Algebra Connections 22

Chapter 4

Lesson 4.1.1 4-3. The parabola should pass through the points (0, 0) and (2, 0) and have vertex (1, –1). 4-4. She should have received two sports cars and ten pieces of furniture. 4-5. 1! 3! (!2x) = !x ! 3! (x + 2) a. x = !0.75

4-6. a. 2x2 +11x ! 4y ! 3 b. !2x2 ! 2x ! 8 c. !y2 + 33y !14 d. 0

4-7. a. y = x2 +1

Lesson 4.1.2 4-8. b. Figure 0 has 2 tiles. c. The pattern is growing by 4 tiles.

d. 100th figure has 402 tiles

4-9. Both patterns grow by 4 tiles. 4-10. a. They each have the same growth factor, but start with a different number of tiles.

b. #1: y = 4x + 2 , #2: y = 4x +1 , #3: y = 4x + 3

c. The growth of 4 corresponds to the 4x in each equation.

d. You would need to know the number of tiles in figure 0. 4-11. b. y = 2x + 3

c. They both have 3 tiles in figure 0, but they grow by different amounts.

d. The growth factor is the coefficient of the ‘x’ term.

4-13. x +1! 2 ! (!2x + 3) = x + 5 ! (2 !1) a. x = 4

4-14. a. 3 b. –1 4-15. ! 83 words 4-16. See table below.

IN (x) 2 10 6 7 –3 5 –10 1000 x

OUT (y) 9 25 17 19 –1 15 –15 2005 2x + 5

a. Multiply x by two and add five

b. y = 2x + 5

4-17. Each sister should receive 1 poster, 72 books and 8 CD’s.

Answer Key 23

Lesson 4.1.3

4-18. a. #1: y = 4x + 2 , #2: y = 4x +1 , #3: y = 4x + 3 , #4: y = 2x + 3

c. In the pattern, the growth is represented by the added tiles. In the rule, the growth shows up as the coefficient of x. In the graph, the growth can be shown as the vertical change between the points for each figure.

d. Some possible responses include: “If the graph goes in the same direction, they all add the same amount.” “The pattern in the table (growth) is the number multiplied by x in the rule.” “The first number for figure 0 is always the last number in the rule.” “If we have the rule, we have a shortcut to graph by starting with figure 0 (end #) and going up by the growth number.”

4-19. a. Figure 4 has 10 tiles.

b. A and B both increase by 2, while C increases by 4.

c. Since the lines are parallel, tile patterns A and B must grow by the same amount for each figure. Since the lines start at different values on the y-axis, Figure 0 for tile patterns A and B must have a different number of tiles (2 and 8, respectively).

d. Since lines A and C have a different “steepness,” tile patterns A and C grow by different amounts for each figure. However, since they start at the same value on the y-axis, Figure 0 for both patterns must have the same number of tiles.

4-21. c. y = 2x + 3 4-22. a. OUT: 5, 0, -3, -4, -3, 0, 5, A parabola opening upward.

4-23. a. 2 b. 0 4-24. 35 cm long and 15 cm wide.

4-25. OUT: -2.5, -2, -1.5, -1, -0.5, 0, 0.5

Lesson 4.1.4

4-26. a. They all have something added and/or multiplied by x.

b. how each figure changes c. The number of tiles in the 0th figure.

4-27. y = 3x +1 4-28. a. y = 5x ! 2 b. i: y = 2x + 5 , ii: y = !3x + 7

4-29. Answers vary 4-30. Answers vary 4-32. a. all numbers b. x = 8

3

4-33. If x is my number, then 5 ! 2x = !1; x = 3 .

4-34. IN: 5, -5, 4, 1.5, 50; OUT: -11 a. Multiply x by three and subtract two

4-35. a. Answers vary b. Each figure has 4 more tiles than the figure before it. 4-36. Figure 0 has 7; figure 1 has 9; the growth factor is 2.

Algebra Connections 24

Lesson 4.1.5 4-37. a. y = 4x + 4 b. y = 9x ! 5 c. y = !2x +18

4-39. a. 4 b. no solution c. 3.5 d. 1

4-40. a. b. c. d.

4-41. See table below. x -4 -3 -2 -1 0 1 2 3 4

y -14 -11 -8 -5 -2 1 4 7 10

b. Yes. These values for the variables make the rule true

4-42. Answers vary. Typical statements include: “Figure 4 has 32 tiles,” “Each figure has 7

more tiles than the previous figure,” “Figure 0 has 4 tiles,” and “The rule is y = 7x + 4 .”

4-43. a. !5x !1 b. 11x !1 c. 5x !1 d. !4x !1

4-44. Answers vary

4-45. a. any number b. 0 c. –6 d. no solution 4-46. a. This should be a horizontal line with positive y that ends at x = 10.

b. This should start with positive y and have 0 slope, then turn to a negative slope, then again to 0 slope at the x- axis.

c. This should start horizontally along the line x = 0, then rise quickly to a higher horizontal line, then descend to a lower positive horizontal line, ending at x = 4 .

4-47. b. Yes, -29 = 3 – 32. 4-48.

Number of

Letters

Cost of

Stamps

10 $3.40

2 $0.68

15 $5.10

7 $2.38

1 $0.34

500 $170

42 $14.28

1 1

2 2

2.5

10

–2 –5

–7

5

–5 –1

–6

–112

8 –14

–6

Answer Key 25

Lesson 4.1.6 4-49. a. y = 2x + 2 b. y = 4x + 3 c. y = 8x + 5

d. y = !2x + 8 e. y = 4x + 8 f. y = !4x +12

4-50. a. (0, 3) b. (0, 0) c. (0, 8) d. (0, –1) 4-51. a. y = 4x + 3 b. y = !3x + 5 c. y = 2

4-52. The pattern shrinks by two tiles between figures. Figure 0 has 13 tiles. 4-55. a. A and C have 5 tiles, B has 15 tiles.

b. C grows fastest at about 13 tiles per figure. It’s steepest.

c. Figure 2; the lines intersect at (2, 20).

d. y = !10x + 30

4-56. a. They each have 5 tiles. b. The second 4-57. a. 4 b. 34

4-58. 93

4 tsp

Lesson 4.1.7 4-59. a. y = 3x + 2 b. y = !3x +14

4-60. Figure 19 4-61. y = 4x ! 8

4-62. a. x: (1.5, 0), y: (0, 3)

b. x and y: (0, 0)

c. no x-intercept, y: (0, 3) d. x: (6, 0), y: (0, 3)

4-64. a. 5 b. –8 c. 76

d. -18

4-65. Figure 6; 40 = 6x + 4 4-66. Josie is correct. 10 inches long

Algebra Connections 26

Lesson 4.2.1 4-67. Evie is the traveling toward Fairbanks so she is the circle graph. Joyla is moving

away from Fairbanks so she is the triangle graph; Between 90 – 100 hours; 1100 miles; Evie’s line is steeper; Extend the line to 1100 miles and look at the time, about 230 hours.

4-68. b. (2, 2)

4-69. a. As San Francisco warms up in the spring, Rio is cooling. b. The cost of machine-made tortillas decreases as more are produced, whereas the

handmade tortillas’ cost stays the same c. Boston’s pollution increases more quickly than Denver’s as the population

increases. d. The hare catches up to the tortoise and then passes it even after starting later than

the tortoise.

4-71. 15 ÷ 8 · 20 = 37.50 minutes

4-72. a. 4896 sq. units, b. 91+13x sq. units c. 900 sq. units d. 336 sq. units

4-73. a. Leslie passes Gale b. 2:00 PM c. 18 miles

d. Gale stopped to rest, flat tire, etc. e. at approximately 6:40 PM

4-74. a. y = 3x !1 b. y = 5x + 2

4-75. a. the opposite of y increased by 8 b. 48 less than two times x

c. the square of the sum of x and 3 d. !6x2 e. x ·x + 5

Lesson 4.2.2 4-76. They will have the same amount of money in 5 weeks

4-78. a. y = 50 + 30x b. y = 75 + 25x

c. When x = 5 is substituted into both equation, y = 200 .

4-79. a. No common points appear in his tables. However, using the pattern in the tables, it is reasonable to predict that the lines intersect when x is between 1 and 2.

b. (1.5, 0)

4-80. a. Science: y = 20 +10x ; Math: y = 50 + 5x b. 6 weeks c. $80

4-81. a. x2 + 5x + 6 b. 4x +10 4-82. y-int: (0, 1), x-int: (12,!0)

4-83. a. (10)(17) = 72 + 30 + 48 + 20 = 170 b. 54

4-84. 1

3x + 74 = 90 ; 48 days later (Thursday, June 19)

Answer Key 27

Lesson 4.2.3

4-85. 7 years 4-86. a. 3x + 5 = x +19 b. x = 7 years c. 26 pounds

4-87. a. Richmond: y = 1900 !15x , Post Falls: y = 1160 + 22x

b. 1900 !15x = 1160 + 22x ; x = 20 years c. 1600 students

4-88. Ficus: y = 6 +1.5x , Oak: y = 2x ; x = 12 years; 24 feet 4-89. The solution of the equation formed is the x-value of the point at which the two rules

have equal y-values.

4-90. 8 dumplings, 12 eggrolls, 4 sweet buns and 4 drinks

4-91. He is correct, because !2(2)+ 4 = 0 and 2 ! 2 = 0 .

4-92. a. (2,1) b. (2,1)

4-93. a. (-2,-7) b. ( 12

, -4)

4-94. a. If x represents weeks and y represents total money, y = 20 + 6x and y = 172 ! 4x .

b. They will have the same amount between weeks 15 and 16.

Lesson 4.2.4

4-95. a. $1105 b. Town B increases by 6000 in one year.

c. They will have the same in 52 weeks d. Answers vary, but the story should include the point of intersection at (4.5, 25). e. $3.50, 24 more

4-96. She is not correct 4-97. a. (5,3) b. (2,-6)

4-98. a. parallel lines b. no solution c. parallel lines do not intersect

4-99. a. y = 12,200 + 300x and y = 21,000 ! 250x ; y = total population and x = number of

years b. They will have the same population in 16 years.

4-100. a. y = 27 + 43x b. After the 22nd day. 4-101. –7

4-102. This parabola should point downward and pass through (0, 0) and (–2, 0). The vertex

is at (–1, 2).

4-103. 1296 4-104. a. (0, 3) b. (0, ! 5

12) c. (0, 12)

4-105. If r is Regina’s favorite number, then 3r = 12 + 2r and r = 12 .

4-106. a. parallel lines b. no solution c. Parallel lines do not intersect.