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3.1e: Exercises - Quadratic Functions

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A: Concepts

Exercise 3.1e.A 

1) Explain the advantage of writing a quadratic function in standard form.

2) How can the vertex of a parabola be used in solving real world problems?

3) Explain why the condition of a0 is imposed in the definition of the quadratic function.

4) What is another name for the standard form of a quadratic function?

5) What two algebraic methods can be used to find the horizontal intercepts of a quadratic function?

Answers to Odd Examples:

1. When written in that form, the vertex can be easily identified.

3. If a=0 then the function becomes a linear function.

5. If possible, we can use factoring. Otherwise, we can use the quadratic formula.Add texts here.

B: Parabola Orientation

Exercise 3.1e.B 

 Determine if the parabola opens up or down.

7.  a.  f(x)=2x26x7
b.  f(x)=6x2+2x+3

 

8.  a.  f(x)=4x2+x4
b.  f(x)=9x224x16

9.  a.  f(x)=3x2+5x1
b.  f(x)=2x24x+5

 

10.  a.  f(x)=x2+3x4
b.  f(x)=4x212x9

11. y=x29x+20
12. y=x212x+32
13. y=2x2+5x+12
14. y=6x2+13x6
15. y=64x2
16. y=3x+9x2

Answers to Odd Examples:
7. a.  down    b.  up 9. a.  down    b.  up 11. Upward 13. Downward 15. Downward

C: Vertex and Axis of Symmetry

Exercise 3.1e.C 

 Determine the vertex.

17. y=(x5)2+3

18. y=2(x1)2+7

19. y=5(x+1)2+6

20. y=3(x+4)2+10

21. y=5(x+8)21

22. y=(x+2)25

 Find the vertex and the axis of symmetry.

23.  f(x)=x2+8x1

24.  f(x)=x2+10x+25

25.  f(x)=x2+2x+5

26.  f(x)=2x28x3

27. y=x2+10x34

28. y=x26x+1

29. y=4x2+12x7

30. y=9x2+6x+2

31. y=4x21

32. y=x216

Answers to Odd Examples

17. (5,3)

19. (1,6)

21. (8,1)

23. Vertex: (4,17), Axis of symmetry: x=4

25.  Vertex: (1,6),  Axis of symmetry: x=1

27. Vertex: (5,9); axis of symmetry: x=5

29. Vertex: (32,2); axis of symmetry: x=32

31. Vertex: (0,1); axis of symmetry: x=0

D: Domain and Range

Exercise 3.1e.D 

 Use the vertex of the graph of the quadratic function and the direction the graph opens to find the domain and range of the function.

33) Vertex (1,2),
opens up.
34) Vertex (1,2)
opens down.
35) Vertex (5,11),
opens down.
36) Vertex (100,100),
opens up.

 Given the following quadratic functions, determine the domain and range.

37. f(x)=3x2+30x+50

38. f(x)=5x210x+1

39. g(x)=2x2+4x+1

40. g(x)=7x214x9

41. f(x)=x2+x1

42. f(x)=x2+3x2

43) f(x)=(x3)2+2

44) f(x)=2(x+3)26

45) f(x)=x2+6x+4

46) f(x)=2x24x+2

47) k(x)=3x26x9

 

Answers to Odd Examples:

33. Domain is (,). Range is [2,).

35. Domain is (,) Range is (,11].

37. Domain: (,); range:[25,)

39. Domain: (,); range: (,3]

41. Domain: (,); range: [54,)

43. Domain is (,). Range is [2,).

45. Domain is (,). Range is [5,).

47. Domain is (,). Range is [12,).

E: Minimum or maximum Value

Exercise 3.1e.E 

 In the following exercises, find the maximum or minimum value of each function.

49.  f(x)=2x2+x1

50.  y=4x2+12x5

51.  y=x26x+15

52.  y=x2+4x5

53.  y=9x2+16

54.  y=4x249

55. y=x26x+1

56. y=x24x+8

57. y=25x210x+5

58. y=16x224x+7

59. y=x2

60. y=19x2

61. . y=20x10x2

62. y=12x+4x2

63. y=3x24x2

64. y=6x28x+5

65. y=x25x+1

66. y=1xx2

 Determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.

68. y(x)=2x2+10x+12

69. f(x)=2x210x+4

70. f(x)=x2+4x+3

71. f(x)=4x2+x1

72. h(t)=4t2+6t1

73) f(x)=12x2+3x+1

74) f(x)=13x22x+3

Answers to Odd Examples:

49. The minimum value is 98 when x=14.

51. The minimum value is 6 when x=3.

53. The maximum value is 16 when x=0.

55. Maximum: y=10

57. Minimum: y=4 when x=15

59. Maximum: y=0

61. Maximum: y=10

63. Minimum: y=103

65. Minimum: y=214

69. Minimum is 172 and occurs at 52.
      Axis of symmetry is x=52.

71. Minimum is 1716 and occurs at 18.
Axis of symmetry is x=18.

73. Minimum is 72 and occurs at 3.
Axis of symmetry is x=3.

F: Intercepts

Exercise 3.1e.F 

 Determine the x- and y-intercepts of each function.

75.  f(x)=x2+7x+6

76.  f(x)=x2+10x11

77.  f(x)=x2+8x+12

78.  f(x)=x2+5x+6

79.  f(x)=x2+8x19

80.  f(x)=3x2+x1

81.  f(x)=x2+6x+13

82.  f(x)=x2+8x+12

83.  f(x)=4x220x+25

84.  f(x)=x214x49

85.  f(x)=x26x9

86.  f(x)=4x2+4x+1

87. y=x2+4x12

88. y=x213x+12

89. y=2x2+5x3

90. y=3x24x4

91. y=5x23x+2

92. y=6x2+11x4

93. y=4x227

94. y=9x250

95. y=x2x+1

96. y=x26x+4

97) g(x)=x(x4)20

98) g(x)=x(x2)10

Answers to Odd Examples:

75. y-intercept: (0,6); x-intercept(s): (1,0),(6,0)

77. y-intercept: (0,12); x-intercept(s): (2,0),(6,0)

79. y-intercept: (0,19); x-intercept(s): none

81. y-intercept: (0,13); x-intercept(s): none

83. y-intercept: (0,25); x-intercept(s): (52,0)

85. y-intercept: (0,9); x-intercept(s): (3,0)

 

87. x-intercepts: (6,0),(2,0); y-intercept: (0,12)

89. x-intercepts: (3,0),(12,0); y-intercept: (0,3)

91. x-intercepts: (1,0),(25,0); y-intercept: (0,2)

93. x-intercepts: (332,0),(332,0); y-intercept: (0,27)

95. x-intercepts: none; y-intercept: (0,1)

97. x-intercepts: 2+26,226; y-intercept: (0,20)

G: Graph Quadratic Functions

Exercise 3.1e.G 

  Sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts.

99) f(x)=x22x

100) f(x)=x26x1

101) f(x)=x25x6

102) f(x)=x27x+3

103) f(x)=2x2+5x8

 

105) f(x)=4x212x3

 

 Sketch each quadratic function below

107. f(x)=x210x

108. f(x)=x2+8x

109. f(x)=x29

110. f(x)=x225

111. f(x)=1x2

112. f(x)=4x2

113. f(x)=x22

114. f(x)=x23

115. f(x)=2x2+3

116. f(x)=2x21

117. f(x)=x21

118. f(x)=x2+1

121. n/a

122.  f(x)=13x23

123. n/a

124.  f(x)=5x2+2

Answers to Odd Examples:
 

99.

CNX_Precalc_Figure_03_02_201.jpg

Vertex (1,1),
Axis of symmetry is x=1.
Intercepts: (0,0),(2,0)

101.

CNX_Precalc_Figure_03_02_203.jpg

Vertex (52,494),
Axis of symmetry is x=52.
Intercepts: (0,6),(1,0),(6,0)

103.

CNX_Precalc_Figure_03_02_205.jpg

Vertex (54,398),
Axis of symmetry is x=54.
Intercepts: (0,8)

105.

CNX_Precalc_Figure_03_02_206.jpg

Vertex  (32,12),  
Axis of symmetry is x=32,
Intercepts: (32±3,0),  (0,3)

107.

Figure 107

109.

Figure 109  

111.

Figure 111

113.

Figure 113  

115.

Figure 115

117.

Figure 117
   

 Sketch each quadratic function below

125.  f(x)=x2+2x7

126. f(x)=x2+2x4

127. f(x)=x22x8

128. f(x)=x22x+15

129. f(x)=x2+3x+4

130. f(x)=x2+3x4

131.  f(x)=x2+4x+3

132.  f(x)=x2+4x12

133. f(x)=x24x+2

134. f(x)=x2+4x5

135.  f(x)=x2+6x+5

136.  f(x)=x26x+8

137. f(x)=x26x+15

138. f(x)=x26x+6

Answers to Odd Examples:

125.

This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 15 to 10. The parabola has a vertex at (1, negative 6). The y-intercept, point (0, negative 7), is plotted. The axis of symmetry, x equals 1, is plotted as a dashed vertical line.
Figure 125

127.

Figure 127 

133.

This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 2, 6), y-intercept of (0, 2), and axis of symmetry shown at x equals negative 2.
Figure 133

129.

Figure 129

135.

This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 3, negative 4). The y-intercept, point (0, 5), is plotted as are the x-intercepts, (negative 5, 0) and (negative 1, 0).
Figure 135 

131.

This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, negative 1). The y-intercept, point (0, 3), is plotted as are the x-intercepts, (negative 3, 0) and (negative 1, 0).
Figure 133 

137.

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (3, 6), y-intercept of (0, 10), and axis of symmetry shown at x equals 3.
Figure 137

 Sketch each quadratic function below

145. f(x)=2x24x+1

146.  f(x)=3x26x1

147. f(x)=2x2+8x10

148. f(x)=2x24x5

149. f(x)=5x210x+8

150. f(x)=3x212x+7

151.  f(x)=3x2+18x+20

152. f(x)=3x2+6x+1

153. f(x)=4x2+12x9

154.  f(x)=4x26x2

155. f(x)=4x2+4x3

156. f(x)=4x24x+3

157. f(x)=2x2+6x3

158.  f(x)=9x2+12x+4

159. f(x)=2x2+4x3

160. f(x)=3x2+2x2

Answers to Odd Examples:

145.

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 1), y-intercept of (0, 1), and axis of symmetry shown at x equals 1.
Figure 145

147.

This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (2, negative 2), y-intercept of (0, negative 10), and axis of symmetry shown at x equals 2.
Figure 147 

149.

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, 3), y-intercept of (0, 8), and axis of symmetry shown at x equals 1.
Figure 149 

151.

This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 3, negative 7). The x-intercepts are plotted at the approximate points (negative 4.5, 0) and (negative 1.5, 0). The axis of symmetry is the vertical line x equals negative 3, plotted as a dashed line.
Figure 151 

153.

Figure 153 

155.

Figure 155 

157.

Figure 157 

159.

Figure 117 

 Sketch each quadratic function below

165. f(x)=(x1)2

166. f(x)=(x+1)2

167. f(x)=(x1)2+5

169. f(x)=(x3)2+4

169. f(x)=(x4)29

170. f(x)=(x6)22

171. f(x)=(x+2)2+1

172. f(x)=(x+3)21

173. f(x)=(x4)23

174. f(x)=(x+5)22

175. f(x)=2(x4)2+22

176. f(x)=2(x+3)213

177. f(x)=2(x+1)2+8

178. f(x)=2(x5)23

179. f(x)=4(x1)22

180. f(x)=3(x+2)2+12

181. f(x)=5(x1)2

182. f(x)=(x+2)2

Answers to Odd Examples:

165.

Figure 165

167.

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (1, 5) and other points (negative 1, 9) and (3, 9).
Figure 167

169.

Figure 169 

171.

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 2, 1) and other points (negative 4, 5) and (0, 5).
Figure 171

173.

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (4, negative 2) and other points (3, negative 2) and (5, negative 2).
Figure 173 

175.

Figure 175 

177.

Figure 177 

179.

Figure 179 

181.

Figure 181 

H: Convert to vertex form

Exercise 3.1e.H 

 Rewrite in Standard (vertex) form y=a(xh)2+k and determine the vertex.

185. y=x214x+24

186. y=x212x+40

187. y=x2+4x12

188. y=x2+6x1

189. y=2x212x3

190. y=3x26x+5

191. y=x2+16x+17

192. y=x2+10x

193. f(x)=x24x+2

194. f(x)=x212x+32

195. g(x)=x2+2x3

196. f(x)=x2x

197. f(x)=x2+5x2

198. h(x)=2x2+8x10

199. k(x)=3x26x9

200. f(x)=2x26x

201. f(x)=3x25x1

203. f(x)=3x212x5

204. f(x)=2x212x+7

205. f(x)=3x2+6x1

206. f(x)=4x216x9

207. f(x)=5x210x+8

208.  f(x)=3x26x1

209. f(x)=2x24x+1

210.  f(x)=16x2+24x+6

Answers  to Odd Examples:

185. y=(x7)225; vertex: (7,25)

187. y=(x+2)216; vertex: (2,16)

189. y=2(x3)221; vertex: (3,21)

191. y=(x8)2+81; vertex: (8,81)

193. f(x)=(x+2)2+6; vertex: (2,6)

195. g(x)=(x+1)24, Vertex (1,4)

197. f(x)=(x+52)2334, Vertex (52,334)

199. k(x)=3(x1)212, Vertex (1,12)

201. f(x)=3(x56)23712, Vertex (56,3712)

203. f(x)=3(x+2)2+7; vertex: (2,7)

205. f(x)=3(x+1)24; vertex: (1,4)

207. f(x)=5(x1)2+3; vertex: (1,3)

209. f(x)=2(x1)21; vertex: (1,1)

 

I: Convert to vertex form

Exercise 3.1e.I 

 In the following exercises, write the quadratic function in f(x)=a(xh)2+k form whose graph is shown.

211.
This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, negative 5) and y-intercept (0, negative 4).

212.
This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (2,4) and y-intercept (0, 8).

213.
This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 3) and y-intercept (0, negative 1).

214.
This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, negative 5) and y-intercept (0, negative 3).

215. 

CNX_Precalc_Figure_03_02_207.jpg

216.

CNX_Precalc_Figure_03_02_208.jpg

217.

CNX_Precalc_Figure_03_02_209.jpg

218. 

CNX_Precalc_Figure_03_02_210.jpg 

219. 

CNX_Precalc_Figure_03_02_211n.jpg

220. 

CNX_Precalc_Figure_03_02_212.jpg

221.

This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (4, 0), y-intercept of (0, negative 16), and axis of symmetry shown at x equals 4.

 
Answers to Odd Examples

211. f(x)=(x+1)25

213. f(x)=2(x1)23

215. f(x)=x24x+1

217. f(x)=2x2+8x1

219. f(x)=12x23x+72

221. f(x)=(x4)2+0

J: Construct an equation from points

Exercise 3.1e.J 

 Use the vertex (h,k) and a point on the graph (x,y) to find the general form of the equation of the quadratic function.

223) (h,k)=(2,0),(x,y)=(4,4)

224) (h,k)=(2,1),(x,y)=(4,3)

225) (h,k)=(0,1),(x,y)=(2,5)

226) (h,k)=(2,3),(x,y)=(5,12)

227) (h,k)=(5,3),(x,y)=(2,9)

228) (h,k)=(3,2),(x,y)=(10,1)

229) (h,k)=(0,1),(x,y)=(1,0)

230) (h,k)=(1,0),(x,y)=(0,1)

 Write the equation of the quadratic function that contains the given point and has the same shape as the given function.

231) Contains (1,1) and has shape of f(x)=2x2. Vertex is on the y-axis.

232) Contains (1,4) and has the shape of f(x)=2x2. Vertex is on the y-axis.

233) Contains (2,3) and has the shape of f(x)=3x2. Vertex is on the y-axis.

234) Contains (1,3) and has the shape of f(x)=x2. Vertex is on the y-axis.

235) Contains (4,3) and has the shape of f(x)=5x2. Vertex is on the y-axis.

236) Contains (1,6) has the shape of f(x)=3x2. Vertex has x-coordinate of 1.

Answers to Odd Examples:

223. f(x)=x24x+4

225. f(x)=x2+1

227. f(x)=649x2+6049x+29749

229. f(x)=x2+1

231. f(x)=2x21

233. f(x)=3x29

235. f(x)=5x277


3.1e: Exercises - Quadratic Functions is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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