2009年天津中考数学压轴题精选精析
（2009年天津）26．（本小题10分）
已知函数y1?x，y2?x?bx?c，?，?为方程y1?y2?0的两个根，点M?1，T?在函数y2的图象上． （Ⅰ）若??13，??122，求函数y2的解析式；
（Ⅱ）在（Ⅰ）的条件下，若函数y1与y2的图象的两个交点为A，B，当△ABM的面积为
112时，求t的值；
（Ⅲ）若0?????1，当0?t?1时，试确定T，?，?三者之间的大小关系，并说明理由．
（2009年天津26题解析）本小题满分10分.
解（Ⅰ）?y1?x，y2?x?bx?c，y1?y2?0，
?x??b?1?x?c?0. ··································································································· 1分
22将??213，??12分别代入x??b?1?x?c?0，得
2211?1??1??b?1??c?0，?b?1??c?0， ????????32?3??2?解得b?16，c?16.
2?函数y2的解析式为y2?x?56x?16． ····································································· 3分
（Ⅱ）由已知，得AB?262，设△ABM的高为h，
?S△ABM?12AB·h?12h?1123，即2h?1144.
根据题意，t?T?16162h，
56161144由T?t?562t?，得?t?11442t??.
当t?2t?16??时，解得t1?t2?512；
当t?256t?16?1144时，解得t3?5?122，t4?5?122.
?t的值为
5?25?2，，. ······················································································ 6分 1212125（Ⅲ）由已知，得
????b??c，????T????t??T????t??222?b??c，T?t?bt?c.
2??t???b?，
??t???b?，
2???????b??c?????b??c?，化简得??????????b?1??0.
?0?????1，得????0， ?????b?1?0.
有??b?1???0，??b?1???0. 又0?t?1，?t???b?0，t???b?0，
?当0?t≤a时，T≤?≤?；
当??t≤?时，??T≤?；
当??t?1时，????T. ································································································· 10分