Alg II Analyzing Functions ~1~ NJCTL.org. Domain and Range Class Work Find the domain and range for each of the following

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1 Domain and Range Class Work Find the domain and range for each of the following. 1. {(1,), (3,4), (5,6)}. {(4,3), (3,), (4,)} 3. {(5,1), (3,1), ( 4,1)} Domain and Range Home Work Find the domain and range for each of the following 14. {(3,1), (,6), (1,4)} 15. {(1,), (,), (1,)} 16. {(, 1), (5, 1), ( 6, 7)} Alg II Analyzing Functions ~1~ NJCTL.org

2 Interval and Inequality Notation Class Work Give the interval and inequality notation for each graph Alg II Analyzing Functions ~~ NJCTL.org

3 Simplify each of the following 37. (x 4 ) 3 x x y 4 4x y 4 3x 3x 3 y 39. (x 3 z ) 3 x 3 y 4 z x 4 z 3 Interval and Inequality Notation Homework Give the interval and inequality notation for each graph Simplify each of the following 50. (x x 3 ) x y 3x 1 4yx 5. (pm 1 q 0 ) 4 m 1 p 3 pq Alg II Analyzing Functions ~3~ NJCTL.org

{(5,1), (3,1), ( 4,1)} 56. 57. 58. 59. 60. 61.

Continuous Home Work Is the relation discrete

4 Discrete vs. Continuous Class Work Is the relation discrete or continuous? If continuous, state the interval of continuity. 53. {(1,), (3,4), (5,6)} 54. {(4,3), (3,), (4,)} 55. {(5,1), (3,1), ( 4,1)} Discrete vs. Continuous Home Work Is the relation discrete or continuous? If continuous, state the interval of continuity. 66. {(3,1), (,6), (1,4)} 67. {(1,), (,), (1,)} 68. {(,1), (5,1), ( 6,7)} Alg II Analyzing Functions ~4~ NJCTL.org

5 Relations and Functions Class Work Is the relation a function? 79. {(1,), (3,4), (5,6)} 80. {(4,3), (3,), (4,)} 81. {(5,1), (3,1), ( 4,1)} Alg II Analyzing Functions ~5~ NJCTL.org

6 Relations and Functions Home Work Is the relation a function? 9. {(3,1), (,6), (1,4)} 93. {(1,), (,), (1,)} 94. {(,1), (5,1), ( 6,7)} Alg II Analyzing Functions ~6~ NJCTL.org

the following 105. f() 106. f(3) 107. g(6) 108. g() 109. f(6) 110.

7 Function Notation Class Work Let f(x) = 3x + 4 and g(x) = x 4, find the following 105. f() 106. f(3) 107. g(6) 108. g() 109. f(6) g() 111. f(4) g(3) 11. g(5) f(5) 113. f(0) 114. g(3) g(a) 116. f(b) Alg II Analyzing Functions ~7~ NJCTL.org

8 Function Notation Home Work Let f(x) = (x 1) and g(x) = x 3, find the following 117. f() 118. f(3) 119. g(6) 10. g() 11. f(6) 1. 1 g() 13. f(4) g(3) 14. g(5) f(5) 15. f(0) 16. g(3) g(a) 18. f(b) Multiply each of the following 19. (4x + 1)(x + 6) 130. (7x 6)(5x + 6) 131. (x + 6x 4)(x 4) Value, Change & Rate of Change Class Work Use the table below from Center for Disease Control (CDC) to answer questions The chart shows stature for age of males. 5th Percentile 10th Percentile 5th Percentile 50th Percentile 75th Percentile 90th Percentile 95th Percentile Age (in Stature (in Stature (in Stature (in Stature (in Stature (in Stature (in Stature (in months) centimeters) centimeters) centimeters) centimeters) centimeters) centimeters) centimeters) What is the height of a boy in the 90 th percentile at age 6.5 months? 133. What is the rate of change for a boy 6.5 months to 7.5 months in 5 th percentile? 134. What is the rate of change for a boy 4 months to 4.5 months in 50 th percentile? 135. What is the average rate of change of a boy who is always in the 75 th percentile from 4 to 8.5 months? 136. What is the average rate of change of a boy who is always in the 10 th percentile from 4 to 8.5 months? Alg II Analyzing Functions ~8~ NJCTL.org

9 Use the graph of Pressure vs. Altitude to answer questions What is the pressure when the altitude is,000 ft? 138. What is altitude when the pressure is 00 hpa? 139. What is the rate of change from,000 ft to 4,000ft? 140. What is the rate of change from 4,000ft to 6,000ft? 141. What is the rate of change from,000 ft to 6,000ft? Factor each of the following 14. 3x x x + 19x x + 53x + 8 Value, Change & Rate of Change Home Work Use the table below from Center for Disease Control (CDC) to answer questions The chart shows stature for age of females. Age (in months) 5th Percentile Stature (in centimeters) 10th Percentile Stature (in centimeters) 5th Percentile Stature (in centimeters) 50th Percentile Stature (in centimeters) 75th Percentile Stature (in centimeters) 90th Percentile Stature (in centimeters) 95th Percentile Stature (in centimeters) What is the height of a girl in the 90 th percentile at age 6.5 months? 146. What is the rate of change for a girl 6.5 months to 7.5 months in 5 th percentile? 147. What is the rate of change for a girl 4 months to 4.5 months in 50 th percentile? 148. What is the average rate of change of a girl who is always in the 75 th percentile from 4 to 8.5 months? 149. What is the average rate of change of a girl who is always in the 10 th percentile from 4 to 8.5 months? Alg II Analyzing Functions ~9~ NJCTL.org

In Questions 150-154, refer to the graph of the participation of the players on the field of a soccer team. P(t) is amount of participation at any given t, time in minutes. 150. What was the amount of participation at t = 7?

10 In Questions , refer to the graph of the participation of the players on the field of a soccer team. P(t) is amount of participation at any given t, time in minutes What was the amount of participation at t = 7? 151. What was the rate of change in participation from t = to t = 3? 15. What was the rate of change in participation from t = to t = 5? 153. At what time was there a participation of 4 players? 154. What was the rate of change in participation from t = 6 to t = 7? Factor each of the following x 35x x + 7x x 7x 6 Maxima and Minima Class Work 158. A box manufacturer wants to make a box with a square base that holds 10,000 in 3 and has a height of more than 1 inch. To minimize materials, what dimensions should the box have? 159. A farmer has 300 of fence and wants to maximize his materials. He wants to make two equal size pens that share a side. What are the dimensions of one pen? 160. An 8 in by 10 in sheet of paper is to have squares removed from its corners so that the remaining edges can be folded into a lid-less box. What is the greatest volume possible? 161. An isosceles triangle is to have an area of 30 cm. What side lengths will minimize the perimeter? 16. A 10 in by 0 in sheet of material will have squares cut out of the corners and two squares cut out from the middle of each long side so that when folded the net forms a box. Calculate the size of the squares that will maximize the volume. Alg II Analyzing Functions ~10~ NJCTL.org

11 Maxima and Minima Home Work 163. A box manufacturer wants to make a box with a square base that holds 0,000 in 3 and has a height more than 1 inch. To minimize materials, what dimensions should the box have? 164. A farmer has 450 of fence and wants to maximize his materials. He wants to make two equal size pens that share a side. What are the dimensions of one pen? 165. An 8 in by 1 in sheet of papers is to have squares taken out of its corners so that the remaining edges can be folded into a lid-less box. What is the greatest possible volume? 166. An isosceles triangle is to have an area of 100 cm. What side lengths minimize the perimeter? 167. A 10 in by 8 in sheet of material will have squares cut out of the corners and two squares cut out from middle of each long side so that when folded, the net forms box. Calculate the size of the squares that will maximize the volume. Increasing and Decreasing Functions Class Work Use the graph of f(x) to answer the following questions Interval(s) on which f(x) is increasing 169. Interval(s) on which f(x) is decreasing x-value of any local maxima 171. x-value of any local minima 17. x-value of any extreme maximum 173. x-value of any extreme minimum 174. Interval(s) on which f(x) is concave up 175. Intervals on which f(x) is concave down Use the table to answer the following questions. The table represents the scores one student received on practice math exams leading up to the SAT s. Week Score During what interval(s) were the scores increasing? 177. State any relative minimum scores. Alg II Analyzing Functions ~11~ NJCTL.org

12 178. What is the concavity of the graph at w = 4? 179. What was the greatest rate of change and when did it occur? Increasing and Decreasing Functions Home Work Use the graph of f(x) to answer the following questions Interval(s) on which f(x) is increasing 181. Interval(s) on which f(x) is decreasing. 18. x-value of any local maxima 183. x-value of any local minima 184. x-value of any extreme maximum 185. x-value of any extreme minimum 186. Interval(s) on which f(x) is concave up 187. Intervals on which f(x) is concave down Use the table to answer the following questions. The table represents the number of assignments one student received in math class for a marking period Week Assignments During what interval(s) was the number of assignments increasing? 189. State any relative minimum assignment weeks What is the concavity of the graph at w = 7? 191. What was the greatest rate of change and when did it occur? End Behaviors Class Work Use each graph to determine if the degree of the polynomial is odd or even and the sign of the lead coefficient y y y x x x Is the equation given an odd function, an even function, or neither? 195. f(x) = 3x 5 + x 3 + 6x 196. g(x) = 5x 4 3x h(x) = x + 1 Alg II Analyzing Functions ~1~ NJCTL.org

13 198. f(x) = 3x g(x) = 5x 3 1 Is the graphed function odd, even, or neither? y y y x x x Simplify each of the following b p x 3 y 3 Simplify and add each of the following End Behaviors Home Work Use each graph to determine if the degree of the polynomial is odd or even and the sign of the lead coefficient y y x x Is the equation given an odd function, an even function, or neither? 1. f(x) = x 7 + x g(x) = x 6 x h(x) = x f(x) = 6x 4 + x 16. g(x) = 7x 3 x Is the graphed function odd even or neither? Alg II Analyzing Functions ~13~ NJCTL.org

3 1-8 -6-4 - 4 6 8-1 - -3 8 6 4-8 -6-4 - 4 6 8 - -4-6 -8 y 17. 18. 19.

384x 4 y 3 Simplify and add each of the following. 3. 1 + 3 3 4. 3 3 7 5.

y-axis, diagonal (y=x), origin, or none. 6. f(x) = 3x 5 + x 3 + 6x 7. g(x) = 5x 4 3x + 8.

14 y y x x Simplify each of the following m 3 n m 4 n. 384x 4 y 3 Simplify and add each of the following Symmetry and Periodicity Classwork State whether the function is has symmetry over the x-axis, y-axis, diagonal (y=x), origin, or none. 6. f(x) = 3x 5 + x 3 + 6x 7. g(x) = 5x 4 3x + 8. h(x) = x f(x) = 3x g(x) = 5x 3 x 31. h(x) = 1 3x y y x x Find the period of the function Alg II Analyzing Functions ~14~ NJCTL.org

g(x) = 7x 3 x 41. h(x) = 6x 4 3x 4. 43. Find the period of the function. 44. 45.

15 Symmetry and Periodicity Homework State whether the function is has symmetry over the x-axis, y-axis, diagonal (y=x), origin, or none. 36. f(x) = x 7 + x g(x) = x 6 x h(x) = 1 x f(x) = 6x 4 + x 40. g(x) = 7x 3 x 41. h(x) = 6x 4 3x Find the period of the function Graphing Classwork Identify the transformations on the function. 46. g(x) = (x + 4) h(x) = 3x 5 Alg II Analyzing Functions ~15~ NJCTL.org

16 48. f(x) = 3x 9 x Identify k for the transformation on f(x) Graphing Homework Identify the transformations on the function. 5. f(x) = 1 (x + 7)4 53. g(x) = 4x 8x x 3 1x 16x 54. f(x) = 3x Find k for the transformation on the function f(x). Alg II Analyzing Functions ~16~ NJCTL.org

17 Alg II Analyzing Functions ~17~ NJCTL.org

18 Domain And Range Classwork 1. D: {1, 3, 5} R: {, 4, 6}. D: {3,4} R: {, 3} 3. D: { 4, 3, 5} R: {1} 4. D: { 3, 1, } R: {, 5, 7} 5. D: {4, 5, 6} R: {6} 6. D: { 4, 0, } R: {3, 4, 5} 7. D: {, 1,, 3} R: {0, 3, 4, 5, 7} Algebra II Analyzing Functions - Solutions NJCTL.org. D: {3, 4} R: {, 3, 4} 3. D: { 4,,, 4, 5} R: { 3,, 4, 5} 4. D: 6 x 6 R: 6 y 6 5. D: 8 x < 0 R: { 6,,, 6} 6. D: All Real Numbers R: {} Interval and Inequality Notation Classwork 7. [1, ) x 1 8. (, 3) x < ( 8, 0) 8 < x < (, 5] x [ 9, ) x [ 4, 0] 4 x (, ) x > 49. ( 6, 0] 6 < x x xy 8. D: {1, } R: {3, 4, 5, 6} 9. D: { 4, 0, 1,, 3} R: {5, 6, 7} 10. D: { 4,, 1, 3} R: { 3,, 3, 5} 11. D: x 4 R: y 0 1. D: x or x > R: All Real Numbers 13. D: All Real Numbers R: y Domain and Range Homework 14. D: {, 1, 3} R: {1, 4, 6} 15. D: {1, } R: {} 16. D: { 6,, 5} R: {1, 7} 17. D: { 1, 0, 1} R: {6, 7, 8} 18. D: {, 4} R: {6, 7, 8} 19. D: { 5, 0, 5} R: {, 1, 0} 0. D: {3, 4, 5, 6} R: {1,, 3, 4} 1. D: {5} R: {0, 1,, 3} 9. [, 6] x [ 3, 1) 3 x < (1, 9) 1 < x < 9 3. (, 0] x [0, ) x [ 8, 4] 8 x ( 5, ) x > (4, ) x > x x 8 y x10 z y 4 Interval and Inequality Notation Homework 40. [ 4, ) x (, ) x < 4. [ 5, 3] 5 x [, 6) x < x6 y 5 z Discrete vs. Continuous Classwork 53. Discrete 54. Discrete 55. Discrete 56. Discrete 57. Discrete 58. Discrete 59. Discrete 60. Discrete 61. Discrete 6. Discrete 63. Continuous [ 4, ) 64. Continuous (, ] or [, ) 65. Continuous All Real Numbers Discrete vs. Continuous Homework 66. Discrete 67. Discrete 68. Discrete 69. Discrete 70. Discrete 71. Discrete 7. Discrete Alg II Analyzing Functions ~18~ NJCTL.org

19 73. Discrete 74. Discrete 75. Discrete 76. Continuous [ 6, 6] 77. Continuous [ 8, 0) 78. Continuous All Real Numbers Relations and Functions Classwork 79. Function 80. Not a Function 81. Function 8. Function 83. Function 84. Not a Function 85. Not a Function 86. Not a Function 87. Function 88. Not a Function 89. Function 90. Function 91. Function Relations and Function Homework 9. Function 93. Function 94. Function 95. Function 96. Not a Function 97. Not a Function 98. Function 99. Not a Function 100. Not a Function 101. Function 10. Not a Function 103. Function 104. Function Function Notation Classwork 105. f() = f(3) = g(6) = 108. g() = 109. f(6) = g() = f(4) g(3) = g(5) f(5) = f(0) = g(3) 3 = g(a) = a f(b) = 6b + 4 Function Notation Homework 117. f() = f(3) = g(6) = g() = f(6) = g() = f(4) g(3) = g(5) f(5) = f(0) = g(3) 3 = g(a) = x f(b) = (b 1) = 4b 4b x + 6x x + 1x x 3 + 8x 3x + 16 Value, Change, & Rate of Change Classwork cm cm mo cm mo cm mo cm mo hpa ft hpa ft hpa ft hpa ft 14. (x + 1)(3x 5) 143. (x + 3)(5x + 4) 144. (x + 7)(7x + 4) Value, Change, & Rate of Change Homework cm cm mo cm mo cm mo cm mo participation min participaiton min min participation min 155. (x 7)(4x 7) 156. (x + 7)(3x 7) (x )(5x + 1) Maxima and Minima Classwork in 1.5 in 1.6 in in leg = cm base = 7.75 cm in 1.05 in Maxima and Minima Homework in 7.1 in 7.1 in in leg = 14.1 cm base = 0 cm in 0.74 in Alg II Analyzing Functions ~19~ NJCTL.org

20 Increasing & Decreasing Functions Classwork 168. [ 5, 3] or [, 1] or [3, 6] 169. [ 7, 5] or [ 1, ] 170. x = 7, 1, x = 5, 17. x = 1, x = 5, 174. [ 7, 3] or [ 1, 3] or [4, 6] 175. [, 1] or [3, 4] 176. [1,4] or [5, 9] in Week in Week Concave Down [, 3] Increasing & Decreasing Functions Classwork 180. [ 7, 5] or [ 3, 0] or [4, 6] 181. [ 5, 3] or [0, 3] 18. x = 5, 0, x = 7, x = x = [ 4, ] 187. [ 7, 4] or [, ] 188. [1, 5] or [7, 9] 189. x = 1, Concave up [1, ] End Behaviors Classwork 19. Even; Negative 193. Odd; Negative 194. Odd; Positive 195. Odd 196. Even 197. Neither 198. Even 199. Neither 00. Odd 01. Even 0. Neither b 04. 4p 5p 05. xy 7xy End Behavior Homework 09. Odd; Negative 10. Even; Positive 11. Odd; Positive 1. Neither 13. Even 14. Even 15. Neither 16. Odd 17. Odd 18. Even 19. Odd 0. 7mn 3mn 1. 10m n. 8x y 6y Symmetry and Periodicity Classwork 6. Origin 7. y axis 8. None 9. None 30. Origin 31. Diagonal 3. y axis 33. Origin Symmetry and Periodicity Homework 36. None 37. y axis 38. y axis 39. None 40. Origin 41. None 4. y axis 43. y axis Graphing Classwork 46. Reflection over the x axis Vertical Stretch Left 4 units Up 15 units 47. Horizontal Shrink Reflection over the y axis Right 5 units 48. Vertical Stretch Left 3 units 49. Reflection over the y axis Left 1 unit Down 1 unit 50. k = 5 for a horizontal translation 51. k = for vertical stretch and reflection of the x axis Graphing Homework 5. Vertical Shrink Left 7 units Down units 53. Vertical Stretch Left 1 unit 54. Horizontal Shrink Up 10 units 55. Vertical Shrink Reflection over the x axis 56. k = 4 for a horizontal translation 57. k = 1; function is the parent function Alg II Analyzing Functions ~0~ NJCTL.org

Name: Unit Test Review Multiple Choice Determine the best answer for each question. 1. Determine the domain of {(1,3), (5,6), (6,8)} Date: Algebra II Analyzing Functions a. {1, 5, 8} b. {1, 5, 6} c.

21 Name: Unit Test Review Multiple Choice Determine the best answer for each question. 1. Determine the domain of {(1,3), (5,6), (6,8)} Date: Algebra II Analyzing Functions a. {1, 5, 8} b. {1, 5, 6} c. {3, 6, 8} d. Set of Reals. Determine the range of f(x) = x + 3. a. [3, ] b. [1, ) c. (1, ) d. [3, ) 3. What is the domain of the graph to the right? a. 10 x 10 b. 10 < x < 10 c. 6 x or 0 x 6 d. 10 x 4 or x 4 or 6 x Which choice represents a discrete set? a. The time it takes people to tie their shoes. b. The amount of rain in a given week. c. The number of people attending a play. d. The number of rotations of a wheel. 5. Which of the following is a function? a. x + y = 4 b. x + y = 4 c. x + y = 4 d. 4x + y = 4 In Questions 6 8, refer to the graph on the right: 6. There is a local minimum at: a. x = 3.5 b. x = 0 c. x = 1 d. There is no local minimum 7. The rate of change from x = to x = 1.5 is the same as the rate of change from a. x = 1 to x = 0. b. x = to x = 3 c. x = 1 to x = 3 d. x = 0.5 to x = 1 8. The graph is concave up on the domain a. (, 1) b. ( 1, 1) c. (1, ) d. (0, ) 9. Given f(x) = (x 6) +, calculate f(3). a. b. 0 c. 9 d In the table to the right, the rate of change between x = 4 and x = 6 is a. b. 1 c. 0.5 d. -1 Alg II Analyzing Functions ~1~ NJCTL.org

22 11. A rancher has 10,000 of fence and wants to use it to make a pen with the maximum area. A barn 40 by 100 is to be used as a corner of the pen. a. A = 5000x x b. A = 5070x x c. 5140x x d. A = 10,000x x In Questions 1 14, consider the following graph to the right: 1. The rate of change from x = 4 to x = 8 is a. 3 b c. 0 d The greatest rate of change is between a. x = 7 and x = 6 b. x = 1 and x = 3 c. x = 1 and x = d. x = 8 and x = The rate of change from x = 5 to x = 4 is the as the rate of change from a. x = 6 to x = 5 b. x = 3 to x = c. x = 6 to x = 7 d. x = 3 to x = 5 In Questions 15 17, refer to the graph below: 15. There is a local max at a. 6 b. 3 c. 1 d In terms of concavity, the point at x = 5 is a. concave up. b. concave down. c. a local minimum. d. none of the above. 17. The rate of change is positive on the interval a. ( 4, 1.5) b. (, 1) c. ( 1, 3) d. (1, 3) 18. Given f(x) = 4x x 3 + 6x, the function is a. an odd function b. an even function c. neither an odd or even function d. both an odd and even function Determine if the statements in questions 19 1 are true or false. 19. The function f(x) = 3x 5 + 4x 3 7x is symmetrical over the x-axis True False 0. Any function with odd exponents in all terms is symmetrical over the y-axis. True False 1. All equations that are symmetrical over the x-axis are functions True False. The period of the graph to the right is: π a. π b. c. π d. 4π Alg II Analyzing Functions ~~ NJCTL.org

23 3. Identify the transformations on the function f(x) = 3x a. shifted up 15; shifted left 3 b. shifted up 3; shifted right 15 c. vertical stretch; shifted up 15 d. vertical shrink; shifted up 15 Extended Response Completely answer each question showing all work. 4. The number of people entering the exciting new amusement park, Math World HD in 3D is given by the following equation, where t is the amount of time, in hours, after the park opens. e(t) { t < t 6 t < t 1 a. If the park opened at 10 am, how many people entered at 1 pm? b. During what hour did the most number of people enter the park? How many people entered during that hour? c. What is the rate of change in people entering from 1 pm to pm? 5. Brenda decides to save her spare change in a jar. The initial amount in the jar, J(0) is $0.00 and after one week it is J(7) = a. How much money did Brenda save? b. At what rate is Brenda saving money? c. If Brenda continued to save at a linear rate, how much would she have on J(14)? 6. Let h(x) = 4 x a. Describe the end behaviors of h(x). b. Describe the concavity of h(x). c. Describe the intervals of increase and decrease of h(x). Alg II Analyzing Functions ~3~ NJCTL.org

24 Analyzing Functions Unit Review - Solutions Multiple Choice 1. B. D 3. D 4. C 5. C 6. B 7. D 8. B 9. B 10. C 11. B 1. B 13. A 14. D 15. D 16. A 17. B 18. C 19. False 0. False 1. False. C 3. C Extended Response 4. a. t = 3 e(3) = (3) + 30 e(3) = = 39 people entered a 1 pm b. The maximum number of people that enter the park is 66 people, which occurs at t = 6 (4 pm). c. ROC = e(4) e() 4 ROC = [(4 + 30) ( + 30)] ROC = = 1 = 6 The rate of change between 1 pm and pm is 6 people. hr 5. a. Brenda saved $3.50 b. ROC = saved period ROC = = 0.5 Brenda is saving $0.50 per day or. c dollars 14 days = 7 dollars day Adding the original $0.00, Brenda would have $7 on J(14) 6. a. Since the degree is even and the lead coefficient is negative, the end behavior would be down and down b. The concavity of h(x) is concave down. c. Increasing: (, 0] Decreasing: [0, ) Alg II Analyzing Functions ~4~ NJCTL.org

Polynomial Functions

Polynomial Functions NOTE: Some problems in this file are used with permission from the engageny.org website of the New York State Department of Education. Various files. Internet. Available from https://www.engageny.org/ccss-library.