Relations and Functions

Midpoint and Distance Formula

Class Work

M is the midpoint of A and B. Use the given information to find the missing point.

1. A(4, 2) and B(3, -8), find M

2. A(5, 7) and B( -2, -9), find M

3. A( 2,0) and B(6, -2), find M

4. A( 3, 7) and M(4,-3), find B

5. M(4, -9) and B( -10, 11) find A

6. B(4, 8) and M(-2, 5), find A

7. Find the distance from A(4, 2) to B(3, -8).

8. Find the distance from A(5, 7) to B(-2, -9).

9. Find the distance from A(2,0) to B(6, -2).

10. The distance from A(2, 3) to B(-6, y) is 10, find y.

11. The distance from A(-4, 7) to B(x, 9) is 7, find x.

Homework

M is the midpoint of A and B. Use the given information to find the missing point.

12. A(4, -2) and B(5, 6), find M

13. A(9, 4) and B(-3, -7), find M

14. A(1, 10) and B(6, -2), find M

15. A( 4, 8) and M(4,-3), find B

16. M(8, 7) and B( -10, 11) find A

17. B(-5, 10) and M(-2, 5), find A

18. Find the distance from A(-3, 9) to B(3, -8).

19. Find the distance from A(5, -9) to B(-2, -9).

20. Find the distance from A(-2,10) to B(-6, 0).

21. The distance from A(2, -3) to B(5, y) is 10, find y.

22. The distance from A(4, 6) to B(2x, 9) is 7, find x.

Circles

Class Work

What are the center and the radius of the following circles?

23. (𝑥 + 2)2 + (𝑦 − 4)2 = 16

24. (𝑥 − 3)2 + (𝑦 − 7)2 = 25

25. (𝑥)2 + (𝑦 + 8)2 = 1

26. (𝑥 − 7)2 + (𝑦 + 1)2 = 17

27. (𝑥 + 6)2 + (𝑦)2 = 32

Write the standard form of the equation for the given information.

28. center (3,2) radius 6

29. center (-4, -7) radius 8

30. center (5, -9) radius 10

31. center (-8, 0) diameter 14

32. center (4,5) and point on the circle (3, -7)

33. diameter with endpoints (6, 4) and (10, -8)

34. center (4, 9) and tangent to the x-axis

35. 𝑥2 + 4𝑥 + 𝑦2 − 8𝑦 = 11

36. 𝑥2 − 10𝑥 + 𝑦2 + 2𝑦 = 11

37. 𝑥2 + 7𝑥 + 𝑦2 = 11

Homework

What are the center and the radius of the following circles?

38. (𝑥 − 9)2 + (𝑦 + 5)2 = 9

39. (𝑥 + 11)2 + (𝑦 − 8)2 = 64

40. (𝑥 + 13)2 + (𝑦 − 3)2 = 144

41. (𝑥 − 2)2 + (𝑦)2 = 19

42. (𝑥 − 6)2 + (𝑦 − 15)2 = 40

Write the standard form of the equation for the given information.

43. center (-2, -4) radius 9

44. center (-3, 3) radius 11

45. center (5, 8) radius 12

46. center (0 , 8) diameter 16

47. center (-4,6) and point on the circle (-2, -8)

48. diameter with endpoints (5, 14) and (11, -8)

49. center (4, 9) and tangent to the y-axis

50. 𝑥2 − 2𝑥 + 𝑦2 + 10𝑦 = 11

51. 𝑥2 + 12𝑥 + 𝑦2 + 20𝑦 = 11

52. 4𝑥2 + 16𝑥 + 4𝑦2 − 8𝑦 = 12

Domain and Range

Class Work

Find the domain and range for each of the following

53. {(1,2), (3,4), (5,6)}

54. {(4,3), (3,2), (4,2)}

55. {(5,1), (3,1), (-4,1)}

56. 57. 58.

59. 60. 61.

62. 63.

64. 65.

Homework

Find the domain and range for each of the following

66. {(3,1), (-2,6), (1,4)}

67. {(1,2), (2,2), (1,2)}

68. {(2,1), (5,1), (-6,7)}

69. 70. 71.

72. 73. 74.

75. 76.

77. 78.

Discrete vs. Continuous

Class Work

Is the relation discrete or continuous?

79. {(1,2), (3,4), (5,6)}

80. {(4,3), (3,2), (4,2)}

81. {(5,1), (3,1), (-4,1)}

82. 83. 84.

85. 86. 87.

88. 89.

90. 91.

Homework

Is the relation discrete or continuous?

92. {(3,1), (-2,6), (1,4)}

93. {(1,2), (2,2), (1,2)}

94. {(2,1), (5,1), (-6,7)}

95. 96. 97.

98. 99. 100.

101. 102.

103. 104.

Relations and Functions

Class Work

Is the relation a function?

105. {(1,2), (3,4), (5,6)}

106. {(4,3), (3,2), (4,2)}

107. {(5,1), (3,1), (-4,1)}

108. 109. 110.

111. 112. 113.

114. 115.

116. 117.

Homework

Is the relation a function?

118. {(3,1), (-2,6), (1,4)}

119. {(1,2), (2,2), (1,2)}

120. {(2,1), (5,1), (-6,7)}

121. 122. 123.

124. 125. 126.

127. 128.

129. 130.

Transformations

Class Work

In each exercise the function is given. Describe the transformation of the parent function. Graph the

function.

131. 𝑓(𝑥) = 𝑥2 + 4

132. 𝑔(𝑥) = 𝑥2 − 2

133. ℎ(𝑥) = |𝑥| − 6

134. 𝑦(𝑥) = |𝑥| + 3

135. 𝑓(𝑥) = (𝑥 − 2)2

136. 𝑔(𝑥) = (𝑥 + 3)2

137. ℎ(𝑥) = |𝑥 + 5|

138. 𝑦(𝑥) = |𝑥 − 6|

139. 𝑓(𝑥) = −𝑥2

140. 𝑔(𝑥) = (−𝑥)2

141. ℎ(𝑥) = |−𝑥|

142. 𝑦(𝑥) = −|𝑥|

143. 𝑓(𝑥) = 3𝑥2

144. 𝑔(𝑥) = (2𝑥)2

145. ℎ(𝑥) = |4𝑥|

146. 𝑦(𝑥) = .5|𝑥|

147. 𝑓(𝑥) = (1

2𝑥)

2

148. 𝑔(𝑥) = . 3𝑥2

149. ℎ(𝑥) = |. 2𝑥|

150. 𝑦(𝑥) = 7|𝑥|

151. 𝑓(𝑥) = −2(𝑥 + 1)2

152. 𝑔(𝑥) = (𝑥 − 3)2 + 4

153. ℎ(𝑥) = −3|𝑥 + 4| − 1

154. 𝑦(𝑥) = |2𝑥| + 7

Vertex Form of a Parabola

Class Work

Find the vertex, direction of openness, and axis of symmetry for each of the following.

155. 𝑦 = 1(𝑥 − 2)2 + 10

156. 𝑦 = −2(𝑥 + 5)2 − 4

157. 𝑦 = −6

5(𝑥 − 6)2 + 8

158. 𝑦 = 2(𝑥 + 7)2

159. 𝑦 = 7(𝑥 − 3)2 + 4

160. 𝑦 = −3(𝑥 + 2)2 − 5

161. 𝑦 =7

−11(𝑥 − 4)2 + 6

162. 𝑦 = 4(𝑥)2 − 4

Convert the following equations from standard form to vertex form.

163. 𝑦 = 𝑥2 + 6𝑥 + 2

164. 𝑦 = 𝑥2 − 10𝑥 + 20

165. 𝑦 = 𝑥2 + 8𝑥 − 12

166. 𝑦 = 𝑥2 + 5𝑥 + 3

167. 𝑦 = 𝑥2 + 𝑥 − 1

168. 𝑦 = 𝑥2 + 4𝑥 + 4

169. 𝑦 = 3𝑥2 + 6𝑥 − 2

170. 𝑦 = 2𝑥2 − 12𝑥 − 4

171. 𝑦 = −6𝑥2 + 12𝑥 + 2

172. 𝑦 = −5𝑥2 − 10𝑥 − 3

173. 𝑦 = 4𝑥2 − 6𝑥 + 2

174. 𝑦 = −2𝑥2 + 6𝑥 + 2

Homework

Find the vertex, direction of openness, and axis of symmetry for each of the following.

175. 𝑦 = −2(𝑥 − 3)2 + 1

176. 𝑦 = 3(𝑥 + 6)2

177. 𝑦 = −7

2(𝑥 + 4)2 + 1

178. 𝑦 = 4(𝑥 − 8)2 + 11

179. 𝑦 = −(𝑥 + 2)2 − 3

180. 𝑦 = −5(𝑥 + 1)2 − 2

181. 𝑦 = 3(𝑥 − 3)2 + 8

182. 𝑦 = 2(𝑥)2 − 3

Convert the following equations from standard form to vertex form.

183. 𝑦 = 𝑥2 + 8𝑥 + 2

184. 𝑦 = 𝑥2 − 12𝑥 + 20

185. 𝑦 = 𝑥2 + 4𝑥 − 12

186. 𝑦 = 𝑥2 + 3𝑥 + 3

187. 𝑦 = 𝑥2 + 7𝑥 − 1

188. 𝑦 = 𝑥2 + 6𝑥 + 9

189. 𝑦 = 4𝑥2 + 12𝑥 − 2

190. 𝑦 = −6𝑥2 + 24𝑥 − 4

191. 𝑦 = 3𝑥2 − 18𝑥 + 2

192. 𝑦 = −2𝑥2 − 10𝑥 − 3

193. 𝑦 = 6𝑥2 + 6𝑥 + 2

194. 𝑦 = −4𝑥2 − 12𝑥 + 2

Operations with Functions

Class Work

Given 𝑓(𝑥) = 3𝑥2 − 4 and 𝑔(𝑥) = |3𝑥 − 2| − 1, find h(x), h(2), h(0), and h(-2).

195. ℎ(𝑥) = 𝑓(𝑥) + 𝑔(𝑥)

196. ℎ(𝑥) = 𝑓(𝑥)𝑔(𝑥)

197. ℎ(𝑥) =𝑓(𝑥)

𝑔(𝑥)

198. ℎ(𝑥) = 2𝑓(𝑥) − 3𝑔(𝑥)

199. ℎ(𝑥) = 𝑓(𝑥)

(𝑔(𝑥))2

Class Work

Given 𝑓(𝑥) = 3𝑥2 − 4 and 𝑔(𝑥) = |3𝑥 − 2| − 1, find h(x), h(2), h(0), and h(-2).

200. ℎ(𝑥) = 𝑓(𝑥) − 𝑔(𝑥)

201. ℎ(𝑥) = 𝑥𝑓(𝑥)

202. ℎ(𝑥) =𝑔(𝑥)

𝑓(𝑥)

203. ℎ(𝑥) = 5𝑓(𝑥) − 2𝑔(𝑥)

204. ℎ(𝑥) = −𝑓(𝑥)

(𝑔(𝑥))2

Composite Functions

Class Work

Given the following functions, find f(g(x) and f(g(2). Graph f(x), g(x), and f(g(x)) on the same

graph.

205. 𝑓(𝑥) = 3𝑥 − 2; 𝑔(𝑥) = −2𝑥 + 4

206. 𝑓(𝑥) = 𝑥2 + 1; 𝑔(𝑥) = 5𝑥 − 1

207. 𝑓(𝑥) =2

𝑥−2; 𝑔(𝑥) = 2𝑥2 − 9

208. 𝑓(𝑥) =𝑥

𝑥2−1+ 3; 𝑔(𝑥) = √𝑥 + 2

209. 𝑓(𝑥) = [𝑥]; 𝑔(𝑥) = |𝑥 − 1|

Homework

Given the following functions, find f(g(x) and f(g(2). Graph f(x), g(x), and f(g(x)) on the same

graph.

210. 𝑓(𝑥) = −1

2𝑥 + 3; 𝑔(𝑥) = −4𝑥 + 2

211. 𝑓(𝑥) = 2𝑥2 − 5; 𝑔(𝑥) = 2𝑥 + 3

212. 𝑓(𝑥) =−1

𝑥+2; 𝑔(𝑥) = 3𝑥2 − 10

213. 𝑓(𝑥) =𝑥

𝑥2+4+ 3; 𝑔(𝑥) = √2 − 𝑥

214. 𝑓(𝑥) = 𝑥2 − 4; 𝑔(𝑥) = |𝑥 − 1|

Inverse Functions

Class Work

Given f(x), find f-1(x). Show that f(f-1(x)) = f-1(f(x)) = x. Graph f(x) and f-1(x) on the same graph.

Describe the domain and range for f -1(x).

215. 𝑓(𝑥) = 3𝑥 − 2

216. 𝑓(𝑥) = 2𝑥2 + 1

217. 𝑓(𝑥) = 1 − 𝑥2

218. 𝑓(𝑥) = 4 − 𝑥3

219. 𝑓(𝑥) = 4𝑥 − 1

Homework

Given f(x), find f-1(x). Show that f(f-1(x)) = f-1(f(x)) = x. Graph f(x) and f-1(x) on the same graph.

Describe the domain and range for f -1(x).

220. 𝑓(𝑥) = 5𝑥 + 2

221. 𝑓(𝑥) =2

3𝑥2 − 6

222. 𝑓(𝑥) = 3𝑥2

223. 𝑓(𝑥) = 5 − 3𝑥

224. 𝑓(𝑥) = 2x2 + 1

Answers

1. (3.5, -3)

2. (1.5, -1)

3. (4,-1)

4. (5,-13)

5. (18,-29)

6. (-8,2)

7. 10.05

8. 17.46

9. 4.47

10. -3 or 9

11. -4+/-√45

12. (4.5, 2)

13. (3, -1.5)

14. (3.5, 4)

15. (4,-14)

16. (26,3)

17. (1,0)

18. 18.03

19. 7

20. 10.77

21. -3 +/-√91

22. 4+/− √13

2

23. C (-2,4) r=4

24. C (3,7); r=5

25. C (),-8); r=1

26. C (7,-1); r= √17

27. C (-6,0); r =4√2

28. (x-3)2 + (x-2)2 =36

29. (x+4)2 + (Y+7)2 =64

30. (x-5)2 + (y+9)2 = 100

31. (x+8)2 + y2 =49

32. (x-4)2 + (y-5)2 =145

33. (x-8)2 + (y+2)2 =40

34. (x-4)2 + (y-9)2 =81

35. (x+2)2 + (y-4)2 =31

36. (x-5)2 + (y+1)2 =37

37. (x+3.5)2 + y2 =23.25

38. C (9,-5) rr=3

39. C -11, 8) r=8

40. C(-13, 3) r=12

41. C(2,0) r= √19

42. C (6,15) r=2√10

43. (x+2)2 +(Y+4)2 =81

44. (x+3)2 + (y-3)2 =121

45. (x-5)2 + (y-8)2 =144

46. X2 + (y-8)2 =64

47. (x+4)2 + (y-6)2 =200

48. (x-8)2 + (y-3)2 = 130

49. (x-4)2 + ( y-9)2 =130

50. (x-1)2 + (y+5)2 =37

51. (x+6)2 + (y+10)2 =147

52. (x+2)2 + (y-1)2 =8 53. D:{1,3,5} R:{2,4,6} 54. D:{3,4} R:{2,3} 55. D:{-4, 3,5} R:{1} 56. D:{-3,1,2} R:{2,5,7} 57. D:{4,5,6} R:{6} 58. D:{-4,0,2} R:{3,4,5} 59. D:{-2,-1,2,3} R:{0,3,4,5,7} 60. D:{1,2} R:{3,4,5,6} 61. D:{-4,0,1,2,3} R:{5,6,7} 62. D:{-4,-2,1,3} R:{0,3,4,5,7} 63. D:{x>-4} R:{y>0} 64. D:{x<-2 or x>2} R:{Reals} 65. D:{Reals} R:{Reals} 66. D:{-2,1,3} R:{1,4,6} 67. D:{1,2} R:{2} 68. D:{-6,2,5} R:{1,7} 69. D:{-1,0,1} R:{6,7,8} 70. D:{2,4} R:{6,7,8} 71. D:{-5,0,5} R:{-2,-1,0} 72. D:{3,4,5,6} R:{1,2,3,4} 73. D:{5} R:{0,1,2,3} 74. D:{3,4} R:{2,3,4} 75. D:{-4,-2,2,4,5} R:{-3,2,4,5} 76. D:{-6<x<6} R:{-6<y<6} 77. D:{-6<x<0 } R:{-6,-2,2,4} 78. D:{Reals} R:{2} 79. D 80. D 81. D 82. D 83. D 84. D 85. D 86. D 87. D

88. D 89. C 90. C 91. C 92. D 93. D 94. D 95. D 96. D 97. D 98. D 99. D 100. D 101. D 102. C 103. C 104. C 105. yes 106. yes 107. yes 108. yes 109. yes 110. no 111. no 112. no 113. yes 114. no 115. yes 116. yes 117. yes 118. yes 119. no 120. yes 121. yes 122. no 123. no 124. yes 125. no 126. no 127. yes 128. no 129. yes 130. yes 131. up 4

132. down 2

133. down 6

134. up 3

135. right 2

136. left 3

137. left 5

138. right 6

139. reflects over x-axis

140. no change, symmetric about y-axis

141. no change, symmetric about y-axis

142. reflection over x-axis

143. vertical stretch of 3

144. horizontal shrink of 1

2

145. horizontal shrink of 1

4

146. vertical shrink of .5

147. horizontal stretch of 2

148. vertical shrink of .3

149. horizontal stretch of 5

150. vertical stretch of 7

151. 1 left, vertical reflection and stretch of 2

152. 3 right, up 4

153. 4 left, vertical reflection and stretch of 3,

down 1

154. horizontal stretch of 1

2, up of 7

155. (2, 10), up, x=2

156. (-5,-4), down, x=-5

157. (6,8), down, x=6

158. (-7,0), up, x=-7

159. (3,4), up, x=3

160. (-2,-5), down, x=-2

161. (4,6),down, x=4

162. (0,-4),up, x=0

163. 𝑦 = (𝑥 + 3)2 − 7

164. 𝑦 = (𝑥 − 5)2 − 5

165. 𝑦 = (𝑥 + 4)2 − 28

166. 𝑦 = (𝑥 + 2.5)2 − 3.25

167. 𝑦 = (𝑥 + .5)2 − 1.25

168. 𝑦 = (𝑥 + 2)2

169. y=3(x+1)2-5

170. y=2(X-3)2-22

171. y=-6(x-1)2+8

172. y=-5(X+1)2+2

173. y=4(x-.75)2-.25

174. y=-2(x-1.5)2-.25

175. (3.1); down; x=3

176. (-6,0); up; x=-6

177. (-4,1): down; x=-4

178. (8,11); up; x=8

179. (-2,-3); down; x=-2

180. (-1,-2); down; x=-1

181. (3,8); up; x=3

182. (0,-3); up; x=0

183. Y=(x+4)2-14

184. Y=(x-6)2-16

185. Y=(x+2)2-16

186. Y=(x+1.5)2+.75

187. Y=(x+3.5)2-13.25

188. Y=(x+3)2

189. Y=4(x+1.5)2-11

190. Y=-6(x-2)2+20

191. Y=3(x-3)2-25

192. Y=-2(x+2.5)2-15.5

193. Y=6(x+.5)2-.5

194. Y=-4(x+1.5)2+11

195. 3𝑥2 + |3𝑥 − 2| − 5, 11, −3, 15

196. (3𝑥2 − 4)(|3𝑥 − 2| − 1), 24, −4, 56

197. 3x2−4

|3x−2|−1,

8

3, −4,

8

7

198. 6𝑥2 − 3|3𝑥 − 2| − 5, 7, −11, −5,

199. 3x2−4

(|3x−2|−1)2 ,8

9, −4,

8

49

200. 3𝑥2 − |3𝑥 − 2| − 3, 5, −5, 1

201. 3𝑥3 − 4𝑥, 16, 0, −16

202. |3x−2|−1

3x2−4,

3

8,

−1

4,

7

8,

203. 15𝑥2 − 2|3𝑥 − 2| − 18, 34, −22, 26

204. −3x2+4

(|3x−2|−1)2 ,−8

9, −2,

−8

49

f(g(x)) ; f(g(2))

205. -6x +6; -6

206. 25x2 + 10x; 120

207. 2

2𝑥2−11 ;

−2

3

208. x

x+1+ 3; 3

2

3

209. [|x-1|]; 1

210. 2x+2; 6

211. 8𝑥2 + 24𝑥 + 7; 87

212. −1

3x2−8;

−1

4

213. √2−𝑥

6−𝑥+ 3; 3

214. (𝑥 − 1)2 − 4; −3

𝑓−1(𝑥); 𝐷; 𝑅; lim𝑥→−∞

ℎ(𝑥) ; lim𝑥→∞

ℎ(𝑥)

215. 𝑥+2

3; 𝑟𝑒𝑎𝑙𝑠; 𝑟𝑒𝑎𝑙𝑠

216. ±√𝑥−1

2; 𝑥 ≥ 1; 𝑟𝑒𝑎𝑙𝑠

217. ±√1 − 𝑥; 𝑥 ≤ 1; 𝑟𝑒𝑎𝑙𝑠

218. √4 − x3

; reals; reals;

219. x+1

4, reals; reals

220. x−2

5; reals; reals

221. ±√3(𝑥+6)

2; 𝑥 ≥ −6; 𝑟𝑒𝑎𝑙𝑠

222. ±√𝑥

3; 𝑥 ≥ 0; 𝑟𝑒𝑎𝑙𝑠

223. 5−x

3; reals; reals

224. ±√𝑥−1

2; 𝑥 ≥ 1; 𝑟𝑒𝑎𝑙𝑠