# 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?

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. $$y=-3 x+9 x^{2}$$
 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. $$y=x^{2}-16$$
 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) $$k(x)=3x^2−6x−9$$
 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) $$f(x)=−\dfrac{1}{3}x^2−2x+3$$
 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) $$g(x) = x(x−2)-10$$
 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: $$(0,-20)$$

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.  $$f(x)=5 x^{2}+2$$

 99. Vertex $$(1, −1)$$, Axis of symmetry is $$x=1$$. Intercepts: $$(0,0), \; (2,0)$$ 101. 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. Vertex $$\left(\tfrac{5}{4}, −\tfrac{39}{8}\right)$$, Axis of symmetry is $$x=\tfrac{5}{4}$$. Intercepts: $$(0, −8)$$ 105. 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. $$f(x)=x^{2}-6 x+6$$
 125. Figure 125 127. Figure 127  133. Figure 133 129. Figure 129 135. Figure 135 131. Figure 133  137. 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. $$f(x)=3 x^{2}+2 x-2$$
 145. Figure 145 147. Figure 147 149. Figure 149 151. 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. $$f(x)=-(x+2)^{2}$$
 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.  $$f(x)=-16 x^{2}+24 x+6$$
 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.

 211 212 213 214. 215 216 217 218. 219 220 221
 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 $$-1$$.

 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. $$f(x)=5x^2−77$$