3.1e: Exercises - Quadratic Functions

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A: Concepts

Exercise $\PageIndex{A}$

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 $a≠0$ 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 $\PageIndex{B}$

$ \bigstar $ Determine if the parabola opens up or down.

7. a. $f(x)=-2 x^{2}-6 x-7$
$ \quad $ b. $f(x)=6 x^{2}+2 x+3$

8. a. $f(x)=4 x^{2}+x-4$
$ \quad $ b. $f(x)=-9 x^{2}-24 x-16$

9. a. $f(x)=-3 x^{2}+5 x-1$
$ \quad $ b. $f(x)=2 x^{2}-4 x+5$

10. a. $f(x)=x^{2}+3 x-4$
$ \quad $ b. $f(x)=-4 x^{2}-12 x-9$

11. $y=x^{2}-9 x+20$
12. $y=x^{2}-12 x+32$
13. $y=-2 x^{2}+5 x+12$
14. $y=-6 x^{2}+13 x-6$
15. $y=64-x^{2}$
16. $y=-3 x+9 x^{2}$

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 $\PageIndex{C}$

$ \bigstar $ Determine the vertex.

17. $y=-(x-5)^{2}+3$

18. $y=-2(x-1)^{2}+7$

19. $y=5(x+1)^{2}+6$

20. $y=3(x+4)^{2}+10$

21. $y=-5(x+8)^{2}-1$

22. $y=(x+2)^{2}-5$

$ \bigstar $ Find the vertex and the axis of symmetry.

23. $f(x)=x^{2}+8 x-1$

24. $f(x)=x^{2}+10 x+25$

25. $f(x)=-x^{2}+2 x+5$

26. $f(x)=-2 x^{2}-8 x-3$

27. $y=-x^{2}+10 x-34$

28. $y=-x^{2}-6 x+1$

29. $y=-4 x^{2}+12 x-7$

30. $y=-9 x^{2}+6 x+2$

31. $y=4 x^{2}-1$

32. $y=x^{2}-16$

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: $\left(\frac{3}{2}, 2\right)$; axis of symmetry: $x=\frac{3}{2}$

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

D: Domain and Range

Exercise $\PageIndex{D}$

$ \bigstar $ 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.

$ \bigstar $ Given the following quadratic functions, determine the domain and range.

37. $f(x)=3 x^{2}+30 x+50$

38. $f(x)=5 x^{2}-10 x+1$

39. $g(x)=-2 x^{2}+4 x+1$

40. $g(x)=-7 x^{2}-14 x-9$

41. $f(x)=x^{2}+x-1$

42. $f(x)=-x^{2}+3 x-2$

43) $f(x)=(x−3)^2+2$

44) $f(x)=−2(x+3)^2−6$

45) $f(x)=x^2+6x+4$

46) $f(x)=2x^2−4x+2$

47) $k(x)=3x^2−6x−9$

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: $[−\frac{5}{4}, ∞)$

43. Domain is $(−∞,∞)$. Range is $[2,∞)$.

45. Domain is $(−∞,∞)$. Range is $[−5,∞)$.

47. Domain is $(−∞,∞)$. Range is $[−12,∞)$.

E: Minimum or maximum Value

Exercise $\PageIndex{E}$

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

49. $f(x)=2 x^{2}+x-1$

50. $y=-4 x^{2}+12 x-5$

51. $y=x^{2}-6 x+15$

52. $y=-x^{2}+4 x-5$

53. $y=-9 x^{2}+16$

54. $y=4 x^{2}-49$

55. $y=-x^{2}-6 x+1$

56. $y=-x^{2}-4 x+8$

57. $y=25 x^{2}-10 x+5$

58. $y=16 x^{2}-24 x+7$

59. $y=-x^{2}$

60. $y=1-9 x^{2}$

61. . $y=20 x-10 x^{2}$

62. $y=12 x+4 x^{2}$

63. $y=3 x^{2}-4 x-2$

64. $y=6 x^{2}-8 x+5$

65. $y=x^{2}-5 x+1$

66. $y=1-x-x^{2}$

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

68. $y(x)=2x^2+10x+12$

69. $f(x)=2x^2−10x+4$

70. $f(x)=−x^2+4x+3$

71. $f(x)=4x^2+x−1$

72. $h(t)=−4t^2+6t−1$

73) $f(x)=\dfrac{1}{2}x^2+3x+1$

74) $f(x)=−\dfrac{1}{3}x^2−2x+3$

Answers to Odd Examples:

49. The minimum value is $−\frac{9}{8}$ when $x=−\frac{1}{4}$.

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 = \frac{1}{5}$

59. Maximum: $y = 0$

61. Maximum: $y = 10$

63. Minimum: $y = −\frac{10}{3}$

65. Minimum: $y = −\frac{21}{4}$

69. Minimum is $−\frac{17}{2}$ and occurs at $\frac{5}{2}$.
Axis of symmetry is $x=\frac{5}{2}$.

71. Minimum is $−\frac{17}{16}$ and occurs at $−\tfrac{1}{8}$.
$ \qquad $ Axis of symmetry is $x=−\frac{1}{8}$.

73. Minimum is $−\frac{7}{2}$ and occurs at $−3$.
$ \qquad $ Axis of symmetry is $x=−3$.

F: Intercepts

Exercise $\PageIndex{F}$

$ \bigstar $ Determine the $x$- and $y$-intercepts of each function.

75. $f(x)=x^{2}+7 x+6$

76. $f(x)=x^{2}+10 x-11$

77. $f(x)=x^{2}+8 x+12$

78. $f(x)=x^{2}+5 x+6$

79. $f(x)=-x^{2}+8 x-19$

80. $f(x)=-3 x^{2}+x-1$

81. $f(x)=x^{2}+6 x+13$

82. $f(x)=x^{2}+8 x+12$

83. $f(x)=4 x^{2}-20 x+25$

84. $f(x)=-x^{2}-14 x-49$

85. $f(x)=-x^{2}-6 x-9$

86. $f(x)=4 x^{2}+4 x+1$

87. $y=x^{2}+4 x-12$

88. $y=x^{2}-13 x+12$

89. $y=2 x^{2}+5 x-3$

90. $y=3 x^{2}-4 x-4$

91. $y=-5 x^{2}-3 x+2$

92. $y=-6 x^{2}+11 x-4$

93. $y=4 x^{2}-27$

94. $y=9 x^{2}-50$

95. $y=x^{2}-x+1$

96. $y=x^{2}-6 x+4$

97) $g(x) = x(x−4)-20$

98) $g(x) = x(x−2)-10$

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): $(\frac{5}{2},0)$

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

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

89. $x$-intercepts: $(-3,0),\left(\frac{1}{2}, 0\right)$; $y$-intercept: $(0,-3)$

91. $x$-intercepts: $(-1,0),\left(\frac{2}{5}, 0\right)$; $y$-intercept: $(0,2)$

93. $x$-intercepts: $\left(-\frac{3 \sqrt{3}}{2}, 0\right),\left(\frac{3 \sqrt{3}}{2}, 0\right)$; $y$-intercept: $(0,-27)$

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

97. $x$-intercepts: ${2+2 \sqrt{6}, 2−2\sqrt{6}}$; $y$-intercept: $(0,-20)$

G: Graph Quadratic Functions

Exercise $\PageIndex{G}$

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

99) $f(x)=x^2−2x$

100) $f(x)=x^2−6x−1$

101) $f(x)=x^2−5x−6$

102) $f(x)=x^2−7x+3$

103) $f(x)=−2x^2+5x−8$

105) $f(x)=4x^2−12x−3$

$ \bigstar $ Sketch each quadratic function below

107. $f(x)=x^{2}-10 x$

108. $f(x)=x^{2}+8 x$

109. $f(x)=x^{2}-9$

110. $f(x)=x^{2}-25$

111. $f(x)=1-x^{2}$

112. $f(x)=4-x^{2}$

113. $f(x)=x^{2}-2$

114. $f(x)=x^{2}-3$

115. $f(x)=-2 x^{2}+3$

116. $f(x)=-2 x^{2}-1$

117. $f(x)=x^{2}-1$

118. $f(x)=x^{2}+1$

121. n/a

122. $f(x)=\tfrac{1}{3} x^{2}-3$

123. n/a

124. $f(x)=5 x^{2}+2$

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 $\left(\tfrac{5}{2},\tfrac{−49}{4}\right)$,
Axis of symmetry is $x=\tfrac{5}{2}$.
Intercepts: $(0,−6), \; (−1,0), \; (6,0)$

103.

$CNX_Precalc_Figure_03_02_205.jpg$

Vertex $\left(\tfrac{5}{4}, −\tfrac{39}{8}\right)$,
Axis of symmetry is $x=\tfrac{5}{4}$.
Intercepts: $(0, −8)$

105.

$CNX_Precalc_Figure_03_02_206.jpg$

Vertex $\left(\tfrac{3}{2}, −12 \right),$
Axis of symmetry is $x=\frac{3}{2},$
Intercepts: $ \left( \tfrac{3}{2} \pm \sqrt{3} , 0 \right), $ $(0, -3)$

107. Figure 107	109. Figure 109	111. Figure 111	113. Figure 113
115. Figure 115	117. Figure 117

$ \bigstar $ Sketch each quadratic function below

125. $f(x)=-x^{2}+2 x-7$

126. $f(x)=-x^{2}+2 x-4$

127. $f(x)=x^{2}-2 x-8$

128. $f(x)=-x^{2}-2 x+15$

129. $f(x)=x^{2}+3 x+4$

130. $f(x)=-x^{2}+3 x-4$

131. $f(x)=x^{2}+4 x+3$

132. $f(x)=x^{2}+4 x-12$

133. $f(x)=-x^{2}-4 x+2$

134. $f(x)=-x^{2}+4 x-5$

135. $f(x)=x^{2}+6 x+5$

136. $f(x)=x^{2}-6 x+8$

137. $f(x)=x^{2}-6 x+15$

138. $f(x)=x^{2}-6 x+6$

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.

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.

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.

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

$ \bigstar $ Sketch each quadratic function below

145. $f(x)=2 x^{2}-4 x+1$

146. $f(x)=3 x^{2}-6 x-1$

147. $f(x)=-2 x^{2}+8 x-10$

148. $f(x)=-2 x^{2}-4 x-5$

149. $f(x)=5 x^{2}-10 x+8$

150. $f(x)=3 x^{2}-12 x+7$

151. $f(x)=3 x^{2}+18 x+20$

152. $f(x)=-3 x^{2}+6 x+1$

153. $f(x)=-4 x^{2}+12 x-9$

154. $f(x)=-4 x^{2}-6 x-2$

155. $f(x)=-4 x^{2}+4 x-3$

156. $f(x)=-4 x^{2}-4 x+3$

157. $f(x)=-2 x^{2}+6 x-3$

158. $f(x)=9 x^{2}+12 x+4$

159. $f(x)=2 x^{2}+4 x-3$

160. $f(x)=3 x^{2}+2 x-2$

Answers to Odd Examples:

145. Figure 145	147. Figure 147	149. 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

$ \bigstar $ Sketch each quadratic function below

165. $f(x)=(x-1)^{2}$

166. $f(x)=(x+1)^{2}$

167. $f(x)=(x-1)^{2}+5$

169. $f(x)=(x-3)^{2}+4$

169. $f(x)=(x-4)^{2}-9$

170. $f(x)=(x-6)^{2}-2$

171. $f(x)=(x+2)^{2}+1$

172. $f(x)=(x+3)^{2}-1$

173. $f(x)=(x-4)^{2}-3$

174. $f(x)=(x+5)^{2}-2$

175. $f(x)=-2(x-4)^{2}+22$

176. $f(x)=2(x+3)^{2}-13$

177. $f(x)=-2(x+1)^{2}+8$

178. $f(x)=-2(x-5)^{2}-3$

179. $f(x)=-4(x-1)^{2}-2$

180. $f(x)=-3(x+2)^{2}+12$

181. $f(x)=-5(x-1)^{2}$

182. $f(x)=-(x+2)^{2}$

Answers to Odd Examples:

165. Figure 165	167. Figure 167	169. Figure 169
171. Figure 171	173. Figure 173	175. Figure 175
177. Figure 177	179. Figure 179	181. Figure 181

H: Convert to vertex form

Exercise $\PageIndex{H}$

$ \bigstar $ Rewrite in Standard (vertex) form $y=a(x-h)^{2}+k$ and determine the vertex.

185. $y=x^{2}-14 x+24$

186. $y=x^{2}-12 x+40$

187. $y=x^{2}+4 x-12$

188. $y=x^{2}+6 x-1$

189. $y=2 x^{2}-12 x-3$

190. $y=3 x^{2}-6 x+5$

191. $y=-x^{2}+16 x+17$

192. $y=-x^{2}+10 x$

193. $f(x)=-x^{2}-4 x+2$

194. $f(x)=x^2−12x+32$

195. $g(x)=x^2+2x−3$

196. $f(x)=x^2−x$

197. $f(x)=x^2+5x−2$

198. $h(x)=2x^2+8x−10$

199. $k(x)=3x^2−6x−9$

200. $f(x)=2x^2−6x$

201. $f(x)=3x^2−5x−1$

203. $f(x)=-3 x^{2}-12 x-5$

204. $f(x)=2 x^{2}-12 x+7$

205. $f(x)=3 x^{2}+6 x-1$

206. $f(x)=-4 x^{2}-16 x-9$

207. $f(x)=5 x^{2}-10 x+8$

208. $f(x)=3 x^{2}-6 x-1$

209. $f(x)=2 x^{2}-4 x+1$

210. $f(x)=-16 x^{2}+24 x+6$

Answers to Odd Examples:

185. $y=(x-7)^{2}-25$; vertex: $(7, -25)$

187. $y=(x+2)^{2}-16$; vertex: $(-2, -16)$

189. $y=2(x-3)^{2}-21$; vertex: $(3, -21)$

191. $y=-(x-8)^{2}+81$; vertex: $(8, 81)$

193. $f(x)=-(x+2)^{2}+6$; vertex: $(-2,6)$

195. $g(x)=(x+1)^2−4$, Vertex $(−1,−4)$

197. $f(x)=\left(x+\frac{5}{2}\right)^2−\frac{33}{4}$, Vertex $\left(−\frac{5}{2},−\frac{33}{4}\right)$

199. $k(x)=3(x−1)^2−12$, Vertex $(1,−12)$

201. $f(x)=3\left(x−\frac{5}{6}\right)^2−\frac{37}{12}$, Vertex $\left(\frac{5}{6},−\frac{37}{12}\right)$

203. $f(x)=-3(x+2)^{2}+7$; vertex: $(-2,7)$

205. $f(x)=3(x+1)^{2}-4$; vertex: $( -1,-4)$

207. $f(x)=5(x-1)^{2}+3$; vertex: $(1,3)$

209. $f(x)=2(x-1)^{2}-1$; vertex: $( 1,-1)$

I: Convert to vertex form

Exercise $\PageIndex{I}$

$ \bigstar $ In the following exercises, write the quadratic function in $f(x)=a(x−h)^{2}+k$ 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. $f(x)=(x+1)^{2}-5$

213. $f(x)=2(x-1)^{2}-3$

215. $f(x)=x^2−4x+1$

217. $f(x)= -2x^2+8x-1 $

219. $f(x)= \frac{1}{2}x^2-3x+\frac{7}{2} $

221. $f(x)=-(x-4)^{2}+0$

J: Construct an equation from points

Exercise $\PageIndex{J}$

$ \bigstar $ 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) $(h,k)=(1,0), \; (x,y)=(0,1)$

$ \bigstar $ 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 $f(x)=2x^2$. Vertex is on the $y$-axis.

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

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

234) Contains $(1,−3)$ and has the shape of $f(x)=−x^2$. Vertex is on the $y$-axis.

235) Contains $(4,3)$ and has the shape of $f(x)=5x^2$. Vertex is on the $y$-axis.

236) Contains $(1,−6)$ has the shape of $f(x)=3x^2$. Vertex has $x$-coordinate of $-1$.

Answers to Odd Examples:

223. $f(x)=x^2−4x+4$

225. $f(x)=x^2+1$

227. $f(x)=\frac{6}{49}x^2+\frac{60}{49}x+\frac{297}{49}$

229. $f(x)=−x^2+1$

231. $f(x)=2x^2−1$

233. $f(x)=3x^2−9$

235. $f(x)=5x^2−77$