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Exploring Quadratic Functions and Discriminant: Parabola with No Real Roots

Exploring Quadratic Functions and Discriminant: Parabola with No Real Roots
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2015-09-06 10:47
FunctionCoordinate GeometryInequalityGraphing
This GeoGebra example demonstrates how to construct the graph of a quadratic function f(x)=ax²+bx+c and verify that the parabola has no intersection with the x-axis when the discriminant b²-4ac < 0 (no real roots). The core mathematical concepts include the graphical representation of quadratic functions, the geometric meaning of the discriminant, and the relationship between roots and function graphs. Construction details: Textfield command creates user input interfaces, Sequence command generates parameter sequences, and If conditional commands determine discriminant signs and display corresponding feedback, involving parameter objects a, b, c and function object f. Ideal for middle and high school mathematics courses on functions and equations, as well as an interactive tool for students to independently explore quadratic function properties.

GeoGebra Commands & Parameters

  1. Textfield(userCondition) → окноВвода4
  2. Sequence(Sequence[Sequence[{i, j, k}, i, -5, 5], j, -5, 5], k, -5, 5) → allArguments
  3. If(abs(A3 - A2) ≤ 2, B2, B2 + 1) → B3
  4. If(abs(A4 - A3) ≤ 2, B3, B3 + 1) → B4
  5. If(abs(A5 - A4) ≤ 2, B4, B4 + 1) → B5
  6. If(abs(A6 - A5) ≤ 2, B5, B5 + 1) → B6
  7. If(abs(A7 - A6) ≤ 2, B6, B6 + 1) → B7
  8. If(abs(A8 - A7) ≤ 2, B7, B7 + 1) → B8
  9. If(abs(A9 - A8) ≤ 2, B8, B8 + 1) → B9
  10. If(abs(A10 - A9) ≤ 2, B9, B9 + 1) → B10
  11. If(abs(A11 - A10) ≤ 2, B10, B10 + 1) → B11
  12. If(abs(A12 - A11) ≤ 2, B11, B11 + 1) → B12
  13. If(abs(A13 - A12) ≤ 2, B12, B12 + 1) → B13
  14. If(abs(A14 - A13) ≤ 2, B13, B13 + 1) → B14
  15. If(abs(A15 - A14) ≤ 2, B14, B14 + 1) → B15
  16. If(abs(A16 - A15) ≤ 2, B15, B15 + 1) → B16
  17. If(abs(A17 - A16) ≤ 2, B16, B16 + 1) → B17
  18. If(abs(A18 - A17) ≤ 2, B17, B17 + 1) → B18
  19. If(abs(A19 - A18) ≤ 2, B18, B18 + 1) → B19
  20. If(abs(A20 - A19) ≤ 2, B19, B19 + 1) → B20
  21. If(abs(A21 - A20) ≤ 2, B20, B20 + 1) → B21
  22. If(abs(A22 - A21) ≤ 2, B21, B21 + 1) → B22
  23. If(abs(A23 - A22) ≤ 2, B22, B22 + 1) → B23
  24. If(abs(A24 - A23) ≤ 2, B23, B23 + 1) → B24
  25. If(abs(A25 - A24) ≤ 2, B24, B24 + 1) → B25
  26. If(abs(A26 - A25) ≤ 2, B25, B25 + 1) → B26
  27. If(abs(A27 - A26) ≤ 2, B26, B26 + 1) → B27
  28. If(abs(A28 - A27) ≤ 2, B27, B27 + 1) → B28
  29. If(abs(A29 - A28) ≤ 2, B28, B28 + 1) → B29
  30. If(abs(A30 - A29) ≤ 2, B29, B29 + 1) → B30
  31. If(abs(A31 - A30) ≤ 2, B30, B30 + 1) → B31
  32. If(abs(A32 - A31) ≤ 2, B31, B31 + 1) → B32
  33. If(abs(A33 - A32) ≤ 2, B32, B32 + 1) → B33
  34. If(abs(A34 - A33) ≤ 2, B33, B33 + 1) → B34
  35. If(abs(A35 - A34) ≤ 2, B34, B34 + 1) → B35
  36. If(abs(A36 - A35) ≤ 2, B35, B35 + 1) → B36
  37. If(abs(A37 - A36) ≤ 2, B36, B36 + 1) → B37
  38. If(abs(A38 - A37) ≤ 2, B37, B37 + 1) → B38
  39. If(abs(A39 - A38) ≤ 2, B38, B38 + 1) → B39
  40. If(abs(A40 - A39) ≤ 2, B39, B39 + 1) → B40
  41. If(abs(A41 - A40) ≤ 2, B40, B40 + 1) → B41
  42. If(abs(A42 - A41) ≤ 2, B41, B41 + 1) → B42
  43. If(abs(A43 - A42) ≤ 2, B42, B42 + 1) → B43
  44. If(abs(A44 - A43) ≤ 2, B43, B43 + 1) → B44
  45. If(abs(A45 - A44) ≤ 2, B44, B44 + 1) → B45
  46. If(abs(A46 - A45) ≤ 2, B45, B45 + 1) → B46
  47. If(abs(A47 - A46) ≤ 2, B46, B46 + 1) → B47
  48. If(abs(A48 - A47) ≤ 2, B47, B47 + 1) → B48
  49. If(abs(A49 - A48) ≤ 2, B48, B48 + 1) → B49
  50. If(abs(A50 - A49) ≤ 2, B49, B49 + 1) → B50
  51. If(abs(A51 - A50) ≤ 2, B50, B50 + 1) → B51
  52. If(abs(A53 - A52) ≤ 2, B52, B52 + 1) → B53
  53. If(abs(A54 - A53) ≤ 2, B53, B53 + 1) → B54
  54. If(abs(A55 - A54) ≤ 2, B54, B54 + 1) → B55
  55. If(abs(A56 - A55) ≤ 2, B55, B55 + 1) → B56
  56. If(abs(A57 - A56) ≤ 2, B56, B56 + 1) → B57
  57. If(abs(A58 - A57) ≤ 2, B57, B57 + 1) → B58
  58. If(abs(A59 - A58) ≤ 2, B58, B58 + 1) → B59
  59. If(abs(A60 - A59) ≤ 2, B59, B59 + 1) → B60
  60. If(abs(A61 - A60) ≤ 2, B60, B60 + 1) → B61
  61. If(abs(A62 - A61) ≤ 2, B61, B61 + 1) → B62
  62. If(abs(A63 - A62) ≤ 2, B62, B62 + 1) → B63
  63. If(abs(A64 - A63) ≤ 2, B63, B63 + 1) → B64
  64. If(abs(A65 - A64) ≤ 2, B64, B64 + 1) → B65
  65. If(abs(A66 - A65) ≤ 2, B65, B65 + 1) → B66
  66. If(abs(A67 - A66) ≤ 2, B66, B66 + 1) → B67
  67. If(abs(A68 - A67) ≤ 2, B67, B67 + 1) → B68
  68. If(abs(A69 - A68) ≤ 2, B68, B68 + 1) → B69
  69. If(abs(A70 - A69) ≤ 2, B69, B69 + 1) → B70
  70. If(abs(A71 - A70) ≤ 2, B70, B70 + 1) → B71
  71. If(abs(A72 - A71) ≤ 2, B71, B71 + 1) → B72
  72. If(abs(A73 - A72) ≤ 2, B72, B72 + 1) → B73
  73. If(abs(A74 - A73) ≤ 2, B73, B73 + 1) → B74
  74. If(abs(A75 - A74) ≤ 2, B74, B74 + 1) → B75
  75. If(abs(A76 - A75) ≤ 2, B75, B75 + 1) → B76
  76. If(abs(A77 - A76) ≤ 2, B76, B76 + 1) → B77
  77. If(abs(A78 - A77) ≤ 2, B77, B77 + 1) → B78
  78. If(abs(A79 - A78) ≤ 2, B78, B78 + 1) → B79
  79. If(abs(A80 - A79) ≤ 2, B79, B79 + 1) → B80
  80. If(abs(A81 - A80) ≤ 2, B80, B80 + 1) → B81
  81. If(abs(A82 - A81) ≤ 2, B81, B81 + 1) → B82
  82. If(abs(A83 - A82) ≤ 2, B82, B82 + 1) → B83
  83. If(abs(A84 - A83) ≤ 2, B83, B83 + 1) → B84
  84. If(abs(A85 - A84) ≤ 2, B84, B84 + 1) → B85
  85. If(abs(A86 - A85) ≤ 2, B85, B85 + 1) → B86
  86. If(abs(A87 - A86) ≤ 2, B86, B86 + 1) → B87
  87. If(abs(A88 - A87) ≤ 2, B87, B87 + 1) → B88
  88. If(abs(A89 - A88) ≤ 2, B88, B88 + 1) → B89
  89. If(abs(A90 - A89) ≤ 2, B89, B89 + 1) → B90
  90. If(abs(A91 - A90) ≤ 2, B90, B90 + 1) → B91
  91. If(abs(A92 - A91) ≤ 2, B91, B91 + 1) → B92
  92. If(abs(A93 - A92) ≤ 2, B92, B92 + 1) → B93
  93. If(abs(A94 - A93) ≤ 2, B93, B93 + 1) → B94
  94. If(abs(A95 - A94) ≤ 2, B94, B94 + 1) → B95
  95. If(abs(A96 - A95) ≤ 2, B95, B95 + 1) → B96
  96. If(abs(A97 - A96) ≤ 2, B96, B96 + 1) → B97
  97. If(abs(A98 - A97) ≤ 2, B97, B97 + 1) → B98
  98. If(abs(A99 - A98) ≤ 2, B98, B98 + 1) → B99
  99. If(abs(A100 - A99) ≤ 2, B99, B99 + 1) → B100
  100. If(abs(A101 - A100) ≤ 2, B100, B100 + 1) → B101
  101. If(abs(A103 - A102) ≤ 2, B102, B102 + 1) → B103
  102. If(abs(A104 - A103) ≤ 2, B103, B103 + 1) → B104
  103. If(abs(A105 - A104) ≤ 2, B104, B104 + 1) → B105
  104. If(abs(A106 - A105) ≤ 2, B105, B105 + 1) → B106
  105. If(abs(A107 - A106) ≤ 2, B106, B106 + 1) → B107
  106. If(abs(A108 - A107) ≤ 2, B107, B107 + 1) → B108
  107. If(abs(A109 - A108) ≤ 2, B108, B108 + 1) → B109
  108. If(abs(A110 - A109) ≤ 2, B109, B109 + 1) → B110
  109. If(abs(A111 - A110) ≤ 2, B110, B110 + 1) → B111
  110. If(abs(A112 - A111) ≤ 2, B111, B111 + 1) → B112
  111. If(abs(A113 - A112) ≤ 2, B112, B112 + 1) → B113
  112. If(abs(A114 - A113) ≤ 2, B113, B113 + 1) → B114
  113. If(abs(A115 - A114) ≤ 2, B114, B114 + 1) → B115
  114. If(abs(A116 - A115) ≤ 2, B115, B115 + 1) → B116
  115. If(abs(A117 - A116) ≤ 2, B116, B116 + 1) → B117
  116. If(abs(A118 - A117) ≤ 2, B117, B117 + 1) → B118
  117. If(abs(A119 - A118) ≤ 2, B118, B118 + 1) → B119
  118. If(abs(A120 - A119) ≤ 2, B119, B119 + 1) → B120
  119. If(abs(A121 - A120) ≤ 2, B120, B120 + 1) → B121
  120. If(abs(A122 - A121) ≤ 2, B121, B121 + 1) → B122
  121. If(abs(A123 - A122) ≤ 2, B122, B122 + 1) → B123
  122. If(abs(A124 - A123) ≤ 2, B123, B123 + 1) → B124
  123. If(abs(A125 - A124) ≤ 2, B124, B124 + 1) → B125
  124. If(abs(A126 - A125) ≤ 2, B125, B125 + 1) → B126
  125. If(abs(A127 - A126) ≤ 2, B126, B126 + 1) → B127
  126. If(abs(A128 - A127) ≤ 2, B127, B127 + 1) → B128
  127. If(abs(A129 - A128) ≤ 2, B128, B128 + 1) → B129
  128. If(abs(A130 - A129) ≤ 2, B129, B129 + 1) → B130
  129. If(abs(A131 - A130) ≤ 2, B130, B130 + 1) → B131
  130. If(abs(A132 - A131) ≤ 2, B131, B131 + 1) → B132
  131. If(abs(A133 - A132) ≤ 2, B132, B132 + 1) → B133
  132. If(abs(A134 - A133) ≤ 2, B133, B133 + 1) → B134
  133. If(abs(A135 - A134) ≤ 2, B134, B134 + 1) → B135
  134. If(abs(A136 - A135) ≤ 2, B135, B135 + 1) → B136
  135. If(abs(A137 - A136) ≤ 2, B136, B136 + 1) → B137
  136. If(abs(A139 - A138) ≤ 2, B138, B138 + 1) → B139
  137. If(abs(A140 - A139) ≤ 2, B139, B139 + 1) → B140
  138. If(abs(A141 - A140) ≤ 2, B140, B140 + 1) → B141
  139. If(abs(A142 - A141) ≤ 2, B141, B141 + 1) → B142
  140. If(abs(A143 - A142) ≤ 2, B142, B142 + 1) → B143
  141. If(abs(A144 - A143) ≤ 2, B143, B143 + 1) → B144
  142. If(abs(A145 - A144) ≤ 2, B144, B144 + 1) → B145
  143. If(abs(A146 - A145) ≤ 2, B145, B145 + 1) → B146
  144. If(abs(A147 - A146) ≤ 2, B146, B146 + 1) → B147
  145. If(abs(A148 - A147) ≤ 2, B147, B147 + 1) → B148
  146. If(abs(A149 - A148) ≤ 2, B148, B148 + 1) → B149
  147. If(abs(A150 - A149) ≤ 2, B149, B149 + 1) → B150
  148. If(abs(A153 - A152) ≤ 2, B152, B152 + 1) → B153
  149. If(abs(A154 - A153) ≤ 2, B153, B153 + 1) → B154
  150. If(abs(A155 - A154) ≤ 2, B154, B154 + 1) → B155
  151. If(abs(A156 - A155) ≤ 2, B155, B155 + 1) → B156
  152. If(abs(A157 - A156) ≤ 2, B156, B156 + 1) → B157
  153. If(abs(A158 - A157) ≤ 2, B157, B157 + 1) → B158
  154. If(abs(A159 - A158) ≤ 2, B158, B158 + 1) → B159
  155. If(abs(A160 - A159) ≤ 2, B159, B159 + 1) → B160
  156. If(abs(A161 - A160) ≤ 2, B160, B160 + 1) → B161
  157. If(abs(A162 - A161) ≤ 2, B161, B161 + 1) → B162
  158. If(abs(A163 - A162) ≤ 2, B162, B162 + 1) → B163
  159. If(abs(A164 - A163) ≤ 2, B163, B163 + 1) → B164
  160. If(abs(A165 - A164) ≤ 2, B164, B164 + 1) → B165
  161. If(abs(A166 - A165) ≤ 2, B165, B165 + 1) → B166
  162. If(abs(A167 - A166) ≤ 2, B166, B166 + 1) → B167
  163. If(abs(A168 - A167) ≤ 2, B167, B167 + 1) → B168
  164. If(abs(A169 - A168) ≤ 2, B168, B168 + 1) → B169
  165. If(abs(A170 - A169) ≤ 2, B169, B169 + 1) → B170
  166. If(abs(A171 - A170) ≤ 2, B170, B170 + 1) → B171
  167. If(abs(A172 - A171) ≤ 2, B171, B171 + 1) → B172
  168. If(abs(A173 - A172) ≤ 2, B172, B172 + 1) → B173
  169. If(abs(A174 - A173) ≤ 2, B173, B173 + 1) → B174
  170. If(abs(A175 - A174) ≤ 2, B174, B174 + 1) → B175
  171. If(abs(A176 - A175) ≤ 2, B175, B175 + 1) → B176
  172. If(abs(A177 - A176) ≤ 2, B176, B176 + 1) → B177
  173. If(abs(A178 - A177) ≤ 2, B177, B177 + 1) → B178
  174. If(abs(A179 - A178) ≤ 2, B178, B178 + 1) → B179
  175. If(abs(A180 - A179) ≤ 2, B179, B179 + 1) → B180
  176. If(abs(A181 - A180) ≤ 2, B180, B180 + 1) → B181
  177. If(abs(A182 - A181) ≤ 2, B181, B181 + 1) → B182
  178. If(abs(A183 - A182) ≤ 2, B182, B182 + 1) → B183
  179. If(abs(A184 - A183) ≤ 2, B183, B183 + 1) → B184
  180. If(abs(A185 - A184) ≤ 2, B184, B184 + 1) → B185
  181. If(abs(A186 - A185) ≤ 2, B185, B185 + 1) → B186
  182. If(abs(A187 - A186) ≤ 2, B186, B186 + 1) → B187
  183. If(abs(A189 - A188) ≤ 2, B188, B188 + 1) → B189
  184. If(abs(A190 - A189) ≤ 2, B189, B189 + 1) → B190
  185. If(abs(A191 - A190) ≤ 2, B190, B190 + 1) → B191
  186. If(abs(A192 - A191) ≤ 2, B191, B191 + 1) → B192
  187. If(abs(A193 - A192) ≤ 2, B192, B192 + 1) → B193
  188. If(abs(A194 - A193) ≤ 2, B193, B193 + 1) → B194
  189. If(abs(A195 - A194) ≤ 2, B194, B194 + 1) → B195
  190. If(abs(A196 - A195) ≤ 2, B195, B195 + 1) → B196
  191. If(abs(A197 - A196) ≤ 2, B196, B196 + 1) → B197
  192. If(abs(A198 - A197) ≤ 2, B197, B197 + 1) → B198
  193. If(abs(A199 - A198) ≤ 2, B198, B198 + 1) → B199
  194. If(abs(A200 - A199) ≤ 2, B199, B199 + 1) → B200
  195. If(abs(A201 - A200) ≤ 2, B200, B200 + 1) → B201
  196. If(abs(A203 - A202) ≤ 2, B202, B202 + 1) → B203
  197. If(abs(A204 - A203) ≤ 2, B203, B203 + 1) → B204
  198. If(abs(A205 - A204) ≤ 2, B204, B204 + 1) → B205
  199. If(abs(A206 - A205) ≤ 2, B205, B205 + 1) → B206
  200. If(abs(A207 - A206) ≤ 2, B206, B206 + 1) → B207
  201. If(abs(A208 - A207) ≤ 2, B207, B207 + 1) → B208
  202. If(abs(A209 - A208) ≤ 2, B208, B208 + 1) → B209
  203. If(abs(A210 - A209) ≤ 2, B209, B209 + 1) → B210
  204. If(abs(A211 - A210) ≤ 2, B210, B210 + 1) → B211
  205. If(abs(A212 - A211) ≤ 2, B211, B211 + 1) → B212
  206. If(abs(A213 - A212) ≤ 2, B212, B212 + 1) → B213
  207. If(abs(A214 - A213) ≤ 2, B213, B213 + 1) → B214
  208. If(abs(A215 - A214) ≤ 2, B214, B214 + 1) → B215
  209. If(abs(A216 - A215) ≤ 2, B215, B215 + 1) → B216
  210. If(abs(A217 - A216) ≤ 2, B216, B216 + 1) → B217
  211. If(abs(A218 - A217) ≤ 2, B217, B217 + 1) → B218
  212. If(abs(A219 - A218) ≤ 2, B218, B218 + 1) → B219
  213. If(abs(A220 - A219) ≤ 2, B219, B219 + 1) → B220
  214. If(abs(A221 - A220) ≤ 2, B220, B220 + 1) → B221
  215. If(abs(A222 - A221) ≤ 2, B221, B221 + 1) → B222
  216. If(abs(A223 - A222) ≤ 2, B222, B222 + 1) → B223
  217. If(abs(A224 - A223) ≤ 2, B223, B223 + 1) → B224
  218. If(abs(A225 - A224) ≤ 2, B224, B224 + 1) → B225
  219. If(abs(A226 - A225) ≤ 2, B225, B225 + 1) → B226
  220. If(abs(A227 - A226) ≤ 2, B226, B226 + 1) → B227
  221. If(abs(A228 - A227) ≤ 2, B227, B227 + 1) → B228
  222. If(abs(A229 - A228) ≤ 2, B228, B228 + 1) → B229
  223. If(abs(A230 - A229) ≤ 2, B229, B229 + 1) → B230
  224. If(abs(A231 - A230) ≤ 2, B230, B230 + 1) → B231
  225. If(abs(A232 - A231) ≤ 2, B231, B231 + 1) → B232
  226. If(abs(A233 - A232) ≤ 2, B232, B232 + 1) → B233
  227. If(abs(A234 - A233) ≤ 2, B233, B233 + 1) → B234
  228. If(abs(A235 - A234) ≤ 2, B234, B234 + 1) → B235
  229. If(abs(A236 - A235) ≤ 2, B235, B235 + 1) → B236
  230. If(abs(A237 - A236) ≤ 2, B236, B236 + 1) → B237
  231. If(abs(A238 - A237) ≤ 2, B237, B237 + 1) → B238
  232. If(abs(A239 - A238) ≤ 2, B238, B238 + 1) → B239
  233. If(abs(A240 - A239) ≤ 2, B239, B239 + 1) → B240
  234. If(abs(A241 - A240) ≤ 2, B240, B240 + 1) → B241
  235. If(abs(A242 - A241) ≤ 2, B241, B241 + 1) → B242
  236. If(abs(A243 - A242) ≤ 2, B242, B242 + 1) → B243
  237. If(abs(A244 - A243) ≤ 2, B243, B243 + 1) → B244
  238. If(abs(A245 - A244) ≤ 2, B244, B244 + 1) → B245
  239. If(abs(A246 - A245) ≤ 2, B245, B245 + 1) → B246
  240. If(abs(A247 - A246) ≤ 2, B246, B246 + 1) → B247
  241. If(abs(A248 - A247) ≤ 2, B247, B247 + 1) → B248
  242. If(abs(A249 - A248) ≤ 2, B248, B248 + 1) → B249
  243. If(abs(A250 - A249) ≤ 2, B249, B249 + 1) → B250
  244. If(abs(A251 - A250) ≤ 2, B250, B250 + 1) → B251

AI Drawing Prompt Reference

Draw a graph of a quadratic function, including a parabola, the x-axis, and three text boxes for inputting coefficients a, b, and c. The parabola should be drawn based on the input coefficients, and when the discriminant b²-4ac is less than 0, the parabola should have no intersection with the x-axis. The construction order is: first draw the x-axis, then draw the parabola, and finally add the input boxes.
The prompt above was compiled from the original drawing commands and is for reference only.