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已知抛物线
![](https://img.yulucn.com/upload/0/6a/06a9d66626ecea1d72de63cc6d8eedbf_thumb.jpg)
.
(1)求证:无论
![](https://img.yulucn.com/upload/1/e6/1e6bb8cd531c4728a6d8a88df7873f5b_thumb.jpg)
为任何实数,抛物线与x轴总有两个交点;
(2)若
![](https://img.yulucn.com/upload/1/e6/1e6bb8cd531c4728a6d8a88df7873f5b_thumb.jpg)
为整数,当关于x的方程
![](https://img.yulucn.com/upload/4/4b/44b09fae0fba490fddd0de05fbbb959a_thumb.jpg)
的两个有理数根都在
![](https://img.yulucn.com/upload/1/4e/14e68f7cb4ada79e93613cab5c147328_thumb.jpg)
与
![](https://img.yulucn.com/upload/d/b1/db166b28ee05c9e2397e39894f5d29d1_thumb.jpg)
之间(不包括-1、
![](https://img.yulucn.com/upload/d/b1/db166b28ee05c9e2397e39894f5d29d1_thumb.jpg)
)时,求
![](https://img.yulucn.com/upload/1/e6/1e6bb8cd531c4728a6d8a88df7873f5b_thumb.jpg)
的值.
(3)在(2)的条件下,将抛物线
![](https://img.yulucn.com/upload/0/6a/06a9d66626ecea1d72de63cc6d8eedbf_thumb.jpg)
在x轴下方的部分沿x轴翻折,图象的其余部分保持不变,得到一个新图象
![](https://img.yulucn.com/upload/2/aa/2aab670bdc85fd75345dac8e5a1bb993_thumb.jpg)
,再将图象
![](https://img.yulucn.com/upload/2/aa/2aab670bdc85fd75345dac8e5a1bb993_thumb.jpg)
向上平移
![](https://img.yulucn.com/upload/6/34/6346da084a57d8d86cdc0234789d01d0_thumb.jpg)
个单位,若图象
![](https://img.yulucn.com/upload/2/aa/2aab670bdc85fd75345dac8e5a1bb993_thumb.jpg)
与过点(0,3)且与x轴平行的直线有4个交点,直接写出n的取值范围是 .
(1)由无论
![](https://img.yulucn.com/upload/1/e6/1e6bb8cd531c4728a6d8a88df7873f5b_thumb.jpg)
为任何实数,都有
![](https://img.yulucn.com/upload/3/93/393a7929840058fc1b18cd4b00ffcf97_thumb.jpg)
即可作出判断;(2)-1;(3)
试题分析:(1)由无论
![](https://img.yulucn.com/upload/1/e6/1e6bb8cd531c4728a6d8a88df7873f5b_thumb.jpg)
为任何实数,都有
![](https://img.yulucn.com/upload/3/93/393a7929840058fc1b18cd4b00ffcf97_thumb.jpg)
即可作出判断;
(2)由题意可知抛物线
![](https://img.yulucn.com/upload/0/6a/06a9d66626ecea1d72de63cc6d8eedbf_thumb.jpg)
的开口向上,与y轴交于(0,-2)点,根据方程
![](https://img.yulucn.com/upload/4/4b/44b09fae0fba490fddd0de05fbbb959a_thumb.jpg)
的两根在-1与
![](https://img.yulucn.com/upload/d/b1/db166b28ee05c9e2397e39894f5d29d1_thumb.jpg)
之间,可得当x=-1和
![](https://img.yulucn.com/upload/b/32/b320142facfc61831224c015ec519ef4_thumb.jpg)
时,
![](https://img.yulucn.com/upload/2/b8/2b804a108d3a026e38f6608ed7bd6bca_thumb.jpg)
.即可求得m的范围,再结合方程的判别式的结果即可作出判断;
(3)根据抛物线的平移规律即函数图象上的点的坐标的特征求解即可.
(1)∵△=
![](https://img.yulucn.com/upload/1/f7/1f72c412b119680ba4909fcda711bf14_thumb.jpg)
,
∴无论
![](https://img.yulucn.com/upload/1/e6/1e6bb8cd531c4728a6d8a88df7873f5b_thumb.jpg)
为任何实数,都有
∴抛物线与x轴总有两个交点;
(2)由题意可知:抛物线
![](https://img.yulucn.com/upload/0/6a/06a9d66626ecea1d72de63cc6d8eedbf_thumb.jpg)
的开口向上,与y轴交于(0,-2)点,
∵方程
![](https://img.yulucn.com/upload/4/4b/44b09fae0fba490fddd0de05fbbb959a_thumb.jpg)
的两根在-1与
![](https://img.yulucn.com/upload/d/b1/db166b28ee05c9e2397e39894f5d29d1_thumb.jpg)
之间,
∴当x=-1和
![](https://img.yulucn.com/upload/b/32/b320142facfc61831224c015ec519ef4_thumb.jpg)
时,
![](https://img.yulucn.com/upload/2/b8/2b804a108d3a026e38f6608ed7bd6bca_thumb.jpg)
.
即
解得
因为m为整数,所以 m=-2,-1,0
当m=-2时,方程的判别式△=28,根为无理数,不合题意
当m=-1时,方程的判别式△=25,根为
![](https://img.yulucn.com/upload/4/84/4845a42bfe16b94b189b4423c3add439_thumb.jpg)
,符合题意
当m=0时,方程的判别式△=24,根为无理数,不合题意
综上所述m=-1;
(3)n的取值范围是
![](https://img.yulucn.com/upload/4/89/48913ebb018373ac7754f27b95737ffa_thumb.jpg)
.
点评:此类问题是初中数学的重点和难点,在中考中极为常见,一般以压轴题形式出现,难度较大.
1年前
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