38:
179:
1276:
1367:
1282:
1287:
90:
2034:
2001:
1971:
1938:
1905:
2051:{{Information |Description=Graphs showing the relationship between the roots, and turning, stationary and inflection points of a cubic polynomial, and its first and second derivatives by CMG Lee. |Source={{own}} |Date= |Author=
1328:– You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
2105:
138:
Graphs showing the relationship between the roots, and turning, stationary and inflection points of a cubic polynomial, and its first and second derivatives by CMG Lee. Thanks to
2141:
Graphs showing the relationship between the roots, and turning, stationary and inflection points of a cubic polynomial, and its first and second derivatives by CMG Lee.
64:
60:
56:
52:
48:
42:
1384:
101:
73:
2110:
1382:; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled
1374:
182:
2033:
2028:
2000:
1995:
1970:
1965:
1937:
1932:
1738:
1650:
1614:
1597:
1561:
2059:
2121:
This file contains additional information, probably added from the digital camera or scanner used to create or digitize it.
1335:
1833:
1796:
1759:
1704:
1670:
1634:
1545:
1517:
1502:
1489:
1379:
1731:
1581:
1462:
1445:
2056:
1297:
1509:
2124:
If the file has been modified from its original state, some details may not fully reflect the modified file.
2071:
The following pages on the
English Knowledge (XXG) use this file (pages on other projects are not listed):
2019:
1956:
1923:
1904:
1899:
37:
1334:– If you remix, transform, or build upon the material, you must distribute your contributions under the
178:
2100:
1294:
139:
109:
1361:
1372:
Permission is granted to copy, distribute and/or modify this document under the terms of the
2090:
2085:
2075:
1524:
2080:
1269:
I, the copyright holder of this work, hereby publish it under the following licenses:
96:
426:## http://wolframalpha.com/input?i=solve+2r1*r2-2r1*x%2Br2^2-4r2*x%2B3x^2%3D0+for+x
2055:|Permission= |other_versions={{source thumb|cubic_graph_special_points.svg}} }}
2095:
2052:
2044:
2011:
1981:
1948:
1915:
144:
an algorithm to exactly draw a cubic polynomial segment with a cubic Bezier.
163:
429:## http://wolframalpha.com/input?i=differentiate+(x-r2)(x-(2r1%2Br2)%2F3)
432:## http://wolframalpha.com/input?i=solve+2x-2r1%2F3-4r2%2F3%3D0+for+x
2106:
Knowledge (XXG):Reference desk/Archives/Mathematics/2024 February 2
1872:
Click on a date/time to view the file as it appeared at that time.
423:## http://wolframalpha.com/input?i=differentiate+(x-r1)(x-r2)(x-r2)
441:## Cubic bezier control points from left to right, to be populated
420:## http://wolframalpha.com/input?i=expand+%28x-r1%29%28x-r2%29%5E2
177:
1922:
Shrink font and simplify labels // Editing SVG source code using
84:
438:## http://www.wolframalpha.com/input?i=factorise+3x^2-12x%2B9
1598:
Creative
Commons Attribution-ShareAlike 4.0 International
1429:
Add a one-line explanation of what this file represents
143:
78:(SVG file, nominally 512 × 512 pixels, file size: 7 KB)
198:
Python script to find distinct non-zero integer values
1615:
1378:, Version 1.2 or any later version published by the
128:
108:
Commons is a freely licensed media file repository.
435:## http://wolframalpha.com/input?i=y%3D(x-4)(x-1)^2
2018:Fix missing dots // Editing SVG source code using
1345:https://creativecommons.org/licenses/by-sa/4.0
89:
1351:Creative Commons Attribution-Share Alike 4.0
1275:
8:
1955:Align text // Editing SVG source code using
1311:– to copy, distribute and transmit the work
41:Size of this PNG preview of this SVG file:
2126:
1874:
1425:
1410:You may select the license of your choice.
1525:https://commons.wikimedia.org/User:Cmglee
1298:Attribution-Share Alike 4.0 International
1720:62626688038d3e1672b4ffed83c881346c2476d2
2152:
2144:
2136:
2128:
2111:Template:Cubic graph special points.svg
2073:
1847:
1831:
1810:
1794:
1773:
1757:
1718:
1702:
1684:
1668:
1648:
1632:
1612:
1595:
1579:
1559:
1543:
1476:
1460:
1443:
1440:
1421:
1414:
134:Cubic graph special points repeated.svg
7:
1390:http://www.gnu.org/copyleft/fdl.html
2133:cubic graph special points repeated
1863:
1286:
1281:
2119:
1434:
1428:
1409:
1272:
1268:
121:
71:
1433:
1419:
1366:
1365:
1322:Under the following conditions:
1293:This file is licensed under the
1285:
1280:
1274:
88:
21:
1416:
14:
2060:Category:Differential calculus
1415:
1396:GNU Free Documentation License
1385:GNU Free Documentation License
1375:GNU Free Documentation License
183:Cubic graph special points.svg
26:
1:
1651:original creation by uploader
31:
1441:Items portrayed in this file
2174:
1336:same or compatible license
1864:
153:
16:
2057:Category:Cubic functions
2020:c:User:Rillke/SVGedit.js
1957:c:User:Rillke/SVGedit.js
1924:c:User:Rillke/SVGedit.js
1418:
1380:Free Software Foundation
1302:
201:
95:This is a file from the
2050:
2017:
1987:
1954:
1921:
169:
159:
149:
131:
99:. Information from its
2029:15:19, 3 February 2024
1996:15:47, 3 February 2024
1966:01:43, 4 February 2024
1933:04:00, 4 February 2024
1900:04:21, 4 February 2024
185:
102:description page there
1988:Assorted improvements
204:#!/usr/bin/env python
181:
2101:User:Ramzuiv/sandbox
1732:determination method
140:en:user:GalacticShoe
65:2,048 × 2,048 pixels
61:1,024 × 1,024 pixels
1317:– to adapt the work
47:Other resolutions:
1503:Wikimedia username
1490:author name string
1254:'^(r_*\d?||pts)$ '
471:## search range =
186:
2161:
2160:
2064:
1582:copyright license
1427:
1407:
1406:
194:
193:
190:
189:
117:
116:
97:Wikimedia Commons
2165:
2127:
2091:Stationary point
2086:Inflection point
2076:Concave function
2041:
2008:
1978:
1945:
1912:
1852:
1815:
1778:
1741:
1736:
1546:copyright status
1527:
1522:
1512:
1507:
1497:
1494:
1403:
1400:
1397:
1394:
1391:
1369:
1368:
1362:
1358:
1355:
1352:
1349:
1346:
1338:as the original.
1295:Creative Commons
1289:
1288:
1284:
1283:
1278:
1277:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1147:
1144:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1063:
1060:
1057:
1054:
1051:
1048:
1045:
1042:
1039:
1036:
1033:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
1009:
1006:
1003:
1000:
997:
994:
991:
988:
985:
982:
979:
976:
973:
970:
967:
964:
961:
958:
955:
952:
949:
946:
943:
940:
937:
934:
931:
928:
925:
922:
919:
916:
913:
910:
907:
904:
901:
898:
895:
892:
889:
886:
883:
880:
877:
874:
871:
868:
865:
862:
859:
856:
853:
850:
847:
844:
841:
838:
835:
832:
829:
826:
823:
820:
817:
814:
811:
808:
805:
802:
799:
796:
793:
790:
787:
784:
781:
778:
775:
772:
769:
766:
763:
760:
757:
754:
751:
748:
745:
742:
739:
736:
733:
730:
727:
724:
721:
718:
715:
712:
709:
706:
703:
700:
697:
694:
691:
688:
685:
682:
679:
676:
673:
670:
667:
664:
661:
658:
655:
652:
649:
646:
643:
640:
637:
634:
631:
628:
625:
622:
619:
616:
613:
610:
607:
604:
601:
598:
595:
592:
589:
586:
583:
580:
577:
574:
571:
568:
565:
562:
559:
556:
553:
550:
547:
544:
541:
538:
535:
532:
529:
526:
523:
520:
517:
514:
511:
508:
505:
502:
499:
496:
493:
490:
487:
484:
481:
478:
475:
472:
469:
466:
463:
460:
457:
454:
451:
448:
445:
442:
439:
436:
433:
430:
427:
424:
421:
418:
415:
412:
409:
406:
403:
400:
397:
394:
391:
388:
385:
382:
379:
376:
373:
370:
367:
364:
361:
358:
355:
352:
349:
346:
343:
340:
337:
334:
331:
328:
325:
322:
319:
316:
313:
310:
307:
304:
301:
298:
295:
292:
289:
286:
283:
280:
277:
274:
271:
268:
265:
262:
259:
256:
253:
250:
247:
244:
241:
238:
235:
232:
229:
226:
223:
220:
217:
214:
211:
208:
205:
174:
173:
155:
135:
129:
113:
92:
91:
85:
79:
68:
57:768 × 768 pixels
53:480 × 480 pixels
49:240 × 240 pixels
43:512 × 512 pixels
2173:
2172:
2168:
2167:
2166:
2164:
2163:
2162:
2115:
2072:
2065:
2047:
2039:
2014:
2006:
1984:
1976:
1951:
1943:
1918:
1910:
1866:
1865:
1862:
1861:
1860:
1859:
1858:
1857:
1856:
1855:
1853:
1850:
1840:
1839:
1838:
1836:
1825:
1824:
1823:
1822:
1821:
1820:
1819:
1818:
1816:
1813:
1803:
1802:
1801:
1799:
1788:
1787:
1786:
1785:
1784:
1783:
1782:
1781:
1779:
1776:
1766:
1765:
1764:
1762:
1751:
1750:
1749:
1748:
1747:
1746:
1745:
1744:
1743:
1742:
1737:
1734:
1725:
1724:
1723:
1721:
1711:
1710:
1709:
1707:
1696:
1695:
1694:
1693:
1692:
1691:
1690:
1689:
1687:
1677:
1676:
1675:
1673:
1662:
1661:
1660:
1659:
1658:
1657:
1656:
1655:
1653:
1641:
1640:
1639:
1637:
1626:
1625:
1624:
1623:
1622:
1621:
1620:
1619:
1617:
1606:
1605:
1604:
1603:
1602:
1600:
1588:
1587:
1586:
1584:
1573:
1572:
1571:
1570:
1569:
1568:
1567:
1566:
1564:
1552:
1551:
1550:
1548:
1537:
1536:
1535:
1534:
1533:
1532:
1531:
1530:
1529:
1528:
1523:
1520:
1514:
1513:
1508:
1505:
1499:
1498:
1495:
1492:
1483:
1482:
1481:
1479:
1469:
1468:
1467:
1465:
1454:
1453:
1452:
1451:
1450:
1448:
1432:
1431:
1430:
1413:
1412:
1411:
1401:
1398:
1395:
1392:
1389:
1360:
1359:
1356:
1353:
1350:
1347:
1344:
1343:
1301:
1290:
1271:
1270:
1265:
1260:
1259:
1256:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1172:
1169:
1166:
1163:
1160:
1157:
1154:
1151:
1148:
1145:
1142:
1139:
1136:
1133:
1130:
1127:
1124:
1121:
1118:
1115:
1112:
1109:
1106:
1103:
1100:
1097:
1094:
1091:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1034:
1031:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
1001:
998:
995:
992:
989:
986:
983:
980:
977:
974:
971:
968:
965:
962:
959:
956:
953:
950:
947:
944:
941:
938:
935:
932:
929:
926:
923:
920:
917:
914:
911:
908:
905:
902:
899:
896:
893:
890:
887:
884:
881:
878:
875:
872:
869:
866:
863:
860:
857:
854:
851:
848:
845:
842:
839:
836:
833:
830:
827:
824:
821:
818:
815:
812:
809:
806:
803:
800:
797:
794:
791:
788:
785:
782:
779:
776:
773:
770:
767:
764:
761:
758:
755:
752:
749:
746:
743:
740:
737:
734:
731:
728:
725:
722:
719:
716:
713:
710:
707:
704:
701:
698:
695:
692:
689:
686:
683:
680:
677:
674:
671:
668:
665:
662:
659:
656:
653:
650:
647:
644:
641:
638:
635:
632:
629:
626:
623:
620:
617:
614:
611:
608:
605:
602:
599:
596:
593:
590:
587:
584:
581:
578:
575:
572:
569:
566:
563:
560:
557:
554:
551:
548:
545:
542:
539:
536:
533:
530:
527:
524:
521:
518:
515:
512:
509:
506:
503:
500:
497:
494:
491:
488:
485:
482:
479:
476:
473:
470:
467:
464:
461:
458:
455:
452:
449:
446:
443:
440:
437:
434:
431:
428:
425:
422:
419:
416:
413:
410:
407:
404:
401:
398:
395:
392:
389:
386:
383:
380:
377:
374:
371:
368:
365:
362:
359:
356:
353:
350:
347:
344:
341:
338:
335:
332:
329:
326:
323:
320:
317:
314:
311:
308:
305:
302:
299:
296:
293:
290:
287:
284:
281:
278:
275:
272:
269:
266:
263:
260:
257:
254:
251:
248:
245:
242:
239:
236:
233:
230:
227:
224:
221:
218:
215:
212:
209:
206:
203:
200:
195:
133:
126:
119:
118:
107:
106:
105:is shown below.
81:
77:
70:
69:
46:
12:
11:
5:
2171:
2169:
2159:
2158:
2155:
2151:
2150:
2147:
2143:
2142:
2139:
2135:
2134:
2131:
2118:
2114:
2113:
2108:
2103:
2098:
2093:
2088:
2083:
2081:Cubic function
2078:
2070:
2069:
2068:
2063:
2062:
2049:
2045:
2042:
2036:
2031:
2026:
2023:
2022:
2016:
2012:
2009:
2003:
1998:
1993:
1990:
1989:
1986:
1982:
1979:
1973:
1968:
1963:
1960:
1959:
1953:
1949:
1946:
1940:
1935:
1930:
1927:
1926:
1920:
1916:
1913:
1907:
1902:
1897:
1893:
1892:
1889:
1886:
1883:
1880:
1877:
1870:
1869:
1854:
1848:
1846:
1845:
1844:
1843:
1842:
1841:
1837:
1832:
1830:
1829:
1828:
1827:
1826:
1817:
1811:
1809:
1808:
1807:
1806:
1805:
1804:
1800:
1795:
1793:
1792:
1791:
1790:
1789:
1780:
1774:
1772:
1771:
1770:
1769:
1768:
1767:
1763:
1758:
1756:
1755:
1754:
1753:
1752:
1730:
1729:
1728:
1727:
1726:
1722:
1719:
1717:
1716:
1715:
1714:
1713:
1712:
1708:
1703:
1701:
1700:
1699:
1698:
1697:
1688:
1685:
1683:
1682:
1681:
1680:
1679:
1678:
1674:
1669:
1667:
1666:
1665:
1664:
1663:
1654:
1649:
1647:
1646:
1645:
1644:
1643:
1642:
1638:
1635:source of file
1633:
1631:
1630:
1629:
1628:
1627:
1618:
1613:
1611:
1610:
1609:
1608:
1607:
1601:
1596:
1594:
1593:
1592:
1591:
1590:
1589:
1585:
1580:
1578:
1577:
1576:
1575:
1574:
1565:
1560:
1558:
1557:
1556:
1555:
1554:
1553:
1549:
1544:
1542:
1541:
1540:
1539:
1538:
1516:
1515:
1501:
1500:
1488:
1487:
1486:
1485:
1484:
1480:
1477:
1475:
1474:
1473:
1472:
1471:
1470:
1466:
1461:
1459:
1458:
1457:
1456:
1455:
1449:
1444:
1442:
1439:
1438:
1437:
1436:
1435:
1424:
1423:
1420:
1417:
1408:
1405:
1404:
1370:
1342:
1341:
1340:
1339:
1329:
1320:
1319:
1318:
1312:
1305:You are free:
1292:
1291:
1273:
1267:
1266:
1264:
1261:
202:
199:
196:
192:
191:
188:
187:
171:
170:Other versions
167:
166:
161:
157:
156:
151:
147:
146:
136:
127:
125:
122:
120:
115:
114:
93:
83:
82:
40:
36:
35:
34:
29:
24:
19:
13:
10:
9:
6:
4:
3:
2:
2170:
2156:
2153:
2148:
2145:
2140:
2137:
2132:
2129:
2125:
2122:
2116:
2112:
2109:
2107:
2104:
2102:
2099:
2097:
2094:
2092:
2089:
2087:
2084:
2082:
2079:
2077:
2074:
2066:
2061:
2058:
2054:
2048:
2043:
2037:
2035:
2032:
2030:
2027:
2025:
2024:
2021:
2015:
2010:
2004:
2002:
1999:
1997:
1994:
1992:
1991:
1985:
1980:
1974:
1972:
1969:
1967:
1964:
1962:
1961:
1958:
1952:
1947:
1941:
1939:
1936:
1934:
1931:
1929:
1928:
1925:
1919:
1914:
1908:
1906:
1903:
1901:
1898:
1895:
1894:
1890:
1887:
1884:
1881:
1878:
1876:
1875:
1873:
1867:
1835:
1798:
1761:
1740:
1733:
1706:
1686:image/svg+xml
1672:
1652:
1636:
1616:
1599:
1583:
1563:
1547:
1526:
1519:
1511:
1504:
1491:
1464:
1447:
1387:
1386:
1381:
1377:
1376:
1371:
1364:
1363:
1348:CC BY-SA 4.0
1337:
1333:
1330:
1327:
1324:
1323:
1321:
1316:
1313:
1310:
1307:
1306:
1304:
1303:
1299:
1296:
1279:
1262:
197:
184:
180:
176:
175:
172:
168:
165:
162:
158:
152:
148:
145:
141:
137:
130:
123:
111:
104:
103:
98:
94:
87:
86:
80:
75:
74:Original file
66:
62:
58:
54:
50:
44:
39:
33:
30:
28:
25:
23:
20:
18:
15:
2123:
2120:
1871:
1868:File history
1383:
1373:
1331:
1325:
1314:
1308:
1239:print_keyval
333:print_keyval
110:You can help
100:
72:
22:File history
2138:Image title
2130:Short title
2096:User:Cmglee
1562:copyrighted
1332:share alike
1326:attribution
132:Description
2067:File usage
2038:512 × 512
2005:512 × 512
1975:512 × 512
1942:512 × 512
1909:512 × 512
1885:Dimensions
1671:media type
1478:some value
714:is_integer
624:is_integer
285:is_integer
27:File usage
1882:Thumbnail
1879:Date/Time
1760:data size
1263:Licensing
2117:Metadata
1705:checksum
1422:Captions
1315:to remix
1309:to share
1300:license.
792:continue
765:continue
675:continue
576:continue
154:Own work
32:Metadata
1896:current
1891:Comment
1463:creator
1446:depicts
1426:English
927:eval_at
909:eval_at
216:eval_at
124:Summary
76:
2154:Height
2053:Cmglee
2046:Cmglee
2040:(6 KB)
2013:Cmglee
2007:(6 KB)
1983:Cmglee
1977:(6 KB)
1950:Cmglee
1944:(7 KB)
1917:Cmglee
1911:(7 KB)
1797:height
1775:6,753
1510:Cmglee
1496:Cmglee
1245:locals
408:re_key
402:search
375:sorted
345:re_key
297:return
228:return
207:import
164:Cmglee
160:Author
150:Source
2146:Width
1851:pixel
1834:width
1814:pixel
1739:SHA-1
528:range
483:range
417:)])))
351:print
2157:100%
2149:100%
1888:User
1849:512
1812:512
1777:byte
1402:true
1399:true
1393:GFDL
1357:true
1354:true
783:<
759:else
669:else
369:join
327:1e-5
324:<
142:for
17:File
1518:URL
1248:(),
1230:3.0
1194:3.0
1122:pts
1119:3.0
1089:pts
1080:3.0
1044:3.0
972:pts
969:3.0
939:pts
921:pts
903:pts
900:pts
891:pts
780:())
777:set
771:len
753:r__
747:int
741:r__
729:r__
726:and
720:r__
708:3.0
678:r__
663:r_1
657:int
651:r_1
639:r_1
636:and
630:r_1
612:r_2
609:3.0
579:r_1
519:for
474:for
444:pts
414:key
390:dic
384:key
381:for
339:dic
330:def
312:int
300:abs
282:def
213:def
1735::
1521::
1506::
1493::
882:r2
876:r2
870:r1
855:r2
849:r1
834:r2
825:r1
819:r2
768:if
732:!=
711:if
699:r2
687:r1
642:!=
621:if
618:r2
600:r2
594:r1
567:==
564:r2
558:r1
555:if
552:):
540:r2
525:in
522:r1
516:):
480:in
477:r2
396:re
393:if
387:in
378:()
360:\t
348:):
321:))
294:):
258:**
240:**
225:):
210:re
63:|
59:|
55:|
51:|
45:.
1388:.
1257:)
1251:r
1242:(
1236:d
1233:+
1227:/
1224:)
1221:q
1218:*
1215:2
1212:+
1209:p
1206:(
1203:*
1200:c
1197:+
1191:/
1188:)
1185:q
1182:*
1179:p
1176:*
1173:2
1170:+
1167:q
1164:*
1161:q
1158:(
1155:*
1152:b
1149:+
1146:q
1143:*
1140:q
1137:*
1134:p
1131:*
1128:a
1125:=
1116:/
1113:)
1110:q
1107:*
1104:2
1101:+
1098:p
1095:(
1092:=
1086:d
1083:+
1077:/
1074:)
1071:q
1068:+
1065:p
1062:*
1059:2
1056:(
1053:*
1050:c
1047:+
1041:/
1038:)
1035:q
1032:*
1029:p
1026:*
1023:2
1020:+
1017:p
1014:*
1011:p
1008:(
1005:*
1002:b
999:+
996:q
993:*
990:p
987:*
984:p
981:*
978:a
975:=
966:/
963:)
960:q
957:+
954:p
951:*
948:2
945:(
942:=
936:)
933:q
930:(
924:=
918:)
915:p
912:(
906:=
897:=
894:q
888:=
885:p
879:*
873:*
867:-
864:=
861:d
858:)
852:+
846:*
843:2
840:(
837:*
831:=
828:c
822:-
816:*
813:2
810:-
807:=
804:b
801:1
798:=
795:a
789::
786:3
774:(
762::
756:)
750:(
744:=
738::
735:0
723:)
717:(
705:/
702:)
696:*
693:2
690:+
684:(
681:=
672::
666:)
660:(
654:=
648::
645:0
633:)
627:(
615:=
606:/
603:)
597:+
591:*
588:2
585:(
582:=
573::
570:0
561:*
549:1
546:-
543:,
537:,
534:n
531:(
513:1
510:-
507:,
504:n
501:-
498:1
495:-
492:,
489:n
486:(
468:4
465:=
462:n
459:]
456:,
453:,
450:,
447:=
411:,
405:(
399:.
372:(
366:.
363:'
357:'
354:(
342:,
336:(
318:x
315:(
309:-
306:x
303:(
291:x
288:(
279:d
276:+
273:x
270:*
267:c
264:+
261:2
255:x
252:*
249:b
246:+
243:3
237:x
234:*
231:a
222:x
219:(
112:.
67:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
