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1 Domain and Range Class Work Find the domain and range for each of the following 1. {(1,2), (3,4), (5,6)} 2. {(4,3), (3,2), (4,2)} 3. {(5,1), (3,1), (-4,1)}

2 Homework Find the domain and range for each of the following 14. {(3,1), (-2,6), (1,4)} 15. {(1,2), (2,2), (1,2)} 16. {(2,1), (5,1), (-6,7)}

3 Discrete vs. Continuous Class Work Is the relation discrete or continuous? 47. {(1,2), (3,4), (5,6)} 48. {(4,3), (3,2), (4,2)} 49. {(5,1), (3,1), (-4,1)}

4 Homework Is the relation discrete or continuous? 60. {(3,1), (-2,6), (1,4)} 61. {(1,2), (2,2), (1,2)} 62. {(2,1), (5,1), (-6,7)}

5 Relations and Functions Class Work Is the relation a function? 73. {(1,2), (3,4), (5,6)} 74. {(4,3), (3,2), (4,2)} 75. {(5,1), (3,1), (-4,1)}

6 Homework Is the relation a function? 86. {(3,1), (-2,6), (1,4)} 87. {(1,2), (2,2), (1,2)} 88. {(2,1), (5,1), (-6,7)}

7 Evaluating Functions Class Work Let f(x)= 3x+4 and g(x)= x-4, find the following 99. f(2) 100. f(3) 101. g(6) 102. g(2) f(6) g(2) 105. f(4) g(3) 106. g(5) f(5) 107. f(0) g(3) g(a) 110. f(2b) Class Work Let f(x)= (x-1) 2 and g(x)= 2x-3, find the following 111. f(2) 112. f(3) 113. g(6) 114. g(2) f(6) g(2) 117. f(4) g(3) 118. g(5) f(5) 119. f(0) g(3) g(a) 122. f(2b) Graphing Linear Equations Chapter Problems Graph using a table Classwork For the equations below, make a table with at least 3 ordered pairs, plot the points and connect them to form the line y = 3x y = -2x y = x y = 1 2 x y = x + 1 Homework For the equations below, make a table with at least 3 ordered pairs, plot the points and connect them to form the line y = -x y = 2x + 1

130. y = 1 x 4 131. y = -2x 2 132. y = - 1 x + 4 3 Graph using the slope and y-intercept Classwork 133. Use lines A, B, C and D to answers the questions. a. What is the y-intercept of each line? b.

8 130. y = 1 x y = -2x y = - 1 x Graph using the slope and y-intercept Classwork 133. Use lines A, B, C and D to answers the questions. a. What is the y-intercept of each line? b. Is the slope of each line positive, negative, zero or undefined? 134. What is the slope of lines E, F, G and H? 135. What are the equations of lines E, F,G and H? Homework 136. Use lines I, J, K and L to answers the questions. a. What is the y-intercept of each line? b. Is the slope of each line positive, negative, zero or undefined? 137. What are the slopes of lines M, N, O and P? 138. What is the equation of lines M, N, O and P? Graph using intercepts Classwork Rewrite the equations in standard form. Graph the intercepts and then the line that passes through them y = 3x y = -2x y = x y = 2 5 x y = - 3x 144. y = x + 5

9 145. y 4 = 2(x 5) 146. y + 5 = -3(x 4) 147. y + 6 = 1 (x 6) y 3 = (x + 7) Homework Rewrite the equations in standard form. Graph the intercepts and then the line that passes through them y = 6x y = -3x y = 1 6 x y = x y = x y = -7x 155. y 5 = 2(x 3) 156. y + 2 = -4(x 2) 157. y 3 = 3 (x 3) y + 1 = 2(x + 1) Horizontal & Vertical Lines Classwork Determine if the following equations are horizontal, vertical, neither or cannot be determined y = x = x + 4y = x 21 = x + 2y = 3x 4 Homework Determine if the following equations are horizontal, vertical, neither or cannot be determined x = y = x + -4y = x 3y = -6x x = -24 Slope Formula Classwork Find the slope of the line through each of the following two points (-12,-5), (0,-8) 170. (12,-18),(11,12) 171. (-18,-20),(-18,-15) 172. (-20,-4),(-12,-10) 173. (8,10),(0,14) 174. (6,9),(3,-9)

10 175. (1,2),(5,7) 176. (3,-3),(12,-2) 177. (-4,-8),(-1,1) 178. (4,7),(-3,7) Homework Find the slope of the line through each of the following two points (3,-9),(1,1) 180. (7,4),(3,8) 181. (-3,0),(5,12) 182. (8,-2),(12,-2) 183. (6,-3),(2,9) 184. (-3,7),(-4,8) 185. (5,9),(5,-8) 186. (-5, 0.5),(-6,3) 187. (-7,1),(7,8) 188. (-2,1),(5,7) Parallel and Perpendicular Lines Classwork Match the following parallel equations from each column y = 3x y = -2x y = 1 3 x x 2 = 7y (y + x) = y = x x + 17 = 2y y 6x = (3y 9x) = y = 12x + 28 Match the following perpendicular equations from each column y = -4x y = 3x x + y = y 5 = 1 (x + 4) (4y + 8x) = y + 3 = -2(x 5) y = x x = 14y y + 4 = 1 (x 4) x + 3y = Write an equation that would be parallel to the line y = 4x Write an equation that would be perpendicular to the line 4x + 6y = -12

11 Homework Match the following parallel equations from each column y = 3x y = 16x x + 12y = y 4 = 4 (x 11) = 22x 11y Match the following perpendicular equations from each column y = -12x y 4 = 1 (x + 7) (2x 8y) = x 5y = y = -3x x 6y = y + 5 = 2(x 13) y + x = (15y 12x) = x = -4y y 9 = -4(x 2) x + 21y = (14y 7x) = y = -5x x + 30y + 20 = Write an equation that would be parallel to the line 6y = -24y Write an equation that would be perpendicular to the line y 4 = 2 (x + 4) 5 Graph using point slope Classwork Graph each equation 233. y 2 = (x 3) 234. y + 5 = 2(x + 2) 235. y 4 = -3(x 3) 236. y + 7 = 1 (x 1) y 3 = - 3 (x + 5) 4 Homework Graph each equation 238. y 2 = (x + 4) 239. y + 5 = -2(x 4) 240. y 1 = 4(x 2) 241. y 5 = - 1 (x 1) y 0 = 1 (x 3) 5 How to write equations from given information Classwork 243. Write an equation in point slope form for the line through the given point with the given slope. a. (3,4); m = 6 b. (-2,-7); m = c. (7,-4); m = -3

12 d. (4,0); m = 1 e. (-4,-4); m = A line passes through the given points. First write an equation for the line in point-slope form. Then rewrite the equation in slope-intercept form. a. (-1,0), (1,2) b. (3,5), (0,0) c. (6,-2), (9,-8) d. (-1,-5), (-7,-6) e. (-3,4), (3,-2) 245. Write the equation of the line through the given points in standard form. a. (6,3), (-4, 2) b. (12,3), (-8,-4) c. (15,-4), (8, -3) d. (7,2), (8,5) e. (20,-10), (-30, 0) 246. A line has an x-intercept of 8 and y-intercept of 12. a. Write an equation for the line. b. Write an equation that is parallel to this line. c. Write an equation that is perpendicular to the line from part a Write the equation of the line through (-3,-2) and is parallel to the line y = -2x Write the equation of the line through (7,4) and perpendicular to y = 1 2 x 5 Homework 249. Write an equation in point slope form for the line through the given point with the given slope. a. (5,-4); m = -2 b. (-2, -3); m = 4 c. (7,4); m = 1 4 d. (9,-9); m = 3 e. (6,0); m = A line passes through the given points. First write an equation for the line in point-slope form. Then rewrite the equation in slope-intercept form. a. (-10,-50), (5,25) b. (0,10), (10,-20) c. (0.5,9), (50,-90) d. (8,9), (5,-6) e. (-7, 4), (-3,6) 251. Write the equation of the line through the given points in standard form. a. (1,4), (-1,1) b. (5,-3), (3,4) c. (2,4), (-3,-6) d. (5,3), (4,5)

13 e. (0,0), (-1,-2) 252. A line has an x-intercept of -4 and y-intercept of 16. a. Write an equation for the line. b. Write an equation that is parallel to this line. c. Write an equation that is perpendicular to the line from part a Write the equation of the line through (8,5) and is parallel to the line y = x Write the equation of the line through (-2,5) and perpendicular to y = x + 3 Scatter Plots and Lines of Best Fit Classwork 255. Predict the test score of someone who spends 48 minutes studying. test scores Time spent studying 256 Predict the test score of someone who spends 34 minutes studying. test scores Time spent studying 257. Draw a scatter plot from the following data: Size of shoe Height (inches)

258. Consider the scatter graph to answer the following: Which two points would give the line of best fit? A and B A and C D and B There is no pattern 259.

14 258. Consider the scatter graph to answer the following: Which two points would give the line of best fit? A and B A and C D and B There is no pattern 259.Consider the scatter graph to answer the following: Which two points would give the line of best fit? A and B B and C C and D There is no pattern Homework 260. Using the scatter graph, predict the mile time of someone who spends 6 hours a week training. 11:55 10:43 Mile time 9:31 8:19 7:07 5: Time spent training

261. Using the scatter graph, predict the mile time of someone who spends 12 hours a week training. 11:55 10:43 Mile time 9:31 8:19 7:07 5:55 0 5 10 15 20 25 Time spent training 262.

15 261. Using the scatter graph, predict the mile time of someone who spends 12 hours a week training. 11:55 10:43 Mile time 9:31 8:19 7:07 5: Time spent training 262. Draw a scatter graph from the following data, Time spent studying (min) Grade Consider the scatter graph to answer the following: Which point would most likely be on the line best fit? A B C There would be no line of best fit 264. Consider the scatter graph to answer the following: Which two points would give the line of best fit? A and D A and C B and D There is no pattern

16 Determining the Prediction Equation Class Work 265. Use the two points (7,14) and (15,27) to write an equation for the line of best fit If the prediction equation is y=.5t+60, where t represents time in minutes, what will the person get on his test if he studies for 45 minutes? 267. If the prediction equation to determine a test grade is y=.5t+60, and someone received an 80 on the test, how long did they study for? Consider the scatter graph to answer : 268. What is the slope of the line of best fit that passes through (3.4, 7) and (8, 3)? 269. What is the y-intercept of the line of best fit that passes through (3.4, 7) and (8, 3)? 270. Consider the scatter graph to answer the following: The equation for the line of best fit is y = -1.06x Determine the value for x=15? Is this an interpolation or extrapolation? Homework , Using the scatter graph below use the two points (3.4, 7) and (9, 1) to write an equation for the line of best fit , , 7 4, , , , If the prediction equation for a test grade is y=.52t+65, where t represents the time in minutes, what grade will someone earn if they study for 30 minutes.

17 273. If the prediction equation for a test grade is y=.52t+65, where t represents the time in minutes, how long did someone study for if they received an 83 on the exam? Consider the scatter graph to answer : 274. What is the slope of the line of best fit passing through (2.7, 11.1) and (9.4, 3.7)? 275. What is the y-intercept of the line of best fit passing through (2.7, 11.1) and (9.4, 3.7)? 276. Consider the scatter graph. The equation for the line of best fit is y= -.98x Determine the value for x=5. Is this an interpolation or extrapolation? Absolute Value Functions Class Work On a separate sheet of paper, graph the following. Sate the domain and range of each y = x 278. y = x y = x y = x y = x y = x y = x y = x y = x y = x y = x y =- x 289. y= 3 x 290. y = -2 x y =.5 x 6-2 Homework On a separate sheet of paper, graph the following. Sate the domain and range of each y = x y = x y = x y = x + 3

18 296. y = x y = x y = x y = x y = x y = x y = -x 303. y= -3 x 304. y = 2 x y = -.5 x Greatest Integer Function Class Work Evaluate the following [2.2] 307. [3.5] 308. [5.9] 309. [7.98] 310. [8] 311. [-3.9] 312. [-3.4] 313. [0] 314. [ ] 315. [ ] On a separate sheet of paper graph 316. y = [x + 1] 317. f(x)= 2[x] 318. g(x)=-[x] 319. h(x)= [x] -3 Homework Evaluate the following [3.4] 321. [3.8] 322. [7.95] 323. [9.98] 324. [10] 325. [-3.8] 326. [-2.3] 327. [0.1] 328. [3(2.1)] 329. [-2(4.2)] On a separate sheet of paper graph 330. y = [x - 1] 331. f(x)= 3[x] 332. g(x)=[-x]

19 333. h(x)= [x] + 3 Piecewise Functions Class Work a. f(-2) b. f(0) c. f(4) d. state the domain and range of f e. graph f 336. a. f(-2) b. f(2) c. f(4) d. state the domain and range of f e. graph f a. f(-2) b. f(1) c. f(4) d. state the domain and range of f e. graph f a. f(-2) b. f(0) c. f(4) d. state the domain and range of f e. graph f a. f(-2) b. f(2) c. f(4)

20 d. state the domain and range of f e. graph Homework a. f(-2) b. f(0) c. f(4) d. state the domain and range of f e. graph f 341. a. f(-2) b. f(2) c. f(4) d. state the domain and range of f e. graph f a. f(-2) b. f(1) c. f(4) d. state the domain and range of f e. graph f a. f(-2) b. f(0) c. f(4) d. state the domain and range of f e. graph f a. f(-2) b. f(2) c. f(4)

d. state the domain and range of f e. graph Graphing Linear Inequalities Class Work Graph the following inequalities 344. y > 2x + 1 345. y 3x + 2 346. y 1 2 x 4 347. y < 3 4 x 348. 2x 3y > 6 349.

21 d. state the domain and range of f e. graph Graphing Linear Inequalities Class Work Graph the following inequalities 344. y > 2x y 3x y 1 2 x y < 3 4 x x 3y > x + 4y 10 Write the equation for the inequality graphed Class Work Graph the following inequalities 352. y > 3x y x y 1 2 x y < 3 2 x x y > x + 2y 11 Write the equation for the inequality graphed

22 Linear Relations- Multiple Choice 1. Find the domain of {(1,3), (5,6), (6,8)} A. {1, 5, 8} B. {1, 5, 6} C. {3, 6, 8} D. Set of Reals 2. Find the range of f(x)= x A. [3, ] B. [1, ) C.(1, ) D. [3, ) 3. What is domain of the following graph? A. {x -10< x< 10} B. {x -10< x< 10} C. {x -6< x< -2 or 0< x< 6} D. {x -10< x< -4 or -2< x< 4 or 6< x< 10} 4. Which choice represents a discrete set? A. the time it takes people to tie their shoes B. amount of rain in a given week C. number of people attending a play D. the number of rotations of a wheel 5. Which of the following is a function? A. x 2 + y 2 = 4 B. x + y 2 = 4 C. x 2 + y = 4 D. 4x 2 + y 2 = 4 6. Given f(x) = 2(x-6) 2 +2, find f(3) A. 2 B. 20 C. 29 D The x-intercept of 4x + 3y = 12 is A. 4 B. (4,0) C. 3 D. (3,0) 8. The slope of 4x + 3y = 12 is A. 4 B. -4 C. 4 3 D. 4 3

23 9. The equation of the line containing the points in the table is A. y= x + 3 B. y= 2x + 3 C. y= 1 2 x + 3 D. Points in table are not collinear X Y A line parallel to y = 2x + 6 is A. y= 2x -1 B. y-4 = 2(x+3) C. 10x 5y = 7 D. All of the above 11. The slope of a line perpendicular to the line thru (3,6) and (5,2) is A. -2 B. 1 2 C. 1 2 D An example of a line with no slope is A. y = x B. y = 3 C. x = 2 D. y = 0 x 13. A line thru (3,-2) and (4,6) has equation A. y 6 = 8(x 4) B. y 6 = 4(x 4) C. y + 2 = 8(x + 3) D. Both A and C 14. The equation of a line perpendicular to f(x) is A. y= -2x +4 B. y= 2x + 5 C. y= 1 2 x 3 D. y= 1 2 x 15. Using the graph of f(x), find f(6). A. -1 B. -2 C. -3 D. Undefined

24 In 16 and 17, consider the piecewise function 16. Find g(3) A. -9 B. 1 C. -9 or 1 D. Does not exist 17. The slope at x=1 is A. -3 B. -1 C. 1 D Find [3.75] A. 3 B. 3.7 C. 3.8 D Find [-4.14] A. -5 B C D When graphing y > 2x -3 A. solid boundary and shade above B. dotted boundary and shade above C. solid boundary and shade below D. dotted boundary and shade below 21. Which point could be used as a test point to decide where to shade when graphing y > 4x? A. (0,0) B. (-2,-8) C. ( 3, 10) D. both B and C Extended Response 1. An employer offers $12 an hour for the first eight hours of work and 1.5 times that rate for overtime. a. Create a piecewise function that models this situation. b. Make a graph of the equation in part A (Let the domain be [0,15]) c. How much does a person make if they work 10 hours? d. If a raise of $2 was given, describe how the piecewise function would change. 2. Cal C. s grandmother offers him $5 for every A he receives on his report card. a. If he takes 8 classes, what are the domain and range? b. Is the answer in part A discrete or continuous? Is it a function? c. If his grandfather gives him $10 for finishing the marking period, write an equation for how much he can make with one report card.

25 3. Line segment connects AB A(4,2) and B(7,6). a. What is the slope of? AB b. AB is the side of a rectangle, what are the slopes of the other three sides. c. Write the equation of AB. 4. Given y - 3 = 1 (x 8) 2 a. Write the equation in slope y-intercept form b. Write the equation in standard form c. The line is rotated 90 about the point (8,3), what is the equation of the new line? 1) D:{1,3,5} R:{2,4,6} 2) D:{3,4} R:{2,3} 3) D:{-4, 3,5} R:{1} 4) D:{-3,1,2} R:{2,5,7} 5) D:{4,5,6} R:{6} 6) D:{-4,0,2} R:{3,4,5} 7) D:{-2,-1,2,3} R:{0,3,4,5,7} 8) D:{1,2} R:{3,4,5,6} 9) D:{-4,0,1,2,3} R:{5,6,7} 10) D:{-4,-2,1,3} R:{0,3,4,5,7} 11) D:{x>-4} R:{y>0} 12) D:{x<-2 or x>2} R:{Reals} 13) D:{Reals} R:{Reals} 14) D:{-2,1,3} R:{1,4,6} 15) D:{1,2} R:{2} 16) D:{-6,2,5} R:{1,7} 17) D:{-1,0,1} R:{6,7,8} 18) D:{2,4} R:{6,7,8} 19) D:{-5,0,5} R:{-2,-1,0} 20) D:{3,4,5,6} R:{1,2,3,4} 21) D:{5} R:{0,1,2,3} 22) D:{3,4} R:{2,3,4} 23) D:{-4,-2,2,4,5} R:{-3,2,4,5} 24) D:{-6<x<6} R:{-6<y<6} 25) D:{-6<x<0 } R:{-6,-2,2,4} 26) D:{Reals} R:{2} 47) D 48) D 49) D 50) D 51) D 52) D 53) D 54) D 55) D 56) D 57) C 58) C 59) C 60) D 61) D 62) D Answers 63) D 64) D 65) D 66) D 67) D 68) D 69) D 70) C 71) C 72) C 73) yes 74) yes 75) yes 76) yes 77) yes 78) no 79) no 80) no 81) yes 82) no 83) yes 84) yes 85) yes 86) yes 87) no 88) yes 89) yes 90) no 91) no 92) yes 93) no 94) no 95) yes 96) no 97) yes 98) yes 99) ) ) 2 102) 2 103) ) 1

26 105) ) ) ) 1 109) a-4 110) 6b+4 111) 1 112) 4 113) 9 114) 1 115) ).5 117) 6 118) 9 119) 1 120) ) 2a-3 122) (2b-1) 2 = 4b 2-4b+1 123) (0, -4), (4/3, 0), (1, -1) 124) (0, 4), (2, 0), (1, 2) 125) (0, -3), (3, 0), (1, -2) 126) (0, 4), (-8, 0), (1, 4.5) 127) (0, 1), (3/2, 0), (1, 1/3) 128) (0, -2), (-2, 0), (1, -3) 129) (0, 1), (-.5, 0), (1, 3) 130) (0, 0), (1, ¼), (2, ½) 131) (0, -2), (-1, 0), (2, -6) 132) (0, 4), (12, 0), (1, 11/3) 133) a. A(0, 0), B(0, 6), C(0, -5), D(0, - 2) b. A: negative, B: positive, C: positive, D: zero 134) E: -1/2, F: -2, G: undefined, H: 1 135) E: -(x/2)+1, F: -2x+4, G: x=8, H: x-7 136) a. I(0, 8), J(0, 2), K(0, -1), L(0, -8) b. I: zero, J: negative, K: positive, L: positive 137) M: -1, N: 3/2, O: 1, P: -1/3 138) M: -x+5, N: (3x)/2, O: y=-4, P: -(x/3)-6 139) -3x + y = x + y = (2/5)x + y = x + y = (2/3)x + y = x + y = x + y = x + y = 7

27 147. (x/4) + y = -(15/2) 152. x + y = x + y = (2/7)x + y = x + y = x + y = x + y = x + y = (1/6)x + y = x + y = 6

28 157. (3/7)x + y = 12/ x + y = Horizontal 160. Vertical 161. Neither 162. Vertical 163. Horizontal 164. Vertical 165. Horizontal 166. Neither 167. Horizontal 168. Vertical 169. ¼ Undefined / / / / / / Undefined / ½ / Multiple Answers ex: y = 4x Multiple Answers ex: y = (6/4)x y = y = -(5/2)x

30 a. y-4=6(x-3) b. y+7=-(3/2)(x+2) c. y+4=-3(x-7) d. y=x-4 e. y+4=(1/3)(x+4) a. y-1=x-2, y=x-1 b. y-5=(5/3)(x-3), y=(5/3)x c. y+8=-2(x-9), y=-2x+10 d. y+6=(1/6)(x+7), y=(1/6)x-(29/6) e. y+2=-1(x-3), y=-x+1 a. (x/10)+y=(12/5) b. (7/20)x+y=-(6/5) c. (x/7)+y=-(13/7) d. -3x+y=-19 e. (x/5)+y= a. y=-(3/2)x+12 b. y=-(3/2)x c. y=(2/3)x y=-2x y=-2x a. y+4=-2(x-5) b. y+3=4(x+2) c. y-4=(1/4)(x-7) d. y+9=3(x-9) e. y=-(2/3)(x-6) 250. a. y+50=5(x+10), y=5x b. y-10=-3x, y=-3x+10 c. y-9=-2(x-.5), y=-2x+10 d. y-9=5(x-8), y=5x-31 e. y-4=.5(x+7), y=.5x+(15/2) 251. a. (3/2)x+y=-.5 b. (7/2)x+y=14.5 c. -2x+y=0 d. 2x+y=13 e. -2x+y= a. y=4x+16 b. y=4x+4 c. y=-(1/4)x y=x y=(3/2)x ~ ~ A and C 259. no pattern 260. ~7: ~7 minutes A 264. B and D 265. y = 13 8 x minutes ; extrapolated ~35 minutes ; interpolated

287. 288. 289. 290. 306. 2 307. 3 308. 5 309. 7 310. 8 311. -4 312. -4 313. 0 314. 7 315. -1 316. 317. 291. 292. 318. 319. 293. 294. 295. 296. 297. 298. 320. 3 321. 3 322. 7 323. 9 324. 10 325.

31 a. 0 e. b. 0 c. -4 d. D:Reals; R:(, 2) 335. a.4 e. b.4 c.12 d.d: Reals; R:{ 2 (6, )} 336. a. 2 e. b. 1 c. 4 d. D: Reals; R:[0, )

-4 d. D: Reals; R:{(, 1) 3} 339. a. -5 e. b. 0 c. 4 d.

D: Reals; R:{(, 0] 1} 343. a. 4 e. b. 2 c. 4 d.

32 337. a. -2 e. b. 0 c. 4 d. D: Reals; R:{neg Integers [0, )} 338. a. -5 e. b. 3 c. -4 d. D: Reals; R:{(, 1) 3} 339. a. -5 e. b. 0 c. 4 d. D: Reals; R:{(, 1) [0, )} 340. a. -2 e. b. -2 c. -4 d. D: Reals; R:(, 2] 341. a. 1 e. b. 2 c. 2 d. D: Reals; R:[0, ) 342. a. 0 e. b. 2 c. -4 d. D: Reals; R:{(, 0] 1} 343. a. 4 e. b. 2 c. 4 d. D: Reals; R :[0, ) 350. y < 3x y 1 x y > y 1 x

Algebra I Exam Review

Name Algebra I Exam Review Assigned on Assignment 1/17 Final Exam Practice Units 1 and Problems 1-4 1/18 Final Exam Practice Units and 4 Problems 5-5 1/19 Practice Final Exam Multiple Choice 1-16 1/ Practice