15 or 15; es Selected Answers 3

4 Lesson a: b: c: d: a: 2 b: 30 c: 13 d: a: 4 b: 2 c: 2 d: a: =! 2 9 b: no solution c: = 3 11 d: = a: b: 51 tiles. Add 5 tiles to get the net figure. Figure 0 Figure a: b: The graph is a curve, going up. As increases, increases. c: Answers will var. d: Eponential a: 1 b: 0 c: 2 d: a: = 3 b: = 1 c: = 1.5 d: = a: = 5 b: = 3 c: = The graph is a parabola opening up. The verte is at ( 4, 9) and is a minimum. It has a vertical line of smmetr through the verte. The -intercepts are ( 7, 0) and ( 1, 0). The -intercept is (0, 7). 4 Core Connections Algebra

5 Lesson V-shaped graph, opening upward. As increases, decreases left to right until = 2, then increases. -intercepts: ( 3, 0) and ( 1, 0). -intercept: (0, 1). Minimum output of 1. Special point (verte) at ( 2, 1). Smetric across the line = a: 1 b: 2 c: 11 d: a: = 2 b: = c: = 0 d: no solution Answers will var a: b: c: d: 1 2 a ab a+b b Lesson a: 1 b: 9 c: t a: 3 b: 12 c: 3 d: See table and graph below. The graph is flat S-shaped and increasing everwhere (left to right); -intercept is (8, 0); -intercept is (0, 2); an value can be input, and an value can be the output; there is no maimum or minimum; (0, 2) is a special point because that is where the S turns direction; there are no lines of smmetr No a: 0.75 b: 99 c: 2 d:! Selected Answers 5

6 Lesson , 5, a: = 7 b: = 1 c: = 9 d: = a: 7 b: 20 c: 3 d: See graph at right. It is a parabola opening down. The verte and maimum are at (0, 3). The -intercepts are approimatel ( 1.75, 0) and (1.75, 0). There is a vertical line of smmetr through (0, 3) a: 8 b: 1 c: 2 d: no solution Lesson a: Not a function because more than one -value is assigned for between 1 and 1 inclusive. b: Appears to be a function c: Not a function because there are two different -values for = 7. d: Function a: -intercepts ( 1, 0) and (1, 0), -intercepts (0, 1) and (0, 4) b: -intercept (19, 0), -intercept (0, 3) c: -intercepts ( 2, 0) and (4, 0), -intercept (0, 10) d: -intercepts ( 1, 0) and (1, 0), -intercept (0, 1) a: 2 b: a: es b: 6 6 c: a: = 8 b: = 144 c: = 3 or = 5 6 Core Connections Algebra

7 Lesson = a: 5 b: 1 c: 132 d: a: 2 b: 4 c: 5, 2, 0, 2, 4 d: 2 e: a and b: The are functions because each onl has one output for each input. c: Not a function. d: (a) D: all real numbers, R: 1!! 3; (b) D: all real numbers, R:! 0 ; (c) D:! "2, R: all real numbers All graphs have lines of smmetr. Graph (a) has multiple vertical lines of smmetr, one at each maimum and minimum; graph (b) has one line of smmetr at = 1; graph (c) has one line of smmetr at = 1. Lesson Answers will var. See graph at right a: 10 b: 3 c: 3 d:! Answers will var = No solution; ou cannot divide b zero m = 1 3 Selected Answers 7

8 Lesson f () = a: Line a: = 2! 2, Line b: = See graph at right. = 4 3! Answers will var ! 5 because of the denominator cannot be 0. Lesson a: m = 1 2 b: (0, 4) c: a: m = 2 b: m = 0.5 c: Undefined d: m = No; when = 12, = 102, so it would have 102 tiles a: m = 5 3, b = (0, 4) b: m =! 4 7, b = (0, 3) c: m = 0, b = (0, 5) a: 18 b: 4 c: undefined d: 5 Lesson a: 4 b: 16 c: = d: It would get steeper a: 4 3 b: (0, 5) c: = 4 3! a: = 12 b: = 0 c: = 8 d: no solution = Graphs (a) and (b) have a domain of all numbers, while graphs (a) and (c) have a range of all numbers. Graphs (a) and (b) are functions. 8 Core Connections Algebra

9 Lesson = a: b: c: d: a: The dependent variable is distance in meters and the independent variable is time in seconds. b: See graph at right. Mark won the race, finishing in 5 seconds. c: Barbara: = , Mark: = 4 d: 5 meters ever 2 seconds, or 5 2 meters per second. e: 2 seconds after the start of the race, when each is 6 meters from the starting line. Distance (m) Mark Carlos Time (sec) Barbara intercept: (2, 0), -intercept: (0, 10) m = (a ) (c) (b ) a: = 2 +1 b: : (0.5, 0), : (0, 1) a: 1 b: 0 c: 2 d: a: GF = 5, Fig 0 = 3 b: GF = 2, Fig 0 = 3 c: GF = 3, Fig 0 = 14 d: GF = 5, Fig 0 = 3 Selected Answers 9

10 Lesson a: (4, 0) and (0, 2) b: (8, 0) and (0, 4) a: 1 b: 1 2 c: 3 2 d: 1 5 e: The line travels downward from the left to right, so m = a: b: c: d: = IN () OUT () a: 4 b: 3 c: 1 d: 2 Lesson a: = b: Answers will var a: b: c: d: a: (3.5, 0) and (0, 2.23) b: = a: 3, (0, 5) b:! 5 4, (0, 0) c: 0, (0, 3) d: 4, (0, 7) Core Connections Algebra

11 Lesson a: The slope represents the change in height of a candle per minute, m = 0 cm per minute. b: The slope represents the gallons per month of water being removed from a storage tank, m = 900 gallons per month = a: 8 b: 1 c: 2 d: 17 e: 45 f: A = 50w +100, A = (50)(52) +100 = a: Das () Height cm () b:!3 cm 2 das or!1.5 cm/da c: =! Etension Activit The equation in part (b) has no solution. There are the same number of -terms on each side of the equation Rena is correct a: 2-99.!5 pounds 2 months or!2.5 pounds/month b: = = Selected Answers 11

12 Lesson a: 1 h 2 b: 7 c: 9k 10 d: n 8 e: 8 3 f: a: incorrect, 100 b: correct c: incorrect, 8m 6 n Let = number of weeks a: 9 b: 4 c: 1 d: a: 7 10 b: c: d: 3 Lesson b, c, d, f a: 1 4 b: 1 c: = 1 25 d: a: = 3 b: = 6 c: = 2 d: = a: m = 1 3 b: = Let = number of weeks. 1500! 35 = 915; = 17 weeks = Core Connections Algebra

13 Lesson a: b: c: d: a: = 8 b: = A pair of parallel lines Let = the number of votes for Candidate B, ( 15,000) = 109, ,000 votes a: = b: c: d: a: b: c: d: Possible equation: 2 + (!2)! (4! ) = 2 + (!3)! (! 2) or 2 + (!2)! (!)! 4 = 2 +!3!! (!2) a: = 8 b: c: It is the point where the lines intersects the -ais on the graph. It is the -value when = 0 in the table a: 16 b: 2 c: undefined d: B a: 15 2 b: 8 c: 6 2 d: 7 Selected Answers 13

14 Lesson = 190 square units (2 + 4)( + 2) = a: Multipl b 6. b: = 15 c: = a: m = 3 b: (0, 2) c: = , = 18, , = 40 months The -intercepts are (1.5, 0) and ( 1, 0); the -intercept is (0, 3) Lesson a: ( +1)( + 3) = b: (2 +1)( + 2) = a: 238 square units b: 112 square units a: = 6 b: = 16 c: = 15 d: = ounces = a: b: c: 17 d: Core Connections Algebra

15 Lesson a: b: 3m 2 4m 15 c: d: = a: After 3 hours b: ( 2, 1) See graph at right The are not. An odd number added to an odd number is an even number a: 15 b: 64 p 6 q 3 c: 3m 8 Lesson a: b: c: ! 6! Yes, for even numbers. On a number line, if ou start at an multiple of two and add a multiple of two (an even number), ou will alwas be stepping up the number line in multiples of two; ou will alwas land on an even number. No for odd numbers. For eample, = 8; the sum of two odd numbers is not alwas odd ( 5)( + 3) = a: = 8 or = 2 b: = ±7 c: = 1 or = 3 d: no solution a: b: c: d: a: 12 b: 59 c: 7 d: 9 e: 13 f: 5 Selected Answers 15

16 Lesson a: = ±7 b: = ±16 c: = 3, 17 d: = ± a: = b: = c: r = d t d: r = C 2! a: b: c: d: a: = 11 b: = See graph at right. (2, 5) a: 4 3 b: c: 6 6 d: a: = 10 or =!16 b: = 9 2,! 11 2 c: =! 1 3 or =! 1 3 d: no solution a: b: a: 2( + 5) = b: (2 + 3)( + 5) = a: = 5 b: = 2 c: = 0 d: = a: b: c: d: a: = +5 2 b: w = p 9 3 c: m = (4n+10) 2 d: = 3 16 Core Connections Algebra

17 Lesson a: = 0 b: = 8 c: = 1 d: = 3, = = a: b: c: 7 5 = 12 5 d: 8 3 = e: f: a: 6(13 21) = b: ( + 3)( 5) = c: 4( ) = d: (3 2)( + 4) = a: 15 3 b: c: 5 d: 8 3 Lesson (Da 1) 4-8. Approimatel f = a where f is the final eam score (in percent) and a is the AP score; about 79%. See graph at right a: no solution b: = ( 1, 3) Final % AP Score Cadel is correct because he followed the eponent rules. Jorge is incorrect; the problem onl contains multiplication, so there are not two terms and the Distributive Propert cannot be used. Lauren did not follow the eponent rules a: 3(! 4) = 3 2!12 b: (3 + 5)(! 4) = 3 2! 7! No; 2 is a prime number and it is even. Selected Answers 17

18 Lesson (Da 2) If = the length, 2() + 2(3!1) = 30, width is 4 in., length is 11 in Lakeisha, Samantha, Carl, Barbara, and Kendra She combined terms from opposite sides of the equation. Instead, line 4 should read 2 = 14, so = 7 is the solution This statement is sometimes true. It is true when = 0, but otherwise it is false because the Distributive Propert states that a(b + c) = ab + ac = a: b: Lesson a: t 4; 2(t 4) b: 150 c c: 14.95c v If Nina has n nickels, then 5n (2n) = 84, and n = 5 nickels See table and graph below. -intercepts ( 2, 0) and (5, 0) and -intercept (0, 10) a: not a function, D:!3 " " 3, R: 3!! 3 b: a function, D: 2!! 3, R: 2!! = 1; It will create a fraction with a denominator of zero, which is undefined a: 15 b: 4 c: 3 d: m 3 18 Core Connections Algebra

19 Lesson A ver strong positive non-linear association with no apparent outliers a: a = 0 b: m = 2 c: = 10 d: t = a: ii b: 4 touchdowns and 9 field goals a: b: Yes; ( 3, 3) and ( 2, 1) both make ( 3, 3) 3 3 this equation true Kat is correct; the 6 1 should be substituted for because the are equal a: 1 8 b: b4 c: 9.66!10 1 d: 1.225!10 7 Lesson Yes; each point makes the equation true a: (3, 5) b: Answers will var a: h = 2c 3 b: 3h +1.5c = 201 c: 28 corndogs were sold a: b: c: d: Yes; adding equal values to both sides of an equalit preserves the equalit a: = 2.2 b: = 6 c: = 10.5 d: = 0 Selected Answers 19

20 Lesson a: ( 5, 1) b: (3, 1) c: no solution a: There are infinite solutions. b: The two lines coincide. c: Since the two lines coincide, the will have an infinite number of points of intersection. Thus, the sstem has infinite solutions a: Let p represent the number of pizza slices and b represent the number of burritos sold. Then 2.50 p + 3b = 358 and p = 2b! $ a: b: c: d: a: Moderatel strong negative linear association with no apparent outliers. b: About 25mpg Lesson a: (3, 1) b: (0, 4) c: (10, 2) d: ( 4, 5) These lines coincide. There are infinite points of intersection a: = 4 or = 4 b: = 7.9 or = 1.5 c: = 5 6 or = d: = or = The are both correct. The lines coincide = 2 + 5, 105 tiles a: b = m b: = b m c: t = 1 pr d: t = A p pr 20 Core Connections Algebra

21 Lesson a: ( 0, 1 3 ) b: ( 6, 2) c: no solution d: (11, 5) n = p and n + p = 168 ; 56 nectarines are needed a: Yes, because these epressions are equal. b: 5(3) + = 32, = 2, = a: 127 b: 10 c: 4 d: a: See graph at right. u = p where p is the price in dollars and u is the number of unpopped kernels. b:! 21 kernels # Unpopped a: m = 12 b: = 24 c: = 16 5 Price ($) Lesson (Da 1) a: all numbers b: 1 ( 3, 2 3 ) c: (1, 2) d: (8, 7) a: It is a line. b: Answers will var. c: = 3 + 2; Yes, because the points are the same = 2 + 6; 206 tiles See graph at right. ( 1, 0) and (2, 0) Mr. Greer distributed incorrectl. The correct solution is = a: See graph at right. = 94! 6.7 where is the test score and is the number of tired behaviors observed. b:! 61 Selected Answers 21

22 Lesson (Da 2) n + d = 30 and 0.05n d = 2.60, so n = 8. There are 8 nickels (a), (b), and (d) = IN () OUT () a: 25 8 b: 6 c: 1.2!10 9 d: 8! Answers will var a: 2 b: 9 c: 3 d: 1 e: 3 f: 5 Lesson (Da 3) C a: no solution b: = 5, = These epressions are equivalent because of the Commutative Properties of Addition and Multiplication a: b: a: = 5 b: = 2 3 c: no solution d: = , 18, and Core Connections Algebra

23 Lesson a: Rabbits-108; 324 b: Rabbits-12; a: (1, 2) b: ( 3, 2) 5-8. a: 6 b: 2 c: 2 3 d: undefined e: = b 3 a a: b: 7 c: 1 4 d: a: Answers will var. b: = 1 c: = a: curved b: Time (das) Number of cells ,000 Number of cells a: 5 2 = 25 b: 3 51 c: 1 d: 1.6!10 11 Time (das) Jackie squared the binomials incorrectl. It should be: = , = 2 +1, 8 = 10, and = a: m = 5 b: a = 4! 7 " a: = b: = a: 1 4 b: 3 4 Selected Answers 23

24 Lesson a: ( 1, 2) b: (3, 1) a: b = t an b: = 3(b + a) c: = m d: = m a: 2 2 b: m 3 n 3 c: 1 m 3 d: 4!10 6 n a: domain: all numbers, range:! 1 b: domain: all numbers, range:! 1 c: domain: all numbers, range:! 0 d: domain: all numbers, range:! There is no association between number of pets and age a: = b: 12.5 inches Lesson a: Answers will var. b: Approimatel 228 cm. Since DeShawna measured to the nearest centimeter, a prediction rounded to the nearest centimeter would be reasonable. c: Approimatel 72 cm. d: Approimatel 166 meters. e: Approimatel 138 meters, approimatel 14 meters a: 10(0.555) = 5.55 ft b: 10(0.555)(0.555) = 3.08 ft c: 10(0.555) 5 = ft a: (1, 1) b: ( 1, 3) a: 144, 156, 168, 180 b: 264 stamps c: t(n) = 12n d: n = 31.67; She will not be able to fill her book eactl, because 500 is not a multiple of 12 more than 120. The book will be filled after 32 months The are not on the same line; m AB = 1 5, m BC = 1 3, m AC = a: = b: = Core Connections Algebra

25 Lesson a: m = 3 b: m = 6 c: m = 5 d: m = a: 3 b: = ounces a: 15 cm b: 15 2! cm a and b: Answers will var a: Eponential, because the ratio of one rebound to the net is roughl constant! 0.6. b: Roughl geometric, because it has a multiplier a: 1 b:5 c: 10! 3.16 d: undefined a: 1.03 b: 0.8z c: a: 500 liters b: liters ( 1, 7) b: 1 ( 2, 2) = a: 4 7 b: 1 c: 2 = 1 2 d: 6 3 e: 1.28!104 f: 8!10 3 Selected Answers 25

26 Lesson a: Yes, the 90 th term or t(90) = 447 b: No c: Yes, the 152 nd term or t(152) = 447 d: No e: No, n = 64 is not in the domain Answers will var a: m = 3, t(n) = 3n + 1 b: m = 5, t(n) = 5n 2 c: m = 5, t(n) = 5n + 29 d: m = 2.5, t(n) = 2.5n a: Answers will var. b: $ m = 13, b = b + 3h = 339, b = h + 15; 48 bouquets and 33 hearts Lesson a: 3.5, 1, 2.5, 6 b: Evaluate the equation for n = t(n) = 3n. t(n +1) = t(n) + 3; t(1) = = a: b: Let = weight in kg; = ; 5 months a: no solution b: (7, 2) 26 Core Connections Algebra

27 Lesson a: eponential, multipl b 12 b: linear, add (2, 4) c: other (quadratic) d: eponential, multipl b a: 3, 6, 12, 24, 48 b: 8, 3, 2, 7, 12 c: 2, 1 2, 2, 1 2, Moderate negative linear association with no outliers. The data appear to be in two clusters, probabl indicating two classes of vehicles a: (4 +1)(4 +1) = b: (4 + 5)(2! 3) = 8!12 +10! a: = 2 3 b: = 3 1 c: = d:! Lesson (Da 1) a: = 2! 4 Month () Population () b: = 5!(1.2) Year () Population () ~8.6 ~10.4 ~12.4 ~ a: 1.03 b: 0.75 c: 0.87 d: Technicall, Mathias can never leave, either because he will never reach the door or because he cannot avoid breaking the rules. The equation for this situation is = 100(0.5), where is the number of minutes that have passed and is the distance (in meters) from the door a: 8m 5 b: 2 3 c: 2 35 d: a: #1 is arithmetic, #2 is neither, #3 is geometric b: #1 the generator is to add 3, #3 the generator is to multipl b 1 2 Selected Answers 27

28 Lesson (Da 2) a: 3, 1, 1, 3, 5 b: 3, 6, 12, 24, a: = 16 5 b: no solution c: = 4 or 5 d: = a: 12, 7, 2,! 3,! 8; t(n) = 17! 5n b: 32, 16, 8, 4, 2; a n = 64( 1 2 )n B a: = +2 b: 28 grams Lesson No; the 5 th term is 160, and the 6 th term is 320. Justifications will var Yes;! a: Sequence 1: 10, 14, 18, 22, add 4, t(n) = 4n 2 Sequence 2: 0, 12, 24, 36, subtract 12, t(n) = 12n + 36 Sequence 3: 9, 13, 17, 21, add 4, t(n) = 4n 3 b: Yes, Sequence 1: 18, 54, 162, 486, multipl b 3, t(n) = 2 3 (3)n Sequence 2: 6, 3, 1.5, 0.75, multipl b 1 2, t(n) = 48( 1 2 )n Sequence 3: 25, 125, 625, 3125, multipl b 5, t(n) = 1 5 (5)n c: Answers var a: 4 b: 6 c: 8 d: a: = 23500(0.85), worth $ b: = (1.12), population 138,570, t(n) =!188n a: all numbers b: 1, 2, 3, c:! 0 d:1, 2, 3, 4, 28 Core Connections Algebra

29 Lesson a: Strong positive linear association with one apparent outlier at 2.3g. b: She reversed the coordinates of (4.5, 2.3) when she graphed the data. c: An increase of 1 cm length is epected to increase the weight b 0.25 g. d: (11.5)! 4.3g e: We predict that when the pencil is so short there is no paint left, the pencil is epected to weigh 1.4g a: arithmetic b: t(n) = 3+ 4n c: n = 26.5, so no a: (15, 2) b: ( 3, 4) 6-7. a: 6 4 b: c: 2 4 d: a: = b: b = ac c: = d: t 2 = 2g a 6-9. a: 43 b: Lesson The predicted price for a 2800 sq ft home in Smallville is $264,800 while in Fancville it is $804,400. The selling price is much closer to what was predicted in Smallville, so she should predict that the home is in Smallville a: geometric b: 5 5 = 3125 c: a n = 5 n a n = t(n) =!2 + 6n ounces a: W = V LH b: = 2( 3) c: R = E I d: = (3, 2) Selected Answers 29

30 Lesson a: The form is linear, the direction is negative, the strength is moderate, and there are no apparent outliers. b: About 5 1.6; 2.6 das c: 3.3! 2.6 = 0.7 das. The cold actuall lasted 0.7 das longer than was predicted b the linear model. d: The -intercept of 5 means that we epect a person who has not taken an supplement to have a cold that lasts five das; more generall, the average cold is five das long a n = t(n) = 4! 3 n a: = b: (12, 0) a: ( + 3+ ) = b: ( + 8)( + 3) = See graph at right. (!2, 0), (0, 2), "!2, " a: = 3!10 5 = 3 5! 2 30 Core Connections Algebra

31 Lesson (Da 1) a: The slope means that for ever increase of one ounce in the patt size ou can epect to see a price increase of $0.74. The -intercept would be the cost of the hamburger with no meat. The -intercept of $0.23 seems low for the cost of the bun and other fiings, but is not entirel unreasonable. b: One would epect to pa (3) = $2.46 for a hamburger with a 3 oz patt while the cost of the given 3 oz patt is $3.20, so it has a residual of $3.20 $2.46 = $0.74. The 3 oz burger costs $0.74 more that predicted b the LSRL model. c: The LSRL model would show the epected cost of a 16 oz burger to be (16) = $ oz represents an etrapolation of the LSRL model, however $14.70 is more than $2 overpriced a: 1.05 b: 20(1.05) 5 = $25.52 c: t(n) = 20(1.05) n a: (2, 4) b: ( 3, 2 3 ) a: 1 b: 2 c: undefined d: m =! 2 3, (3, 0), (0, 2); See graph at right a: Room temperature. The hot water will approach room temperature but will never cool more than that. b: The asmptote would be lower, but still parallel to the -ais. If the temperature outside was below zero, the asmptote would be below the -ais. Selected Answers 31

32 Lesson (Da 2) a: Answers will var. b: The -intercept is halfwa between and 7.67, so the equation is g = 9.47! 0.14d. c: For each additional mile from church, we epect families to pa $140 less for groceries this ear. d: $ a: See scatterplot at right. = b: See table below; sum of the squares is in 2 Distance from wall (in) Residual (in) a: = 2 b: = a: 0.85 b: 1500(0.85) 4! $783 c: a n = 1500(0.85) n a: D: 2!! 2, R: 3!! 2 b: D: = 2, R:!" < < " c: D:! "2, R:!" < < " d: Onl graph (a) a: a n = 20 3n b: a n = 40 ( 1 2 ) n Core Connections Algebra

33 Lesson a: = b: = and = 4.95!1.58, based on answer from part (a). c: 0 to 1.4 das. The measurements had one decimal place. d: Between 4.6 and 6.2 das. The -intercept is the number of das a cold will last for a person who takes no supplements. e: Answers will var. f: A negative residual is desirable because it means the actual cold was shorter than was predicted b the model a n = t(n) = 32( 1 2 )n a: 3 4 b: The graph is a parabola opening upward. From left to right the graph decreases until = 2 and then increases. The verte is at (2, 1). The -intercepts are at (1, 0) and (3, 0). The -intercept is at (0, 3). The line of smmetr is at = 2. The domain is all real numbers and the range is! a: (5! 3)(2! 4 + 5) = 10 2! !15 b: Answers will var a: b: c: d: a: = 7 b: = 1 c: = 9 d: = a + p = 11, 0.60a p = $5.60 ; 7 apples and 4 pears a: = 4 3 b: Yes; Substitute 3 for and 4 for She should add 1 first, since the addition is placed inside the absolute value smbol, which acts as a grouping smbol a: There is no solution, so the lines do not intersect. b: = c: Yes; both lines have the same slope = 2 1 Selected Answers 33

34 Lesson a: b: = c: d: Yes, the residual plot appears randoml scattered with no apparent pattern. e: Predicted weight is (16.8) = 5.5g, residual is 6.0! 5.5 = 0.5g. The measurements had one decimal place. f: A positive residual means the pencil weighed more than was predicted b the LSRL model a: = 2 b: = a: 92 4 b: 22 c: Multiplier of 1.03, 3% increase emploees a: 1 60 b: c: 1 24 d: Core Connections Algebra

35 Lesson a: A ver strong positive linear association with no outliers. See graph at right. b: See plot below right. Yes, the residual plot shows random scatter with no apparent pattern. c: r = a ver strong positive linear association a: With each additional degree of temperature, we predict an increase of 410 park visitors. b: The residuals are positive, so we epect the actual values to be greater than the predicted values. The predictions from the model ma be too low. c: The residual is about 17 thousand people; the LSRL predicts thousand people. actual = 7; the actual number of people in attendance was about 17,900. d: The predicted attendance is between 11,800 and 25,800 people. e: Answers will var a: a 4 = a = 23 b: a 5 = a = 29 c: 5, 11, 17, 23, a: 2a 2! 5ab! 3b 2 b: Hourl Wage ($) ($) Eperience (ears) Eperience (ears) a: = 12 7 b: = 15 Selected Answers 35

36 Lesson r = 0; Answers will var a: With a car readil available these teens might simpl be driving more and the etra time on the road is causing them to be in more crashes. b: Families which can afford the considerable epense of bottled water can also afford better nutrition and better health care u = 4, v = = a: 9 b: 11 c: = 12 or a: b: c: = 3 d: = 2 Lesson (Da 1) a: 81.5% of the variabilit in fuel efficienc can be eplained b a linear relationship with weight. b: The negative slope means there is a negative association. An increase of 1000 pounds in weight is epected to decrease the fuel efficienc b 8.4 miles per gallon a: Answers will var. b: Answers will var a: 5, 10, 20, 40, 80 b: a n = 5 2 ( 2)n a n = t(n) = n + 2, a n = t(n) =! 1 3 n !14 = , = 11 months 36 Core Connections Algebra

37 Lesson (Da 2) a: m = 2 7, b = 2 b: m = 1 3, b = 6 c: m = 5, b = 1 d: m = 3, b = All equations are equivalent and have the same solution: = Answers will var a: 3 b: 2 c: 3.24 d: There is no real solution because ou cannot take the square root of a negative number a: 1 b: 3 c: : (0, 0) and (4, 0), : (0, 0), verte: (2, 4) Selected Answers 37

38 Lesson (Da 1) 7-7. a: If s is the price of a can of soup and b is the cost of a loaf of bread, then Khalil s purchase can be represented b 4s + 3b = $11.67 and Ronda s b 8s + b = $ b: soup = $1.35, bread = $ Sometimes true; true onl when = a: It can be geometric, because if each term is multiplied b 1 2, the net term is generated. b: See graph at right. c: No, because the sequence approaches zero, and half of a positive number is still positive a: 90 cm b: cm c: t(n) = 160(0.75) n a: = b: Yes. There is random scatter in the residual plot with no apparent pattern. c: r = and R 2 = 90.7%. 90.7% of the variabilit in the length of a cold can be eplained b a linear relationship with the amount of time taking supplements. d: Answers will var a: z 8 b: r = 90 ; 2.7 pounds s 6 t 3 c: 6m2 +11m! 7 d: 2! Core Connections Algebra

39 Lesson (Da 2) a: b: c: a: a 1 = 108, a n+1 = a n +12 b: a 1 = 2 5, a n+1 = 2a n c: t(n) = 3780! 39n d: t(n) = 585(0.2) n a: 1.25 b: 0.82 c: 1.39 d: a: No, b observation a curved regression line ma be better. See graph at right. b: Eponential growth. c: m = 8.187!1.338 d, where m is the percentage of mold, and d is the number of das. Hannah predicted the mold covered 20% of a sandwich on Wednesda. Hannah measured to the nearest percent. % Mold Da a: 94 ears b: From 1966 to 1999, 429 marbles were added, which means there were 13 marbles added per ear. c: 17 d: t(n) = n e: In the ear 2058, when the marble collection is 153 ears old, it will contain more than 2000 marbles a: b: c: d: Selected Answers 39

40 Lesson = 1.2(3.3) b: = 5! Answers will var The are all parabolas, with = 2 2 rising most rapidl and = most slowl. See solution graph at right weeks a: arithmetic t(n) = 3n 2 b: neither c: geometric, r = 2 d: arithmetic, t(n) = 7n 2 e: arithmetic, t(n) = n + ( 1) f: geometric, r = There is a weak negative linear association: as dietar fiber is increased, blood cholesterol drops % of the variabilit in blood cholesterol can be eplained b a linear association with dietar fiber. Lesson Simple interest at 20%, let = ears, = amount in the account, = a: = 15!5 b: = 151(0.8) a: 8%, 1.08 b: cost = 150(1.08) 8 = $ c: $ a: = (1.0625) t b: $504, a: Sample solution at right. Answers will var. b: The model made predictions that were closer to the actual values in more recent ears. Swim time(s) a: (4, 1) b: ( 1, 2) c: Part (b) d: Part (a) Height (cm) P(heads) = 1 2 ; P(tails) = Core Connections Algebra

41 Lesson (Da 1) See graph at right a: 0.40 b: $32, $2.05 c: V(t) = 80(0.4) t Boes d: It never will e: See graph below right a: Let = oungest child, + ( + 5) + 2 = 57; The children are 13, 18 and 26 ears b: Let = months, = insects, = , = 175 3; 14 months c: Let = amount paid, 8 5 = 3 ; $4.80 d: Let a = # adult tickets, s = # student tickets, 3s + 5a = 1770, s = a + 30; 210 adult and 240 student a: b: 4m 2 + 4m +1 c: d: a: = 14, = 3 b: = 32, = 9 Value Shoes Time (ears) a: b: c: d: Selected Answers 41

42 Lesson (Da 2) a: 3 b: %, 9.6%, = 500(1.008) m a: ( 8, 2) b: 5 ( 3, 1) = a: See graph at right. = Weight of Cereal (g) b: See at right graph below. The U-shaped residual plot indicates a non linear model ma be better. c: See plots below. The residual plot shows no apparent pattern, so the power model is appropriate. d: = Packaging (in 2 ) a: sometimes true (when = 0) b: alwas true c: sometimes true (for all values of and for all ecept = 0) d: never true Residuals (g) Packaging (in 2 ) Weight of Cereal (g) Residuals (g) Packaging (in 2 ) Packaging (in 2 ) 42 Core Connections Algebra

43 Lesson See graph at right = 4(1.75) a: = 500(1.08) b: $ c:! 0,! 500 Cost ($) Both have the same shape as = 2, but one is shifted up 3 units and the other is shifted left 3 units. See graphs at right. Number of Das a: 10 b: a: a = 0 b: m = c: 5 d: 3 c: = 10 d: = 9, 3 Selected Answers 43

44 Lesson a: = 281.4(1.02) 5, million people b: million people c: 34 million people. Population growth has slowed a: a = 6, b = 2 b: a = 2, b = a: 33 5 b: m4 4q a: 2, 6, 18, 54 b: See graph shown above right. domain: non-negative integers c: See graph shown below right. d: The have the same shape, but (b) is discrete and (c) is continuous ( 3, 6) a: See graph shown at right. Weight is ver strongl positivel associated with radius in a non-linear manner with no apparent outliers. b: A power function makes sense, probabl a quadratic function since weight is a function of! r 2. The thickness does not var, so this is not a cubic function. c: See graph at right. w = 0.056r The -intercept of (0, 0) makes sense since a disk with zero radius will not weigh anthing. d: 2.8 g Weight (g) Radius (cm) Lesson a: = 2! 4 b: = 4(0.5) a: a = 3, b = 5 b: a = 2, b = a: 4 b: 2 c: 2 d: Answers will var Equation: = 4 12 ; intercepts: (3, 0) and (0, 12) 44 Core Connections Algebra

45 Lesson a: = 5!1.5 b: = 0.5(0.4) a: 2, 4, 8, 16 b: 2 n c: 1 a n = an a: = 0, 1, 2 and = 2, 0, 1 b: 1!! 1 and 1!! 2 c:! 2 and! 2 d: : all real numbers and! a: 3 2 b: 3 c: 6 d: 2 e: Never; (0.3) f: a: 16 b: 3125 c: a: b: c: d: Selected Answers 45

46 Lesson = 7.68(2.5) a: 228 shoppers b: 58 people per hour c: at 3:00 p.m a: See table at right. The two sequences are the same. b: The coefficient is the first term of the sequence, and the eponent is n 1. c: See table at right. Yes, both forms create the same sequence. d: Because the coefficient is the first term of the sequence instead of the zeroth term. Dwane subtracts one because his equation starts one term later in the sequence, so he needs to multipl or add n one less time. t t(n) t t(n) (3 2) 2 = a: Answers will var. b: See graphs at right. = Answers will var a: ( 2, 5) b: (1, 5) c: ( 12, 14) d: (2, 2) Diameter of Cell (µm) Residuals (µm) Length of Organelle (µm) Length of Organelle (µm) 46 Core Connections Algebra

47 Lesson (2 3)( + 2 4) = a: b: a: t(n) = (n 1) b: t(n) = 30!5 n a: b: c: d: e: f: a: 4( + 2) b: 5( ) c: 2( 4) d: 3( ) a: (0, 8); It is the constant in the equation. b: ( 2, 0) and (4, 0); Students ma notice that the product of the -intercepts equals the constant term. c: (1, 9); Its -coordinate is midwa between the -intercepts a: 1 b:! 7.24 c:! 4.24 Selected Answers 47

48 Lesson a: ( 6)( + 2) b: (2 +1) 2 c: ( 5)(2 +1) d: ( + 4)(3 2) a: -intercepts ( 1, 0) and (3, 0), -intercept: (0, 3) b: -intercept (2, 0), no -intercept c: -intercepts ( 3, 0), ( 1, 0), and (1, 0), -intercept (0, 2) d: -intercept (8, 0), -intercept (0, 20) 8-19 a: t(n) = 1 2 ( 1 2 )n!1 b: t(n) =!7.5! 2(n!1) (0.92) 5! $ a: (6, 9) b: (0 2) a: = b: all real numbers c: c = = Core Connections Algebra

49 Lesson If represents time traveled (in hours) and represents distance between the two trains, then =. When = 111, = 0.75 hours, which is 45 minutes. So, the time when the trains are 111 miles apart is 4:10 p.m a: 9 units b: 15 units c: 10 units d: 121 square units a: (k 2)(k 10) b: (2 + 7)(3 2) c: ( 4) 2 d: (3m +1)(3m 1) a: = 25 b: 16 = 4 c: 1 = d: = a: = 5 b: = 6 c: = 5 or 6 d: = 1 4 e: = 8 f: = 1 4 or a: On average student backpacks get 0.55 pounds lighter with each quarter of high school completed. b: About 44% of the variation in student backpack weight can be eplained b a linear relationship with the length of time spent in high school. c: The largest residual value is about 6.2 pounds and it belongs to the student who has completed 3 quarters of high school. d: (10) = 8.34 lbs e: A different model would be better because it looks like there is a curved pattern in the residual plot. Selected Answers 49

50 Lesson a: (2 + 5)( 1) b: ( 3)( + 2) c: (3 +1)( + 4) d: It is not factorable because no integers have a product of 14 and a sum of a: eplicit b: t(n) =!3+ 4(n!1) or a n =!3+ 4(n!1) c: t(50) = a 50 = 193 d: t(n) = 3! 1 3 (n!1) or a n = 3! 1 3 (n!1) a: In 7 weeks b: Joman will score more with 1170 points, while Jhalil will have a: Michelle is correct. One wa to view this is graphicall: The -intercept alwas has a -coordinate of 0 because it lies on the -ais. b: ( 4, 0) , 46, 47; + ( +1) + ( + 2) = a: 2 b: 3 c: 1 Lesson a: ( + 8)( 8) b: ( 3) 2 c: (2 +1) 2 d: 5( + 3)( 3) a: 1 b: 20 c: 5 t 3 d: a: ( 3, 7) b: (5, 1) a: 4, 8,12,16; t(n) = 4 + 4(n!1) b: 4, 8,16, 32; t(n) = 4(2) n!1 c: Answers will var a: = b: = 24 c: = 2.5 d: = 0 or a: Answers will var. b: The largest residual value is about 17ºF and it belongs to the da after the 69.8ºF da. c: (55) = 60.0ºF d: The upper bound is given b = , and the lower bound is given b =! Mitchell predicts tomorrow s temperature will fall between 42.9ºF and 76.9ºF. Despite the strong relationship between the variables, Mitchell s model is not ver useful. 50 Core Connections Algebra

51 Lesson Verte: (4, 9), -intercepts: (1, 0) and (7, 0), -intercept: (0, 7) a: 3; 7; 6; 2 b: it does not change the value of the number c: It tells us that a = 0. d: All equal 0. e: the result is alwas a: -intercepts (2, 0), ( 4, 0), and (3, 0), -intercept: (0, 18); b: -intercepts (3, 0) and (8, 0), -intercept: (0, 3) c: -intercept (1, 0) and -intercept (0, 4) a: See scatterplot at right. 45 minutes + 77 strokes = 122 b: There is a weak to moderate positive linear association between Diego s run time and the strokes taken for each match. There looks to be an outlier at 92 minutes. c: See graph shown below right. d: Ever minute of improvement in time reduces the number of strokes b 0.7 on average. e: Answers will var. Strookes Time (minutes) a: no solution b: (7, 2) a: The smbol represents greater than or equal to and the smbol > represents greater than. b: 5 > 3 c: 9 d: 2 is less than 7. Selected Answers 51

52 Lesson This parabola should have -intercepts ( 3, 0) and (2, 0) and -intercept (0, 6) a: One is a product and the other is a sum. b: first: = 2 or = 1; second: = a: = 2 or = 8 b: = 3 or = 1 c: = 10 or = 2.5 d: = a: The line = 0 is the -ais, so this sstem is actuall finding where the line 5 2 = 4 crosses the -ais. b: (0, 2) a: 4; Since the verte lies on the line of smmetr, it must lie halfwa between the -intercepts. b: (4, 2) a: 2( 2)( +1) b: 4( 3) a: (3) 3/2 b: 81 1/ c: 17 /3 Lesson a: = 1 or 4 3 b: = 0 or 6 c: = 5 or The result must be the original epression because multipling and factoring are opposite processes; a: = 3 or 2 3 b: = 2 or 5 c: = 3 or 2 d: = 1 2 or See graphs at right a: true b: false c: true d: true e: false f: false a: 1 b:! 1.6 c: 3 52 Core Connections Algebra

53 Lesson a: = b: = c: = 2 7 d: m = 1 2, (0, 4) a:! 1.4 and! 0.3 b: The quadratic is not factorable a: = 4 or 10 b: = 8 or a: 4 b: 10 c: 8 d: a: (1, 1) b: ( 2, 1 2 ) Lesson a: = ( + 3) 2 + 6, ( 3, 6) b: = ( 2) 2 + 5, (2, 5) c: = ( + 4) 2 16, ( 4, 16) d: = ( + 2.5) , ( 2.5, 8.25) a: ( 4, 1 2 ) b: ( 2, 3) c: ( 0, 5 2 ) d: (0, 4) ! 1.088; 8.8% monthl increase intercepts: ( 1, 0) and ( 2, 0), -intercept: (0, 4), solution graph shown at right a: m = 3 4, b = 29 4 b: Yes, it makes the equation a true statement a: p = 3.97v , where p is power (watts) and v is VO 2 ma (ml/kg/min). b: 280 watts. The measurements are rounded to the nearest whole number. c: = 13 watts d: r = The linear association is positive and weak. e: There is a weak positive linear association between power and VO 2 ma, with no apparent outliers. An increase of one ml/kg/min in VO 2 ma is predicted to increase power b 3.92 watts. 26.7% of the variabilit in the power can be eplained b a linear relationship with VO 2 ma. Selected Answers 53

54 Lesson a: (w +14) 2 = 144 ; w = 2 or 26 b: ( + 2.5) 2 = 2.25; = 1 or 4 c: (k! 8) 2 = 81 ; k = 1 or 17 d: (z! 500) 2 = ; z = 65 or line: (a) and (c); parabola: (b) and (d) 9-8. A and D 9-9. a: 10 1/3 b: 15 1/2 c: 18 3/4 d: 5!1/ (2, 5) a n = 1 9 3n!1 or a n = 4 27! 3n Lesson a: = 6 or = 7 b: = 2 3 or = 4 c: = 0 or = 5 d: = 3 or = = 6 or 7; es No a: The parabola should be tangent to the -ais. b: Answers var, but the parabola should not cross the -ais = a: ( + 2) 2 +1, ( 2, 1) b:(! 3) 2! 9, (3, 9) c: minimum $ a: false b: true c: true d: true e: true f: false g: true h: false 54 Core Connections Algebra

55 Lesson a: = 5 b: = 6 or 1 3 c: = 1 or 5 3 d: = ± = 1 3 or = 6; es a: = ( + 3)(!1) = 2 + 2! 3 b: = (! 2)( + 2) = 2! If = the width, then (2 + 5) = 403; the width is 13 cm Both (b) and (c) are solutions a: 3 feet per second b: He travels a net distance of 18 feet, but he travels backwards, because the conveor belt is traveling faster than he is walking a: 3 2 (!1) b: 2(! 2)(! 3) c: 8(! 2)( + 2) d: 2(2! 3)( + 4) Lesson If n = # nickels and q = # of quarters, then 0.05n q = 1.90, n = 2q + 3, and n = 13, so Daria has 13 nickels a: = ±0.08 b: = 2 9 or 4 c: no solution d: 1.4 or While the epressions ma var, each should be equivalent to = a: = 2 b: = 15 c: = 2 d: all real numbers Line L has slope 4, while line M has slope 3. Therefore, line L is steeper a: b: $ Selected Answers 55

56 Lesson Let n = number of countries in North America. Then n + (2n) + (2n + 7) = 122 and n = 23. There are 23 countries in North America (Antigua and Barbuda, Bahamas, Barbados, Belize, Canada, Costa Rica, Cuba, Dominica, Dominican Republic, El Salvador, Grenada, Guatemala, Haiti, Honduras, Jamaica, Meico, Nicaragua, Panama, St. Kitts & Nevis, St. Lucia, St. Vincent & the Grenadines, Trinidad & Tobago, and the United States), 46 countries in Europe, and 53 countries in Africa a: p >!1, b: k < 2, p k c: 1! k or k! 1, h a: k = 1.5 or 2 b: m = 3 or 3 c: w = 2 or 6 d: n = 4± or a: alwas true b: sometimes true c: never true d: sometimes true e: alwas true f: never true a: (5, 0) and (8, 0); Robbie must have backed up 5m from the launch pad and the rocket must have landed 8m awa from him. b: 3 meters a: = 5± or 4.3 b: =!1± or Core Connections Algebra

57 Lesson a: k < 2 b: p 15 c: n > 1 2 d: t > a: There is a strong negative linear association between the pressure and volume of these three gasses. There are no apparent outliers. The residual plot indicates a curved model might be better than the linear model. About 82% of the variation in the volume of the gasses is eplained b a linear relationship with pressure. On average for ever increase of one atmosphere (at a constant temperature) the volume decreases b 1.65 liters. b: The largest residual value is about 2.3 atmospheres and it belongs to ogen at 2 atmospheres of pressure. c: 9.40 liters, 6.10 liters, and 2.82 liters d: A different model would be better. There is a curved pattern in the residual plot. In fact b the ideal gas law pressure and volume have an inverse relationship. After 8.19 atmospheres of pressure the linear model will start predicting negative volumes. Students ma know at some point the gasses will condense into liquids and have much different phsical characteristics The graph should be a line with -intercept (1.5, 0) and -intercept (0, 3) =! 3 5! = 5 3 or =! a: (2! 5) 2 b: not factorable c: 3(! 4) d: 5(! 4)(2 +1) Selected Answers 57

58 Lesson a: < 4 b: 3 c: > 2 d: ! 2700, so! 5. Algeria can order an advertisement up to 5 inches high = 3(2) a: =! 2 7! 2 b: Yes; students can verif b substituting the coordinates into the equation and testing B D a: f () = ( + 3) b: ( 3, 2); See graph at right. c: The parabola has no -intercepts. 58 Core Connections Algebra

59 Lesson A a: b: g! 20, g! 19.5, so! 19 games a: = 1 3 b: = 16 c: = ±5 d: > No; Bernie would pass Wendel after 40 seconds, when each was 90 meters from the starting line. Since the race was onl 70 meters, that would occur after the race was over There are two -intercepts: (0.6, 0) and (!2, 0). Lesson a: 3 b: 1 c: 4 d: a: < 1 b: 6 c: m 2 d: no solution , 8% increase a: (5! 2)( + 3) b: 2(3t!1)(t! 4) c: 6(! 2)( + 2) a + 3c = 27.75, 3a + 2c = 32.25, a = $8.25, c = $ = 3 ± B Selected Answers 59

60 Lesson a:! 6 b: > 1 c: 2! < 7 d:!3 " "! a: false b: false c: true d: false a: The data appears randoml scattered. There is apparentl no association between time running a mile and heart rate. Onl 1% of the variation in heart rate can be eplained b a linear association with time to run a mile. The LSRL is nearl horizontal. There are no outliers. b: Answers will var. Eample responses: D- Ran a fast mile but seemed to be giving little effort. This athlete might alread be in outstanding phsical condition or have an attitude problem. F- Strong run and strong effort. Keep this plaer. N,O- Ran slowl and gave little effort. Along with plaer M, we don t know these plaers potential or motivation. Cut? P- The slowest of the group but with the highest effort. This plaer ma improve substantiall over time a: r c! 15 b: r + c! 25 c: No; the club cannot sell a negative number of items. d: See graph at right. The points represent the possible sales of rulers and compasses that would allow the club to break even or make a profit while falling within the sales limit. 60 Core Connections Algebra

61 Lesson a: Yes, the are equivalent. One wa to determine this is to change both to = m + b form and compare slope and -intercept. b: Students can multipl or divide both sides of either equation to find an equivalent equation. For eample, 2 + = 3 and = 12 are both equivalent equations < 50, 000, less than 14.8 pounds a: m > 5 b: 6 c: > 7 d: no solution Perfect square form: (! 5) 2 = 0 ; = 5; Answers var D a: = 2.75(1.05) ; $4.48 b: = 42, 000(0.75) ; 9967 c: = 25(1.09) ; 9% increase Selected Answers 61

62 Lesson Yes, he can. a: = 2 b: Divide both sides b a: ( 1 3, 2) b: (4, 9) a: Subscribe to Sunda paper and subscribe to local paper. See table at right. b: 77% c: 84.4% Subscribes to weekl local paper Subscribes to Sunda paper es no es no a: See solution graph at right. b: No, it is not; it lies on both boundaries, but the boundar to < is not part of the solution a: = 11 2,!3 b: 4± 28 6! 1.55, " = 3 4! a: See table at right ! 61% purchase washer did not purchase washer b: = 50% purchase drer a:! 2 = 4 b: For each, = 6. c: + 3 = 8, = a: 3 b: 1 c: 4 d: a: $4.10 b: $1.90 did not purchase drer a:! 2 b: >!1 c:! 9 d: > a: (8 + )(8! ) b: (4! 3)(3 + 2) c: (2 + 3) 2 d: not factorable 62 Core Connections Algebra

63 Lesson Answers var The all are equivalent to a: See table at right. b: P(Senior OceanView) = = 25% Not OceanView Ocean View Senior (0.60)(0.10) = Not senior (0.20)(0.90) = =! <! a: 2d! 3 b: 2d! 3 = 19, d = 11 candies = 6(0.8) Lesson a: = 4 b: = 5 or 2 c: = 16 3 d: = a: (0, 3) b: ( 1 2, 0) and (3, 0) a: t = 5 seconds b: 100 feet If and represent the number of minutes he spends delivering the Times and Star, respectivel, then + = 60 and 2 + = 91; = 31 and = 29 ; So he delivers 31 Times and 29 Star papers See solution graph at right = 27( 1 3 ) C Selected Answers 63

64 Lesson a: = 2 b: = 1.5 c: = a: 3+ 4 = 14, = 11 4 b: Rewriting Let represent the amount of mone the oungest child receives. Then = 775 ; $185, $370, and $ terms Marisol: = 2, Mimi: = 3! 3, solution: = 3 hrs, so 6 miles a: 0.463, 53.7% decrease b: = 20(0.463) a: = 39% b: = 56% c: = 77% d: See the relative frequenc table below. Yes, the juniors and seniors are much less likel to be carring a backpack. Freshmen Sophomore Junior Senior Backpack 73% 73% 56% 54% No Backpack 27% 27% 44% 46% 64 Core Connections Algebra

65 Lesson a: = 3 or 11 b: = 14 c: = 2 d: = No, because 1 is not greater than a: 4(! 3) b: 3( +1) 2 c: m(2m +1)(m + 3) d: (3! 2)( + 2) t = number of toppings, 1.19(3) t = 4.55, and t = a: = ( +1) 2! 2 = 2 + 2!1 b: Method 1: (5)!1 = 34 tiles; Method 2: The net term in the pattern is 34 because the terms of the sequence (2, 7, 14, 23) increase b consecutive odd numbers = 9 or = If a = # of adult tickets and s = #of student tickets, then 7a + 5s! Lesson a: = ±9 b: = ± 37 c: = ±!49 d: = ±! a: R b: I c: I d: R a: = 1 3 b: = 35 8 c: = 7 or 3 d: = See graph at right. a: (3, 4) b: D:! " < < ", R:! 4 c: The parabola open upward, so it is a minimum If h = number of hats and t = number of t-shirts then 5h + 8t! =!(! 20) =! ; Its maimum height is 100 feet when = Selected Answers 65

66 Lesson See solution at right a: The quadratic equation (!11) 2 =!4 has no real solutions because when a real number is squared, it must be positive or 0. b: = 11±! a: = 1 or 7 b: = 4 or 8 c: = 3 d: no solution a: < 2 b: 6 c: > 4 d: a: f () = 8( 1 2 ) b: See graph at right a: (2 + 3)(3! 2) b: 2(2! 5)(2 + 5) c: 2( + 8)(! 7) d: (3 + 4) a: = 4± 76 6 ; irrational b: = 5 6, 1 2 ; rational 66 Core Connections Algebra

67 Lesson No; 3(7! 2) = 15 and 15 > Yes, the will intersect; top line: =! , bottom line: = ; the will cross at (12, 7) top line: -intercept (40, 0) and -intercept (0, 10); bottom line: -intercept: ( 9, 0) and -intercept (0, 3) Both (a) and (d) are equivalent. One wa to test is to check that the solution to 4(3!1) + 3 = makes the equation true (the solution is 3 2 ) =! a: See graph at right. b: The verte is at ( 2, 3). c: = A relative frequenc table is shown at right. There is almost no difference between the amount of cheese at Taco Shack and at the competitor. An difference can easil be eplained b NUMBER OF Taco TACOS Shack competitor <15 grams cheese 11% 10% g cheese 60% 61% >25 grams cheese 30% 29% natural sample-to-sample variabilit. No association. The Taco Shack owner does not need to adjust the amount of cheese, and should consider other reasons for the difference in perception a: See graph at right. b:! ("2.3, 0.8) c: = a: I b: R c: I d: I If c = number of cars and t = number of trucks, 2c + 3t! A a: 3 +1 b: 9! 2 c: 9! 6 d: a: See graph at right. b: f () = 10(2.3) c: Anthing with an initial amount of 75 and losing 15% over each time period. Selected Answers 67

68 Lesson a: = 1 5 or 3 b: = 1 2 or 3 c: no real solution, or, =!2±!16 2 d: = 7 or a:! = 2! 4 +1 b:! = (! 2) 2! 3 c: =!(! 2) d: (2, 3); He sold 9 watermelons (6, 20) and ( 1, 6) A See solution graph at right Lesson a: Two: = 1 and =!3 b: Three: <!3,!3 < < 1, and > 1 c:!3 < < 1; a: ( +1) 2 b: (3 +1) 2! 3 c: 3( +1) 2! 9 d: ( + 4) 2! a: two b: one c: none d: one See graph at right a: = 13 b: = 3 c:!5 " " 5 d: <! 2 3 or > a: (! 7)(!1) b: (! 5)( + 3) c: 7( + 3)(! 3) d: (3 + 4)( + 2) 68 Core Connections Algebra

69 Lesson a: 2!1, shift up 2 units b: 4! 3, twice as steep c: 2 +1, shift left 2 units d: 4! 6, twice as steep, -intercept shifts down 3 units, -intercept does not change a: = 2 b: k 3.76 or 1.24 c: 2 < < 10 d: = , 400, 000! 50 > 1, 200, 000, less than 4000 square miles per ear a: -intercepts ( 2, 0) and (0, 0); -intercept (0, 0) b: -intercepts ( 3, 0) and (5, 0); -intercept (0, 3) c: -intercepts ( 1, 0) and (1, 0), -intercept (0, 1) (3, 3) and ( 2, 7) s = a + 150, 3s + 5a = 4730; 685 students a: f () + 2 = 2!1; Shifted up 2 units. b: 2 f () = 2 2! 6 ; Twice as steep and the verte shifts down 3 units. c: f ( + 2) = ( + 2) 2! 3; Shifted left 2 units. d: f (2) = 4 2! 3 ; Four times as steep a: 1 3 b:!1±!3 2 is not a real number because the square root of a negative number is an imaginar number a: 3 b: 2 c: does not eist d: 0 e: 1 f: a: = 2 + 3! 28 b: = c: = A a: Answers var. b: 8; 8 2 Selected Answers 69

70 Lesson a: f!1 () =!3 2 b: g!1 () = a: 1 3 b: =!2 ± 7 " 0.65 or! 4.65 c: all real numbers d: = 2.5 or a: (! 3) 2!1, shift down 1 unit b:!(! 3) 2, reflected over the -ais c: (! 4) 2, shift right 1 unit d: (!! 3) 2, reflected over the -ais a: one b: none c: two d: one after 44 minutes The team president is using the mean, and the fans are using the median. A few large outliers, such as super star plaers, have ver high salaries. Lesson (Da 1) (46.4, 48.5, 50.4, 52.5, 55.9) a: f!1 () = 3( + 2) b: g!1 () = (! 5) 1 2 = 2(! 5) a: = 1 b: > 3 or < 3 c: 0!! 4 3 d: = 4 3 e:!2 " " 3 f: = miles > 2! a: = 1 2 or 2 b: no real solution 70 Core Connections Algebra

71 Lesson (Da 2) a: (1.58, , 3.49, 4.29) b: See solution graph at right. c: The median (center) is at 2.91 points. The shape is smmetric. The IQR (spread) is Q3! Q1 = 3.49! 2.50 = 0.99 points. There are no apparent outliers The graph starts at (3, 1); D:! 3, R:! While there are multiple was to write the equation, one possible wa is = ( + 2)( + 3) +1. However, all equations should be equivalent to = # of Students GPA m m = 5000, m = 8.6; It will be full after 8 months; there will not be enough room for songs in the 9 th month a:!2 < < 2 b:! 2.5 c: = 1 4 d: no solution e: = 12 f:!5 " " a: The verte is at (2, 6). The coefficient of 2 means the graph is pointing downward, so the verte i a maimum. b: The - intercept is at!2(!2) =!2. Along with the verte, and knowing the parabola is pointing downward, there is enough information to make a sketch of the graph. Selected Answers 71

72 Lesson (Da 1) a: See solution graph at right. b: The median is 257 rpm. The graph is single-peaked and skewed. The IQR is Q3! Q1 = 263! 253!10 rpm. 291 rpm is apparentl an outlier. c: The median. Because the data is not smmetrical and has an outlier the mean is not an appropriate measure of center. Frequenc RPM a: t(n) =!3n +10 or t(n) = 7! 3(n!1) b: t(n) = ! 3 n"1 or a n = (3) n a: 2! 5, shift down 2 units b:! , reflected over -ais, stretched c: (! 2) 2! 3, shift right 2 units d: (!2) 2! 3 or 4 2! 3, stretched, and reflected over -ais onto itself a: The verte is ( 1, 5) and the point is a minimum. b: The will be the same after 20 ears, when both are $ a: 25a!22 b 36 b: 5! 3 "1 " represents the initial temperature, 0.7 represents a temperature loss of 30% per unit of time, 72 represents room temperature and a horizontal asmptote of the graph. 72 Core Connections Algebra

73 Lesson (Da 2) a: See graphs at right. The median of the two groups are virtuall identical. Both groups have uniform distributions of ages. Neither group has an outliers. However, the ages in Group 7B are much more widel distributed have much more variabilit than the ages in Group 7A. The IQR for 7A is onl 70! 53 = 17 ears, while the IQR for 7B is more than twice as wide at 77! 39.5 " 38 ears. The minimum for 7A is 20 ears older than the minimum for 7B, and the maimum for 7A is 22 ears ounger than the maimum for 7B. Note that the small number of data points does not allow for bin widths on the histogram much narrower than 20 ears; it is not appropriate to create bin widths of 10 ears. b: Either. Since the data distributions are smmetric and there are no outliers, either measure of center is appropriate a: f!1 () = (+2) 7 b: Yes a:! A Age (ears) 7B Age (ears) b:!10 < < 10 c: < 0 d: <!5, > = 2( +1) a: = ± 4 b: ( 5, 17) c: = 4 or 2 d: = See graph at right.!1± 57 4 " 1.64 or Based on direction and verte of the parabola compared with the slope and -intercept of the line, there are two points of intersection. Selected Answers 73

74 Lesson See graph at right. The distribution of weights is smmetric with no outliers (as determined b the modified boplot). The mean is 40 kg with a standard deviation of 16 kg. The weights are rounded to the nearest whole number a: See boplots at right below. Unequivocall, the farmer should plant in shade. The median crop is about 7 bushels higher in shade. The minimum, maimum, first quartile, and third quartile are all higher in shade. Both distributions are skewed in the same direction. The spread in data (IQR) is almost the same for both tpe of tree the middle bo is the same size for both boplots. The maimum of 127 bushels from one of the shad trees is almost certainl an outlier. b: No. Neither of the boplots are smmetrical; the distributions are skewed. The maimum on the shad plot ma be an outlier a: = 2! 4 b: = 2! 4 c: = 6 2!! ( 2, 10) a: all real numbers b: 5 < < 4 c: no solution d: = a iv, b ii, c v, d i, e iii. b is the onl histogram with a narrow range, so it matches to ii. The two skewed histograms are straightforward to match. c has a uniform distribution, so the quartiles on the boplot must be of even length, as in v. d has a lot of data at the two edges, and the data in the middle is more spread out, so the whiskers of the boplot must be narrow, and the bo must be wide, as in i sunn shad Core Connections Algebra

75 Lesson Edison is correct because 3(2) + 2(!3) = 2 and 5(2)!12 =! See solution graph at right. D:! 3, R:! a: not possible b:!27 c: 8 d: 0 e: See graph at right. The distribution is smmetric with no outliers. The mean is 50.7 cm and the standard deviation is 2.6 cm. The lengths were measured to the nearest tenth of a centimeter (1, 12) and ( 5, 42) If d = number of dimes and q = number of quarters, then q = 2d! 6 and d + q = 147. Then d = 51 and q = 96, so Jessica has 51(0.10) + 96(0.25) = $ f () =!(! 3)( +1) =!(!1) =! a: = 1 b: < 1 or > 7 c: = See graph at right. The verte is at (3, 1); -intercepts (4, 0) and (2, 0); -intercept (0, 2) a: Team 2 works, on average, a little faster the median number of widgets per team member is slightl higher. The distributions for both teams are similarl smmetric. However, the members of Team 1 are much more consistent than Team 2. The variabilit (IQR) of Team 1 is almost half that of Team 2, and Team 1 s range is less too. Neither team had outliers. b: Since both distributions are nearl smmetric with no outliers, it is appropriate to compare standard deviations. Since Team 1 had both IQR and range smaller than Team 2, we would epect that Team 1 has a smaller standard deviation. Selected Answers 75

76 Lesson a: There is a strong positive linear association between the depth of a water well and the cost to install it. There are no apparent outliers. b: On average ever foot deeper ou drill the well the cost increases b $ c: The coefficient of correlation is 0.929, R-squared = % of the variation in the cost of drilling a water well can be eplained b a linear association with its depth. d: $1395 represents the cost of a well that has no depth. It would be roughl the cost of the pump e: (80) = $2567, (150) = $3593, (200) = $4325 f: From part (e), the predicted cost is $2567. Actual $2567 = $363; actual cost was $2930. g: A linear model looks the most appropriate because there is no pattern in the residual plot = 1 2 (! 2) a: 3 b: 1 c: c b! 100, c! 0, b! The parabola has verte (1, 3) and points down. The line has -intercept at (0, 5) and decreases. There are two points of intersection a: (! 9)( + 9) b: ( + 6) 2 c: (2 +1)(2! 3) d: (4! 5)(4 + 5) 76 Core Connections Algebra

77 Lesson No a: es b: no; most inputs have two outputs c: no; =!1 has two outputs a: = 3 ± 21 b: = 2 ± a: none b: two c: two d: one a: The slope of the line of best fit is Jeremiah has been giving coins awa at a rate of about 76 coins a ear. b: In 2010 he had 1295 coins. If c is the number of coins, and is the number of ears since 2010, then c = 1295! 76. When c = 0 coins, 17 ears from now. In 2027 he will have onl 3 coins left a: The IQR for W is more than for Z because the middle of the boplot for W is wider. The standard deviation for Z is greater because overall, including the outliers, the data for Z is spread out more than for W. Since mean is impacted b outliers more than median, the standard deviation (which is based on mean) is impacted b more b the outliers in Chip Z. Mean and standard deviation are not appropriate for Chip Z because the shape is skewed and there are outliers. b: Chip Z appears to be the more energ efficient. It has a lower median use of current. Also, for most of the data sets tested, chip Z uses the same or slightl less current than chip W. Chip Z has smaller IQR: it is more consistent in current usage. However, all these benefits ma be offset b the two high outliers belonging to Chip Z which might indicate a reliabilit problem. Selected Answers 77

78 Lesson (!2.5,! 36.75); Students can complete the square or the can use the fact that due to the parabola s smmetr, the verte must have an -coordinate that is halfwa between the -intercepts a: = 2!or!!2 b: = 8 or! (6, 8) and (5, 3) D: < 2 and > 2, R: < 0 and > 0 Solution graph shown at right (6) = 8.5( + 6), 18 pounds See solution graph at right. The shape is double-peaked and smmetric. There are no outliers. The mean speed is 79.5 mph with a standard deviation of 6.8 mph ( 2, 5) and (6, 21) See solution at right Speed (mph) a: none b: two c: one d: two a: = 12 b: 4! 12 < b < c:!33 " " 27 d: n = 1 5 or intercepts: (! 0.2, 0) and (! 3.1, 0), verte (! 1.7,! "6.3) ; Solution graph is shown at right a: See table and graph at far right. b: = (! 2)(! 5) = 2! c: See graph at right. d: 4 seconds, 256 feet ( 7, 0) ( 1, 0) ( 4, 9) 78 Core Connections Algebra

79 Lesson Based on direction and verte of the parabola compared with the intercepts of the line, there are two points of intersection a:! 4 b: > 20.5 c:!5 " " 1 d: > 19 or <! = 9 or See graph at right homes See graph at right. -intercept: ( 2, 0), -intercept: (0, 2); there is no value for g(1), which creates a break in the graph B =! 3 2! Answers var, but likel answers are 6(m! 2), 2(3m! 6), 3(2m! 4), and 1(6m!12) a: 1 b: 2 c: 2 d: a: 5m 2 + 9m! 2 b:! c: 25 2! d: 6 2! =!3 or 11 Selected Answers 79