一次函数解析式的求法

 

求函数解析式，是初中代数的一个重要内容，下面介绍函数中最基本的函数——一次函数几种常见的解法。

一、待定系数法

待定系数法是求函数解析式的基本方法，其一般步骤为，首先设出所求函数解析式，再根据题设条件列出相应的方程（组），最后将所求待定系数的值代入所设的函数解析式即可。

例1. 已知一次函数的图象经过点A（2，http://b57.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb01275033e1109a400157019c574af94706f2e73cc70b13b865d43d558ad7ee53b58637d0c88fb84777c2b84089be6774274f442b073148951fa9d57bcde5f0f1cc1365b7dfd67&a=58&b=57与y轴的交点，求这个函数的解析式。

解：设一次函数的解析式为http://b66.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb01275a68165d206fadd2cb64898262a71b622d4dbc033346d990966f26d99a543543331d6739479c0b9147fd78cceefa52693fc32ca47814aca2ec47740f19d8a8f6940309898&a=58&b=66，则由题意得交点B的坐标为（0，3），

又http://b70.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb01275256ec23a4eec7ab90c6297454eae9425f0fe169b56b103f51d07a8c56b1e7f957ce32870877fa6410f02fe554dfb7dc8e658ca716a488c5f9ccea44da8acb038f2e76fbe&a=70&b=70一次函数的图象经过点A（2，-1）和点B（0，3），

http://b61.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb0127599a3b0ef458b85d7590313b42bb18bce75d017dc7817a41c586682dfdc2c8bdb3be7e15ba49f791acc8eb6910669594191a1df2a3e8f197c4442fbd1aed4e26321cea428&a=65&b=61

所求的函数解析式为http://b57.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb012757265639d74d76fcf837d3af4e4fc7be9c2d40fcbd5a9dd74e4c995f17ac7724de56d36f187dfd2245ebf5907913e5b887da702f4ab2143122200a08dbc5aaca9c2d900b5&a=61&b=57。

 

例2. 已知http://b58.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb0127559757169c1744d8f5862ffaa45932cffb53913fd31452a03f1c4a0b2cb0e632438a40cc7895dd9f5aeebffea4b382501690a3972eb67e354e1dacbd196fb63d32dd73460&a=66&b=58（其中a,b是常数）成正比例，求证：（1）y是x的一次函数；

（2）如果http://b58.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb012750618390569863efb42e1065b49bec4bfa4f34f44661a0ac9d4010695a1025fabc68a003971b547f8ca70e43a979d8cad49302262fcf2e9acb03bb480d78fa321a92bca7c&a=57&b=58把y表示成x的函数式。

分析：（1）欲证y是x的一次函数，即把y表示成“http://b67.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb01275aec74559121d5069c1239b015519095f552db4ed6562b49ba7312847f7999f3c3101a3ad25b36e6c8d1353502ed08814e6b460369dee4c7735b19a5221a82e0bf0fb9a77&a=66&b=67，经变形可证。

（2）把两组值代入由（1）得到的函数表示式中，求得参数的值。

解：（1）设http://b70.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb01275ce5133afc86f1606a0d581605b7b64bdbea2e65bd291c6b6ca3170a13c0119a4e8a69972de6a951aac08fc9f175ffa85b3a99e05c0d9b4f8fd55709e6c04d6a91c0957b1&a=65&b=70

http://b67.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb012753140b28784e8312b2033428bf744b8140344decb32e47c028949629b93333ae5d85b67a9a36025d77041da891aaa5f5afd0ca14eb73a6c64d3677f3ae924fc8b587d27e5&a=57&b=67

http://b66.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb01275ff56ee2b1eb23c4bc9ff30daf1ab77f91be055ae1201094ce15771694b9cc3533052db2267631f4c11ab3a4816e21ed0a4860cdb5f4975917844ddc5fa35ea5d8dd882b5&a=65&b=66，故y是x的一次函数。

（2）把http://b66.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb01275c2248e7da47964344e87bfb912deb54568c6311fc951cb363cde160d9e35a54582d7f212452a3cb0bab7795e7d4edd84f7534d9d740d848b5080f34776a30ab996fbe7ea&a=65&b=66中，得

http://b65.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb01275e723ac99f16ca9cfaa298d37d2baf0fe76f0fa15ea42b7a890b66e7619a567f30b8d84660be997de2fb3521c551c6eff17e88449e91902666560f78cbf36b2d4b58b4162&a=58&b=65

http://b65.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb01275e159f39f7c3334a9f33a7257ce5fafd46c93f5f17974435c03830b3410fd386cd1f4cd356ae21cbaa44636eb3c80a753ee8a0aaa295477ce15642b4c0e61de3505f65989&a=67&b=65

所求的解析式为http://b66.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb0127552227352de9c37238b29823d5cf8e457afaeb37281e7b2eda354d0d26545481be650cf9b5e2ee9d33d573064c6333754498774ea46953a6a5dcb4abcc2127195dd662a8e&a=57&b=66。

 

二、平移变换法

平移变换法，就是把函数http://b57.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb012756c5be203b931e3cd24ac61e0fa461456a1ee08001715523e41bc7020578117727abd48b0045a33860ca5c181f3a4239f128203b30d2d1a1f95e3095b0a50bb12997c69c7&a=65&b=57的图象。利用这个平移法则可直接写出所求函数图象的解析式。

例3. 将直线http://b57.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb01275edb6ce49d46ce9bef29fe19a91889b0f8c7c3c0ab5d0949daf54d76d6841c2230ac5b9f16eb9719b83ae09052a23f8d7506ccd5fb37b175977c94d3a56a3f15662c8a5e0&a=61&b=57向左平移3个单位，再向上平移一个单位，所得的直线解析式为_______。

解：根据题意及平移变换法则

得http://b70.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb01275faae46b60c533bd67fc280992477df953273d5d6c1b8c350d96166b96f31fe8daa4fb800de4bfbba89e1412c301b03451e903c3d47f4fe11564d4069f8003643f522b997&a=67&b=70

 

三、数形结合法

数形结合法，就是根据问题的需要，既可以把数量关系转化为图形性质去研究也可以把图形性质转化为数量关系来讨论。

例4. 已知两个一次函数http://b61.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb01275136488d4d4c29fd2c7f30636cbd91135aeb0298bfd02590373bdbeb00cb6003b9729a89b6af14c9d5be65786572850ff58d62ce19dd1f844a1667330dfba8c988877d496&a=57&b=61，试用两种不同的方法比较它们同一个自变量对应的函数值的大小。

分析：比较两个一次函数值的大小，可以从图象法，代入法两个角度比较。

http://b57.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb0127540cafa54a52fef0c9a1ef0c023bc36688cab452967e1fad37d7a4acb08748328c0fcc3ded5510144934af6c383ba668d9870abd702ea6c1bd7f461c73c7dc7c659e8fc83&a=57&b=57

解：解法一：（图象法）在同一坐标系中作出一次函数http://b70.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb01275d41cced1ba1122686e5970a50ce5336c5c71d5b74fc2c7809a4d936f951d248d2ac96f5030968d2722b95ed431417bf8a2eba3e2abd7f40c93e4dcce0b1c218fb27598c1&a=61&b=70的图象。

如图，观察可知当http://b70.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb01275a5d3a074704652ad6f77414a77a5f88a0e8def2db87886d968e59a151b6aae2a586f0b341df979b51fc357e9d9642a58ecba2c43289e5c2abb6e7d357d6778535c680146&a=66&b=70与相交于（1，-1），即

http://b66.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb012752960a297026fc5e49374184cce9f3aeffc0f6606b9fe6dcf2da75f2f3e8a6146f95f4310b5193b91d7ce53a07ac671af230050f721b631bed4846ec331e2a1512cb45e2f&a=58&b=66；

当http://b67.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb01275747585ac818ceca657844f389b94343b87584b67f46a7aa769459661e48b140ed46255a42eab11466fae01dbe3585ef9d39f5ee1ce0c112390c5e640a65eec0477939edc&a=61&b=67。

当http://b67.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb012753b3bc0817ce4ec0e08fd0f6229fa4cff24cc64f0725d086940e66df810c6a2980cb89fabe5c0095aca8169e884f5fa78ebf68dd36f63ac2257d613fafcd2e15c3d6f21e6&a=61&b=67。

解法二：（代数法）

当http://b66.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb0127576364164e7518b74f2b41c68f86c13833361d237ea17f9a5350ffc0a347d27873dc206143eb2ec195637b6f756b975a241349b44850790d13f891f176ef65298900909e4&a=67&b=66

当http://b70.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb01275da26932f275a7834b30c64ba253d654c22c037a6cb1111ee15a862013b16a3753efe71c230bd175762f2c18f7eab4994f3af4709d9f0cd188e2e4625271a50f87e8087b2&a=67&b=70

当http://b70.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb01275534efb9f4bf1f1de58c33d72a04d76233d216e56bbd5e6ce767b6481a06e047e3fcc0c30ab84bc1105f6fa1f834d64f00c6ee2a678d9ee06d5b51da614490a7ba370c085&a=57&b=70

由此可见，上述两种解法，分别从数、形两种角度入手，相得益彰。

例5. 如图，正方形ABCD的边长是4，将此正方形置于平面直角坐标系xOy中，使AB在x轴的正半轴上，A点的坐标是（1，0）。

http://b58.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb012755adab421049dbd692a428f1bf2199d0a777429d9067478cb8ac35aad220e425c71c0672b376f7110b4338f59680bf722d228f35c1bb0d52917d957675e140f2039a0bc5d&a=65&b=58

（1）经过点C的直线http://b61.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb01275f50b7aa1cb71a63180f05b6a74ac6ead257c8efbe62bd9f50c16dfb1c5e7329c7d9975fb1a0be115641bff72a768b755efe13dbaa6086799e97fb0b890d95225228ace9e&a=57&b=61与x轴交于点E，求四边形AECD的面积。

（2）若直线l经过点E且将正方形ABCD分成面积相等的两部分，求直线l的方程并在坐标系中画出直线l。

分析：（1）要求四边形ABCD面积，因为正方形ABCD中DC//AE。可见四边形AECD为梯形。

http://b70.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb01275c18fc6d094dc3a1d3c38ebf970d94cc18611f75dc3f62f150620f51f90754ad2da23104336a6e11af29c61b0e54bb86da6685af46581501b2b5518baa9fc8add2326fbe8&a=57&b=70为此只要求AE即可。

（2）要使直线l把正方形面积分成相等两部分，只要直线l过正方形的对称中心，即对角线交点。

解：（1）由http://b57.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb0127573276e1d129488775321658a4dfad3403149b0e1d2f75273ebca4639d60d23398604482fcb6cb9965af1782ff9f01f83fcd189780e60d4b20fc9ee2affb64ea9a342f20e&a=58&b=57。

http://b65.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb01275101747088d80f54a864479597b8d6723efb97d53a126cf05ef5d3edd688696a25143a29d47843a2ecc6f4aeb02e2b5ca063b751e0059bff18890d6603dbc8a7ec4594ace&a=67&b=65。

http://b70.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb01275eba3498a92cb416059f2a245bd23a8123aa6def8d1e48f8ff1b5e55701ca094f3fe59c69d3d0f9987fc6b4e4a2cb202f65fb160494d1c40e96062e51e28c2757c6c25bc6&a=70&b=70四边形AECD为直角梯形，

http://b65.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb012758ac616e1d2146eb78a0b9a56c9f35a34ee3201f6e7c806c145e951c3eaabbe0cefde6070507db50dcacb455e83dc50af7e549c7558df999cc524b770ac40cfbf22470fa9&a=65&b=65（平方单位）

（2）过正方形对称中心的直线，总是将正方形分成面积相等的两部分。

http://b65.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb012750981bad8f6251d97e63209545dc6a0689ebcc66700281b931b674188ec9fb07ffc9880c8cad974d7ee1e551c7e09006fbcf5327edd081ccf4d7f014bb5d53518d9dce275&a=65&b=65过点E及正方形对称中心的直线即为所求的直线l。

连接AC、BD交于G。

则E（2，0），G（3，2）代入的http://b70.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb01275cc2eb3983f85f51e0199d7cb3a2c8e250ca8521bf8a82bfa52c94795fe346c29b7c0a7fcd7c2c998d2880a0fc0fe3fcc19195133c5481bf6fdfde76957a38d790cb22421&a=57&b=70，

解得http://b67.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb01275707ab3fc30e3239e09061b31242cdd0e37fb1e8d318a695e15b7203a86a65901d14d994cd20c96fdfe9dff5dba924ae55461bc048aa29298e634b2e1d886cb33138d212d&a=58&b=67

http://b67.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb012755057f88a026d174aca3f33d2238b61f332d83cb11843a22c4a4f5707de6000066b691a89cc361ab95ff46fe7868ef6edbc718feb06702dce24799d00930410203079b552&a=67&b=67。

 

四、分类讨论法

分类讨论法，就是在题目中未出现图形或具体条件时将会出现多种可能性，因此要分别进行讨论。

例6. 如果一次函数http://b57.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb01275a6062eeebb2b5e647b646cbc29c19ea1606d1890481cdcb5fc968edadaeb214f393c425ec3e95b622c1a1a30a95cc4da2db305ff3b452a31b4f6dab47d934133a8e5945b&a=65&b=57，求此函数的解析式。

分析：由于一次函数的图象是直线，故当http://b57.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb012754ffd1a803061f1dca249ed7ec3642a4a985267a65377006e0d30d945478e7a0ac5075048a2f488b87b7bb1bf3926c68f1ef1f1617e42caf602b1241fcbb7bf8bb475c699&a=57&b=57，这要视k的符号而定。

解：对k的值分两种情况进行讨论：

（1）当http://b61.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb01275786eb329aab5429fa47a7937e2de4f92d291402b9da1e73c62a64958b06e90f393b83ee706a3965ba47c39a5f6e944a8173f6bb9eb19fc9e34d250cc46e48cd9d7b3268d&a=58&b=61。

当http://b67.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb01275193866e59c1bff70054ea4108219127f181b05a437596608cb3fbdb1cd3e876eb0baacac4a56f626d1dacc18f9581cf877776b0ea39659e41f235f3da3bbc267b11ffe82&a=70&b=67

故得http://b57.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb01275076449f7b0f995b813af938efddcb85d969d4b59101499688b7d24f68e5770be69c5e732416d3591fd75fd9be3178d4923b4c688d03b481b065129178992965d2c968131&a=66&b=57

http://b67.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb01275e700d7e5068e395b59414b5f89ad5e7f8e0ee5c7dbb8cd146e8c1fc2dcbbd3642714672e21fcb750118ed76330578ba78f8b363051f18c8ec6bdacb6fe5358c8fc5c8e4c&a=66&b=67。

当http://b65.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb01275ef5451138636b9579637056bec382c0089c46845d054043ff50900f182b49e34a7e9f832e52168a213fad3210cd0d59898377c2896c69524a6c7cafba736d7f8d4e7c340&a=58&b=65。

于是得http://b61.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb012759735d4e20cef5122f5a068142ca44873c8508a0091fc0b61fcffeb7cc4dd0a82444dbf2e1d24c86017bc126770aa0a870800a9f8b031fb7ae4fe4a30123c61dba354b4d9&a=66&b=61

所求解析式为http://b67.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb01275a6f6420108a3f962f2e90689cb649e6d50c65db61151cddfa7d07f1d0f768d9b23ff63381a685d04f3ea47cf11378fc707e4675e9e49068f397e9c65572052a90ab66643&a=67&b=67

综合上述两种情况。符合条件的解析式为

http://b65.photo.store.qq.com/http_imgload.cgi?/rurl4_b=0bfef2fe7830be0d7a72e92b8fb0127547b4bdbe4b00554406a0eeccf2971cc90b5c45e13073390f133599d2685b24150695a91dc1e27d9b4a5cef34b716283a978f341b2ae988d74103b082e730273f51e7b9c5&a=67&b=65

数学问题是千变万化的，但我们总能找着常规，学习用运动变化的观点看待数学问题，这对我们的学习是大有裨益的。